subject:(Fractional Laplacian)

University of Victoria

1. Bowles, Malcolm. Weak Solutions to a Fractional Fokker-Planck Equation via Splitting and Wasserstein Gradient Flow.

Degree: Department of Mathematics and Statistics, 2014, University of Victoria

URL: http://hdl.handle.net/1828/5591

► In this thesis, we study a linear fractional Fokker-Planck equation that models non-local (`fractional') diffusion in the presence of a potential field. The non-locality is… (more)

Subjects/Keywords: splitting; Fractional Laplacian; Wasserstein Gradient Flow

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APA (6^th Edition):

Bowles, M. (2014). Weak Solutions to a Fractional Fokker-Planck Equation via Splitting and Wasserstein Gradient Flow. (Masters Thesis). University of Victoria. Retrieved from http://hdl.handle.net/1828/5591

Chicago Manual of Style (16^th Edition):

Bowles, Malcolm. “Weak Solutions to a Fractional Fokker-Planck Equation via Splitting and Wasserstein Gradient Flow.” 2014. Masters Thesis, University of Victoria. Accessed March 04, 2021. http://hdl.handle.net/1828/5591.

MLA Handbook (7^th Edition):

Bowles, Malcolm. “Weak Solutions to a Fractional Fokker-Planck Equation via Splitting and Wasserstein Gradient Flow.” 2014. Web. 04 Mar 2021.

Vancouver:

Bowles M. Weak Solutions to a Fractional Fokker-Planck Equation via Splitting and Wasserstein Gradient Flow. [Internet] [Masters thesis]. University of Victoria; 2014. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/1828/5591.

Council of Science Editors:

Bowles M. Weak Solutions to a Fractional Fokker-Planck Equation via Splitting and Wasserstein Gradient Flow. [Masters Thesis]. University of Victoria; 2014. Available from: http://hdl.handle.net/1828/5591

Rice University

2. Jiao, Yiyu. Numerical Treatment of Stochastic Dynamic Systems with Fractional Laplacian Terms.

Degree: PhD, Engineering, 2018, Rice University

URL: http://hdl.handle.net/1911/105863

► The fractional Laplacian is an integro-differential operator that is currently widely used in nonlocal models, such as the anomalous diffusion, which arises when a particle… (more)

▼ The fractional Laplacian is an integro-differential operator that is currently widely used in nonlocal models, such as the anomalous diffusion, which arises when a particle moves randomly in the space involving a random process that allows long jumps. Determining the response of such dynamic systems is a daunting task, as general analytical solutions are not available. This thesis proposes approximate and numerical methods for determining the response of dynamic systems containing fractional Laplacian terms. Based on the Riesz-Marchaud and Caputo-type representations of the fractional Laplacian, two Boundary Element Methods (BEM) are introduced to treat fractional dynamic systems. Further, a modal expansion is proposed as a novel expression of the fractional Laplacian. Furthermore, based on the proposed eigenfunctions, statistical linearization procedures are developed to approximate the response statistics. A BEM-based numerical algorithm is first introduced to estimate the solution of the fractional Poisson equation based on the Riesz-Marchaud definition of the fractional Laplacian. Further, the algorithm is applied to a fractional diffusion equation. The properties of the Caputo-type fractional Laplacian are next investigated. Then, a different BEM-based algorithm is developed for time domain simulation of the response of dynamic systems with Caputo-type fractional Laplacian terms. The analog equation is constructed with the unknown load, which is used in calculation of the fractional Laplacian of the response. A discretization and numerical integration scheme are then employed for estimating the response. It is shown that a frequency domain analysis of a nonlinear fractional diffusion equation with stochastic excitation can be conducted by a statistical linearization procedure. The approach is implemented by introducing non-orthogonal eigenfunctions of the fractional Laplacian of the response, which are transformed from the linear modes of the classical diffusion equation solution. Such a representation allows deriving an MDOF nonlinear ordinary differential equation, which is linearized in the mean square sense. Further, a simplified statistical linearization approximation method is proposed. The variance and the power spectral density of the response are then calculated by an iterative procedure. Numerical results pertaining to linear and nonlinear systems exposed to periodic and stochastic excitation are provided to demonstrate the effectiveness of the proposed methods. Advisors/Committee Members: Spanos, Pol D. (advisor).

Subjects/Keywords: Fractional Laplacian; Boundary element method; Statistical linearization; Fractional diffusion equation

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APA (6^th Edition):

Jiao, Y. (2018). Numerical Treatment of Stochastic Dynamic Systems with Fractional Laplacian Terms. (Doctoral Dissertation). Rice University. Retrieved from http://hdl.handle.net/1911/105863

Chicago Manual of Style (16^th Edition):

Jiao, Yiyu. “Numerical Treatment of Stochastic Dynamic Systems with Fractional Laplacian Terms.” 2018. Doctoral Dissertation, Rice University. Accessed March 04, 2021. http://hdl.handle.net/1911/105863.

MLA Handbook (7^th Edition):

Jiao, Yiyu. “Numerical Treatment of Stochastic Dynamic Systems with Fractional Laplacian Terms.” 2018. Web. 04 Mar 2021.

Vancouver:

Jiao Y. Numerical Treatment of Stochastic Dynamic Systems with Fractional Laplacian Terms. [Internet] [Doctoral dissertation]. Rice University; 2018. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/1911/105863.

Council of Science Editors:

Jiao Y. Numerical Treatment of Stochastic Dynamic Systems with Fractional Laplacian Terms. [Doctoral Dissertation]. Rice University; 2018. Available from: http://hdl.handle.net/1911/105863

University of Texas – Austin

3. -6438-8580. Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem.

Degree: PhD, Mathematics, 2019, University of Texas – Austin

URL: http://dx.doi.org/10.26153/tsw/5790

► We study the regularity theory of fractional analogue of k-Hessian operators. We define the fractional k-Hessian operators as concave envelopes of linear fractional order operators.… (more)

Subjects/Keywords: K-Hessian; Non-local operators; Fractional Laplacian; Free boundary problems

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APA (6^th Edition):

-6438-8580. (2019). Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://dx.doi.org/10.26153/tsw/5790

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Chicago Manual of Style (16^th Edition):

-6438-8580. “Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem.” 2019. Doctoral Dissertation, University of Texas – Austin. Accessed March 04, 2021. http://dx.doi.org/10.26153/tsw/5790.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

MLA Handbook (7^th Edition):

-6438-8580. “Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem.” 2019. Web. 04 Mar 2021.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Vancouver:

-6438-8580. Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2019. [cited 2021 Mar 04]. Available from: http://dx.doi.org/10.26153/tsw/5790.

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Council of Science Editors:

-6438-8580. Regularity of fractional analogue of k-Hessian operators and a non-local one-phase free boundary problem. [Doctoral Dissertation]. University of Texas – Austin; 2019. Available from: http://dx.doi.org/10.26153/tsw/5790

Note: this citation may be lacking information needed for this citation format:
Author name may be incomplete

Kent State University

4. Alghamdi, Ohud. Removable Singularities for Holder Continuous Solutions of the Fractional Laplacian.

Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2016, Kent State University

URL: http://rave.ohiolink.edu/etdc/view?acc_num=kent1459422077

► In 1958, L. Carleson showed that a set E is removable for Holder continuous solutions of the Laplacian operator if and only if its (n-2+α)-dimensional… (more)

Subjects/Keywords: Mathematics; Fractional Laplacian; Fourier transform; partition of unity; distribution theory

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APA (6^th Edition):

Alghamdi, O. (2016). Removable Singularities for Holder Continuous Solutions of the Fractional Laplacian. (Masters Thesis). Kent State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=kent1459422077

Chicago Manual of Style (16^th Edition):

Alghamdi, Ohud. “Removable Singularities for Holder Continuous Solutions of the Fractional Laplacian.” 2016. Masters Thesis, Kent State University. Accessed March 04, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=kent1459422077.

MLA Handbook (7^th Edition):

Alghamdi, Ohud. “Removable Singularities for Holder Continuous Solutions of the Fractional Laplacian.” 2016. Web. 04 Mar 2021.

Vancouver:

Alghamdi O. Removable Singularities for Holder Continuous Solutions of the Fractional Laplacian. [Internet] [Masters thesis]. Kent State University; 2016. [cited 2021 Mar 04]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=kent1459422077.

Council of Science Editors:

Alghamdi O. Removable Singularities for Holder Continuous Solutions of the Fractional Laplacian. [Masters Thesis]. Kent State University; 2016. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=kent1459422077

5. Hollifield, Elliott Z. Nonnegative solutions of nonlinear fractional Laplacian equations.

Degree: 2020, NC Docks

URL: http://libres.uncg.edu/ir/uncg/f/Hollifield_uncg_0154D_13115.pdf

► The study of reaction-diffusion equations involving nonlocal diffusion operators has recently flourished. The fractional Laplacian is an example of a nonlocal diffusion operator which allows… (more)

Subjects/Keywords: Laplacian operator; Fractional differential equations; Differential equations, Nonlinear; Reaction-diffusion equations

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APA (6^th Edition):

Hollifield, E. Z. (2020). Nonnegative solutions of nonlinear fractional Laplacian equations. (Thesis). NC Docks. Retrieved from http://libres.uncg.edu/ir/uncg/f/Hollifield_uncg_0154D_13115.pdf

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Hollifield, Elliott Z. “Nonnegative solutions of nonlinear fractional Laplacian equations.” 2020. Thesis, NC Docks. Accessed March 04, 2021. http://libres.uncg.edu/ir/uncg/f/Hollifield_uncg_0154D_13115.pdf.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Hollifield, Elliott Z. “Nonnegative solutions of nonlinear fractional Laplacian equations.” 2020. Web. 04 Mar 2021.

Vancouver:

Hollifield EZ. Nonnegative solutions of nonlinear fractional Laplacian equations. [Internet] [Thesis]. NC Docks; 2020. [cited 2021 Mar 04]. Available from: http://libres.uncg.edu/ir/uncg/f/Hollifield_uncg_0154D_13115.pdf.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Hollifield EZ. Nonnegative solutions of nonlinear fractional Laplacian equations. [Thesis]. NC Docks; 2020. Available from: http://libres.uncg.edu/ir/uncg/f/Hollifield_uncg_0154D_13115.pdf

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

University of South Carolina

6. Peng, Xing. Fractional Chromatic Numbers and Spectra of Graphs.

Degree: PhD, Mathematics, 2012, University of South Carolina

URL: https://scholarcommons.sc.edu/etd/1610

► This dissertation mainly comes from my recent study of fractional chromatic numbers of graphs, spectra of edge-independent random graphs, Laplacian spectra of hypergraphs, and… (more)

Subjects/Keywords: Mathematics; Physical Sciences and Mathematics; chromatic number; edge-independent random graph; fractional chromatic number; Generalized Laplacian matrix; Laplacian matrix

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Peng, X. (2012). Fractional Chromatic Numbers and Spectra of Graphs. (Doctoral Dissertation). University of South Carolina. Retrieved from https://scholarcommons.sc.edu/etd/1610

Chicago Manual of Style (16^th Edition):

Peng, Xing. “Fractional Chromatic Numbers and Spectra of Graphs.” 2012. Doctoral Dissertation, University of South Carolina. Accessed March 04, 2021. https://scholarcommons.sc.edu/etd/1610.

MLA Handbook (7^th Edition):

Peng, Xing. “Fractional Chromatic Numbers and Spectra of Graphs.” 2012. Web. 04 Mar 2021.

Vancouver:

Peng X. Fractional Chromatic Numbers and Spectra of Graphs. [Internet] [Doctoral dissertation]. University of South Carolina; 2012. [cited 2021 Mar 04]. Available from: https://scholarcommons.sc.edu/etd/1610.

Council of Science Editors:

Peng X. Fractional Chromatic Numbers and Spectra of Graphs. [Doctoral Dissertation]. University of South Carolina; 2012. Available from: https://scholarcommons.sc.edu/etd/1610

7. C.D. Bucur. SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY.

Degree: 2017, Università degli Studi di Milano

URL: http://hdl.handle.net/2434/488032

► In this thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and some other types of fractional derivatives. We… (more)

Subjects/Keywords: Fractional Laplacian; nonlocal operators; nonlocal minimal surfaces; fractional derivative; Caputo; Marchaud; Schauder estimates; Settore MAT/05 - Analisi Matematica

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Bucur, C. (2017). SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY. (Thesis). Università degli Studi di Milano. Retrieved from http://hdl.handle.net/2434/488032

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Bucur, C.D.. “SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY.” 2017. Thesis, Università degli Studi di Milano. Accessed March 04, 2021. http://hdl.handle.net/2434/488032.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Bucur, C.D.. “SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY.” 2017. Web. 04 Mar 2021.

Vancouver:

Bucur C. SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY. [Internet] [Thesis]. Università degli Studi di Milano; 2017. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/2434/488032.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Bucur C. SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY. [Thesis]. Università degli Studi di Milano; 2017. Available from: http://hdl.handle.net/2434/488032

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Princeton University

8. Chen, Eric Christopher. Some regularity properties for two equations arising from flows .

Degree: PhD, 2019, Princeton University

URL: http://arks.princeton.edu/ark:/88435/dsp01bz60d014s

► In this thesis, which consists of two parts, we study properties of certain solutions to equations arising from two different flow equations. In the first… (more)

Subjects/Keywords: asymptotically flat; fractional Laplacian; Navier – Stokes; partial regularity; Ricci flow; Sobolev inequality

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Chen, E. C. (2019). Some regularity properties for two equations arising from flows . (Doctoral Dissertation). Princeton University. Retrieved from http://arks.princeton.edu/ark:/88435/dsp01bz60d014s

Chicago Manual of Style (16^th Edition):

Chen, Eric Christopher. “Some regularity properties for two equations arising from flows .” 2019. Doctoral Dissertation, Princeton University. Accessed March 04, 2021. http://arks.princeton.edu/ark:/88435/dsp01bz60d014s.

MLA Handbook (7^th Edition):

Chen, Eric Christopher. “Some regularity properties for two equations arising from flows .” 2019. Web. 04 Mar 2021.

Vancouver:

Chen EC. Some regularity properties for two equations arising from flows . [Internet] [Doctoral dissertation]. Princeton University; 2019. [cited 2021 Mar 04]. Available from: http://arks.princeton.edu/ark:/88435/dsp01bz60d014s.

Council of Science Editors:

Chen EC. Some regularity properties for two equations arising from flows . [Doctoral Dissertation]. Princeton University; 2019. Available from: http://arks.princeton.edu/ark:/88435/dsp01bz60d014s

Princeton University

9. Chen, Eric Christopher. Some regularity properties for two equations arising from flows .

Degree: PhD, 2019, Princeton University

URL: http://arks.princeton.edu/ark:/88435/dsp01x920g0746

► In this thesis, which consists of two parts, we study properties of certain solutions to equations arising from two different flow equations. In the first… (more)

Subjects/Keywords: asymptotically flat; fractional Laplacian; Navier – Stokes; partial regularity; Ricci flow; Sobolev inequality

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Chen, E. C. (2019). Some regularity properties for two equations arising from flows . (Doctoral Dissertation). Princeton University. Retrieved from http://arks.princeton.edu/ark:/88435/dsp01x920g0746

Chicago Manual of Style (16^th Edition):

Chen, Eric Christopher. “Some regularity properties for two equations arising from flows .” 2019. Doctoral Dissertation, Princeton University. Accessed March 04, 2021. http://arks.princeton.edu/ark:/88435/dsp01x920g0746.

MLA Handbook (7^th Edition):

Chen, Eric Christopher. “Some regularity properties for two equations arising from flows .” 2019. Web. 04 Mar 2021.

Vancouver:

Chen EC. Some regularity properties for two equations arising from flows . [Internet] [Doctoral dissertation]. Princeton University; 2019. [cited 2021 Mar 04]. Available from: http://arks.princeton.edu/ark:/88435/dsp01x920g0746.

Council of Science Editors:

Chen EC. Some regularity properties for two equations arising from flows . [Doctoral Dissertation]. Princeton University; 2019. Available from: http://arks.princeton.edu/ark:/88435/dsp01x920g0746

10. Bertsias, Panagiotis. Novel CMOS analog integrated circuits for implementing approximants of the fractional-order laplacian operator.

Degree: 2020, University of Patras; Πανεπιστήμιο Πατρών

URL: http://hdl.handle.net/10442/hedi/47805

►

This Ph.D. Thesis deals with the design of novel CMOS analog integrated circuits, which have been derived through approximation methods of the fractional-order Laplacian operator's… (more)

▼

This Ph.D. Thesis deals with the design of novel CMOS analog integrated circuits, which have been derived through approximation methods of the fractional-order Laplacian operator's implementation. Main attributes of the proposed topologies are the circuit simplicity, due to the small number of the employed MOS transistors, the reduced power consumption, the limited silicon area and the electronic tuning of their characteristics through appropriate dc bias currents.In recent years, fractional calculus has been applied to circuit design, rendering it one of its most interesting interdisciplinary fields. For the implementation of fractional-order circuits, except for the direct, but commercially unavailable, way of using fractional-order capacitors in the standard topologies, an alternative and, simultaneously, efficient method is the approximation of their behavior through passive or active components. The former implies the utilization of appropriately configured RC networks, easily derived, but not electronically programmable, while the latter favors the electronic adjustment at the expense of the increased active component count. Thus, the main objective of the Thesis is to overcome the existing obstacles through two primary axes. The first one includes the implementation of novel simple structures, using implemented fractional-order capacitors or approximated RC networks, and the realization of capacitorless multi-feedback topologies for performing low-order approximations. The second axis comprises the development of new systematic methods for decomposing the initial higher-order approximated transfer functions into a product or sum of simple lower-order active filter terms. The practical value of the proposed concepts is validated through application examples of various scientific fields. The performance of the presented structures is evaluated through simulation results that have been derived, using the Cadence software and the Design Kit provided by the Austria Mikro Systeme (AMS) CMOS 0.35μm technology process.

Η παρούσα Διδακτορική Διατριβή πραγματεύεται τη σχεδίαση καινοτόμων CMOS αναλογικών ολοκληρωμένων κυκλωμάτων, τα οποία έχουν προέλθει μέσω προσεγγιστικών μεθόδων υλοποίησης του Λαπλασιανού τελεστή κλασματικής τάξης. Κύρια χαρακτηριστικά των προτεινόμενων τοπολογιών είναι η κυκλωματική απλότητα, λόγω του μικρού αριθμού των χρησιμοποιούμενων MOS τρανζίστορ, η μειωμένη κατανάλωση ισχύος, η περιορισμένη περιοχή πυριτίου και η ηλεκτρονική ρύθμιση των χαρακτηριστικών τους μέσω κατάλληλων ρευμάτων πόλωσης. Τα τελευταία χρόνια, ο κλασματικός λογισμός έχει εφαρμοστεί στο σχεδιασμό κυκλωμάτων, καθιστώντας τον έναν από τους πιο ενδιαφέροντες διεπιστημονικούς τομείς του. Για την υλοποίηση των κυκλωμάτων κλασματικής τάξης, εκτός από τον άμεσο, αλλά εμπορικά μη διαθέσιμο, τρόπο χρησιμοποίησης πυκνωτών κλασματικής τάξης στις τυπικές τοπολογίες, μία εναλλακτική και, ταυτόχρονα, αποτελεσματική μέθοδος είναι η προσέγγιση της συμπεριφοράς τους μέσω παθητικών ή ενεργών στοιχείων. Το πρώτο συνεπάγεται τη χρήση κατάλληλα…

Subjects/Keywords: Κλασματικός λογισμός; Λαπλασιανός τελεστής κλασματικής τάξης; Κυκλώματα κλασματικής τάξης; Τοιχεία κλασματικής τάξης; Φίλτρα κλασματικής τάξης; CMOS ολοκληρωμένα κυκλώματα; Fractional calculus; Fractional-order Laplacian operator; Fractional-order circuits; Fractional-order elements; Fractional-order filters; CMOS integrated circuits

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Bertsias, P. (2020). Novel CMOS analog integrated circuits for implementing approximants of the fractional-order laplacian operator. (Thesis). University of Patras; Πανεπιστήμιο Πατρών. Retrieved from http://hdl.handle.net/10442/hedi/47805

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Bertsias, Panagiotis. “Novel CMOS analog integrated circuits for implementing approximants of the fractional-order laplacian operator.” 2020. Thesis, University of Patras; Πανεπιστήμιο Πατρών. Accessed March 04, 2021. http://hdl.handle.net/10442/hedi/47805.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Bertsias, Panagiotis. “Novel CMOS analog integrated circuits for implementing approximants of the fractional-order laplacian operator.” 2020. Web. 04 Mar 2021.

Vancouver:

Bertsias P. Novel CMOS analog integrated circuits for implementing approximants of the fractional-order laplacian operator. [Internet] [Thesis]. University of Patras; Πανεπιστήμιο Πατρών; 2020. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/10442/hedi/47805.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Bertsias P. Novel CMOS analog integrated circuits for implementing approximants of the fractional-order laplacian operator. [Thesis]. University of Patras; Πανεπιστήμιο Πατρών; 2020. Available from: http://hdl.handle.net/10442/hedi/47805

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

11. M. Cozzi. QUALITATIVE PROPERTIES OF SOLUTIONS OF NONLINEAR ANISOTROPIC PDES IN LOCAL AND NONLOCAL SETTINGS.

Degree: 2016, Università degli Studi di Milano

URL: http://hdl.handle.net/2434/345873

►

La tesi è dedicata allo studio di varie proprietà qualitative possedute dalle soluzioni di equazioni ellittiche poste nello spazio euclideo. L'attenzione principale del lavoro è… (more)

Subjects/Keywords: anisotropic PDEs; Finsler Laplacian; one-dimensional symmetry; rigidity results; Wulff shape; non-local operators; integro-differential equations; fractional Laplacian; regularity results; Settore MAT/05 - Analisi Matematica

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APA (6^th Edition):

Cozzi, M. (2016). QUALITATIVE PROPERTIES OF SOLUTIONS OF NONLINEAR ANISOTROPIC PDES IN LOCAL AND NONLOCAL SETTINGS. (Thesis). Università degli Studi di Milano. Retrieved from http://hdl.handle.net/2434/345873

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Cozzi, M.. “QUALITATIVE PROPERTIES OF SOLUTIONS OF NONLINEAR ANISOTROPIC PDES IN LOCAL AND NONLOCAL SETTINGS.” 2016. Thesis, Università degli Studi di Milano. Accessed March 04, 2021. http://hdl.handle.net/2434/345873.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Cozzi, M.. “QUALITATIVE PROPERTIES OF SOLUTIONS OF NONLINEAR ANISOTROPIC PDES IN LOCAL AND NONLOCAL SETTINGS.” 2016. Web. 04 Mar 2021.

Vancouver:

Cozzi M. QUALITATIVE PROPERTIES OF SOLUTIONS OF NONLINEAR ANISOTROPIC PDES IN LOCAL AND NONLOCAL SETTINGS. [Internet] [Thesis]. Università degli Studi di Milano; 2016. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/2434/345873.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Cozzi M. QUALITATIVE PROPERTIES OF SOLUTIONS OF NONLINEAR ANISOTROPIC PDES IN LOCAL AND NONLOCAL SETTINGS. [Thesis]. Università degli Studi di Milano; 2016. Available from: http://hdl.handle.net/2434/345873

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Queensland University of Technology

12. Yang, Qianqian. Novel analytical and numerical methods for solving fractional dynamical systems.

Degree: 2010, Queensland University of Technology

URL: https://eprints.qut.edu.au/35750/

► During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and… (more)

Subjects/Keywords: Riemann-Liouville fractional derivative, Grunwald-Letnikov fractional derivative, Caputo fractional derivative, Riesz fractional derivative, fractional Laplacian, anomalous diffusion, fractional diffusion equation, fractional advection-dispersion equation; fractional Fokker-Planck equation, fractional cable equation, numerical method, stability and convergence, L1/L2-approximation, standard/shifted Grunwald method, matrix transform method, fractional method of lines, Lanczos approximation; M-Lanczos approximation, matrix functions, Mittag-Leffler function, Laplace transform of fractional derivative; ODTA

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Yang, Q. (2010). Novel analytical and numerical methods for solving fractional dynamical systems. (Thesis). Queensland University of Technology. Retrieved from https://eprints.qut.edu.au/35750/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Yang, Qianqian. “Novel analytical and numerical methods for solving fractional dynamical systems.” 2010. Thesis, Queensland University of Technology. Accessed March 04, 2021. https://eprints.qut.edu.au/35750/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Yang, Qianqian. “Novel analytical and numerical methods for solving fractional dynamical systems.” 2010. Web. 04 Mar 2021.

Vancouver:

Yang Q. Novel analytical and numerical methods for solving fractional dynamical systems. [Internet] [Thesis]. Queensland University of Technology; 2010. [cited 2021 Mar 04]. Available from: https://eprints.qut.edu.au/35750/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Yang Q. Novel analytical and numerical methods for solving fractional dynamical systems. [Thesis]. Queensland University of Technology; 2010. Available from: https://eprints.qut.edu.au/35750/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Queensland University of Technology

13. Cusimano, Nicole. Fractional models in space for diffusive processes in heterogeneous media with applications in cell motility and electrical signal propagation.

Degree: 2015, Queensland University of Technology

URL: https://eprints.qut.edu.au/84092/

► This work addresses fundamental issues in the mathematical modelling of the diffusive motion of particles in biological and physiological settings. New mathematical results are proved… (more)

Subjects/Keywords: non-local operators in space; fractional Laplacian; reflecting boundary conditions; heterogeneous media; computational biology; computational electrophysiology

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APA (6^th Edition):

Cusimano, N. (2015). Fractional models in space for diffusive processes in heterogeneous media with applications in cell motility and electrical signal propagation. (Thesis). Queensland University of Technology. Retrieved from https://eprints.qut.edu.au/84092/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Cusimano, Nicole. “Fractional models in space for diffusive processes in heterogeneous media with applications in cell motility and electrical signal propagation.” 2015. Thesis, Queensland University of Technology. Accessed March 04, 2021. https://eprints.qut.edu.au/84092/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Cusimano, Nicole. “Fractional models in space for diffusive processes in heterogeneous media with applications in cell motility and electrical signal propagation.” 2015. Web. 04 Mar 2021.

Vancouver:

Cusimano N. Fractional models in space for diffusive processes in heterogeneous media with applications in cell motility and electrical signal propagation. [Internet] [Thesis]. Queensland University of Technology; 2015. [cited 2021 Mar 04]. Available from: https://eprints.qut.edu.au/84092/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Cusimano N. Fractional models in space for diffusive processes in heterogeneous media with applications in cell motility and electrical signal propagation. [Thesis]. Queensland University of Technology; 2015. Available from: https://eprints.qut.edu.au/84092/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

14. Cesbron, Ludovic. On the derivation of non-local diffusion equations in confined spaces.

Degree: PhD, 2017, University of Cambridge

URL: https://www.repository.cam.ac.uk/handle/1810/270355

The subject of the thesis is the derivation of non-local diffusion equations from kinetic models with heavy-tailed equilibrium in velocity. We are particularly interested in confining the kinetic equations and developing methods that allow us, from the confined kinetic models, to derive confined versions of non-local diffusion equations.

Subjects/Keywords: Kinetic theory of gases; Diffusion processes; Diffusion limits; Analysis of PDEs; Vlasov-Fokker-Planck equations; Fractional Laplacian; Vlasov equations

1 more image…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Cesbron, L. (2017). On the derivation of non-local diffusion equations in confined spaces. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/270355

Chicago Manual of Style (16^th Edition):

Cesbron, Ludovic. “On the derivation of non-local diffusion equations in confined spaces.” 2017. Doctoral Dissertation, University of Cambridge. Accessed March 04, 2021. https://www.repository.cam.ac.uk/handle/1810/270355.

MLA Handbook (7^th Edition):

Cesbron, Ludovic. “On the derivation of non-local diffusion equations in confined spaces.” 2017. Web. 04 Mar 2021.

Vancouver:

Cesbron L. On the derivation of non-local diffusion equations in confined spaces. [Internet] [Doctoral dissertation]. University of Cambridge; 2017. [cited 2021 Mar 04]. Available from: https://www.repository.cam.ac.uk/handle/1810/270355.

Council of Science Editors:

Cesbron L. On the derivation of non-local diffusion equations in confined spaces. [Doctoral Dissertation]. University of Cambridge; 2017. Available from: https://www.repository.cam.ac.uk/handle/1810/270355

15. Tarhini, Rana. Équation de films minces fractionnaire pour les fractures hydrauliques : Fractional equation of thin films for hydraulic fractures.

Degree: Docteur es, Mathématiques, 2018, Université Paris-Est

URL: http://www.theses.fr/2018PESC1061

►

Ces travaux concernent deux équations paraboliques, dégénérées et non-locales. La première équation est une équation de films minces fractionnaire et la deuxième est une équation… (more)

▼

Ces travaux concernent deux équations paraboliques, dégénérées et non-locales. La première équation est une équation de films minces fractionnaire et la deuxième est une équation des milieux poreux fractionnaire. La présentation des problèmes, les résultats existants dans la littérature, ainsi que le résumé de nos résultats font l'objet de l'introduction. Le deuxième chapitre est consacré à la présentation de la méthode de De Giorgi utilisée pour montrer la régularité Hölder des solutions des équations elliptiques. On présente de plus les résultats utilisant cette approche dans les cas paraboliques local et non-local. Dans le troisième chapitre, on montre l'existence de solutions faibles d'une équation des films minces fractionnaire. C'est une équation parabolique, dégénérée, non-locale d'ordre α+2 où 0 ≺ α ≺ 2. C'est une généralisation d'une équation étudiée par Imbert et Mellet en 2011 pour α = 1. Pour construire les solutions, on passe par un problème régularisé. En utilisant les injections de Sobolev, on passe à la limite pour trouver des solutions faibles. Vu la différence des injections de Sobolev, on distingue deux cas 0 ≺ α ≺ 1 et 1 ≤ α ≺ 2. Dans les deux cas on démontre que la solution est positive si la condition initiale l'est. Le quatrième chapitre concerne une équation des milieux poreux fractionnaire. On montre la régularité Hölder de solutions faibles positives satisfaisant des estimées d'énergie. D'abord, on montre l'existence de solutions faibles qui satisfont des estimées d'énergie. On distingue deux cas 0 ≺ α ≺ 1 et 1 ≤ α ≺ 2 à cause de problème de divergence. Puis on démontre les lemmes de De Giorgi qui sont des lemmes de réduction de l'oscillation d'en dessus et d'au-dessous. Ces deux lemmes ne suffisent pas pour montrer la régularité Hölder. On a besoin d'améliorer le résultat du lemme de réduction de l'oscillation d'en dessus. Donc, on passe par un lemme des valeurs intermédiaires et on montrer un lemme de réduction de l'oscillation d'en dessus amélioré. Enfin, on montre la régularité Hölder des solutions en utilisant la propriété scaling de ces solutions

In this thesis, we study two degenerate, non-local parabolic equations, a fractional thin film equation and a fractional porous medium equation. The introduction contains a presentation of problems, the previous results in the literature and a brief presentation of our results. In the second chapter, we present a short overview of the De Giorgi method used to prove Hölder regularity of solutions of elliptic equations. Moreover, we present the results using this approach in the local and non-local parabolic cases. In the third chapter we prove existence of weak solutions of a fractional thin film equation. It is a non-local degenerate parabolic equation of order α+2 where 0 ≺ α ≺ 2. It is a generalization of an equation studied by Imbert and Mellet in 2011 for α = 1. To construct these solutions, we consider a regularized problem then we pass to the limit using Sobolev embedding theorem, that's why we distinguish two cases 0 ≺ α ≺ 1 and 1 ≤…

Advisors/Committee Members: Imbert, Cyril (thesis director).

Subjects/Keywords: Equation parabolique dégénérée; Laplacien fractionnaire; Equation non locale; Degenereted parabolic equation; Fractional Laplacian; Non local equation

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Tarhini, R. (2018). Équation de films minces fractionnaire pour les fractures hydrauliques : Fractional equation of thin films for hydraulic fractures. (Doctoral Dissertation). Université Paris-Est. Retrieved from http://www.theses.fr/2018PESC1061

Chicago Manual of Style (16^th Edition):

Tarhini, Rana. “Équation de films minces fractionnaire pour les fractures hydrauliques : Fractional equation of thin films for hydraulic fractures.” 2018. Doctoral Dissertation, Université Paris-Est. Accessed March 04, 2021. http://www.theses.fr/2018PESC1061.

MLA Handbook (7^th Edition):

Tarhini, Rana. “Équation de films minces fractionnaire pour les fractures hydrauliques : Fractional equation of thin films for hydraulic fractures.” 2018. Web. 04 Mar 2021.

Vancouver:

Tarhini R. Équation de films minces fractionnaire pour les fractures hydrauliques : Fractional equation of thin films for hydraulic fractures. [Internet] [Doctoral dissertation]. Université Paris-Est; 2018. [cited 2021 Mar 04]. Available from: http://www.theses.fr/2018PESC1061.

Council of Science Editors:

Tarhini R. Équation de films minces fractionnaire pour les fractures hydrauliques : Fractional equation of thin films for hydraulic fractures. [Doctoral Dissertation]. Université Paris-Est; 2018. Available from: http://www.theses.fr/2018PESC1061

University of Cambridge

16. Cesbron, Ludovic. On the derivation of non-local diffusion equations in confined spaces.

Degree: PhD, 2017, University of Cambridge

URL: https://doi.org/10.17863/CAM.17221 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744418

The subject of the thesis is the derivation of non-local diffusion equations from kinetic models with heavy-tailed equilibrium in velocity. We are particularly interested in confining the kinetic equations and developing methods that allow us, from the confined kinetic models, to derive confined versions of non-local diffusion equations.

Subjects/Keywords: 515; Kinetic theory of gases; Diffusion processes; Diffusion limits; Analysis of PDEs; Vlasov-Fokker-Planck equations; Fractional Laplacian; Vlasov equations

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Cesbron, L. (2017). On the derivation of non-local diffusion equations in confined spaces. (Doctoral Dissertation). University of Cambridge. Retrieved from https://doi.org/10.17863/CAM.17221 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744418

Chicago Manual of Style (16^th Edition):

Cesbron, Ludovic. “On the derivation of non-local diffusion equations in confined spaces.” 2017. Doctoral Dissertation, University of Cambridge. Accessed March 04, 2021. https://doi.org/10.17863/CAM.17221 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744418.

MLA Handbook (7^th Edition):

Cesbron, Ludovic. “On the derivation of non-local diffusion equations in confined spaces.” 2017. Web. 04 Mar 2021.

Vancouver:

Cesbron L. On the derivation of non-local diffusion equations in confined spaces. [Internet] [Doctoral dissertation]. University of Cambridge; 2017. [cited 2021 Mar 04]. Available from: https://doi.org/10.17863/CAM.17221 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744418.

Council of Science Editors:

Cesbron L. On the derivation of non-local diffusion equations in confined spaces. [Doctoral Dissertation]. University of Cambridge; 2017. Available from: https://doi.org/10.17863/CAM.17221 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744418

17. Yang, Ray. Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian.

Degree: PhD, Mathematics, 2011, University of Texas – Austin

URL: http://hdl.handle.net/2152/ETD-UT-2011-08-3926

► We study the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian through the extension technique of Caffarelli and Silvestre.… (more)

Subjects/Keywords: Free boundary; Optimal regularity; Fractional Laplacian

…dimension and examining a particular PDE on the upper half-space, with the fractional Laplacian… …of its minimizers. The extension characterization of the fractional Laplacian has been used… …operator is equivalent to the fractional Laplacian of order σ. lim ya ∂y u = −C(−∆)σ u… …fractional Laplacian is also an integrodifferential operator, with Fourier symbol |ξ|2σ . There is… …directly examining the fractional Laplacian: Proposition 3.4.6. Let w = (−∆)σ u. Assume…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Yang, R. (2011). Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/ETD-UT-2011-08-3926

Chicago Manual of Style (16^th Edition):

Yang, Ray. “Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian.” 2011. Doctoral Dissertation, University of Texas – Austin. Accessed March 04, 2021. http://hdl.handle.net/2152/ETD-UT-2011-08-3926.

MLA Handbook (7^th Edition):

Yang, Ray. “Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian.” 2011. Web. 04 Mar 2021.

Vancouver:

Yang R. Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2011. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/2152/ETD-UT-2011-08-3926.

Council of Science Editors:

Yang R. Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian. [Doctoral Dissertation]. University of Texas – Austin; 2011. Available from: http://hdl.handle.net/2152/ETD-UT-2011-08-3926

Queensland University of Technology

18. Simpson, Daniel Peter. Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion.

Degree: 2008, Queensland University of Technology

URL: https://eprints.qut.edu.au/29751/

► Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this… (more)

▼ Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods…

Subjects/Keywords: Fractional Laplacian, Fractional Poisson equation, Generalised Matern random field, Gaussian Markov random field, Krylov subspace, Lanczos approximation, M-Lanczos approximation, Matrix Functions, Matrix transfer technique, Restarted Lanczos approximation; Stieltjes transform

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Simpson, D. P. (2008). Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion. (Thesis). Queensland University of Technology. Retrieved from https://eprints.qut.edu.au/29751/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Simpson, Daniel Peter. “Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion.” 2008. Thesis, Queensland University of Technology. Accessed March 04, 2021. https://eprints.qut.edu.au/29751/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Simpson, Daniel Peter. “Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion.” 2008. Web. 04 Mar 2021.

Vancouver:

Simpson DP. Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion. [Internet] [Thesis]. Queensland University of Technology; 2008. [cited 2021 Mar 04]. Available from: https://eprints.qut.edu.au/29751/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Simpson DP. Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion. [Thesis]. Queensland University of Technology; 2008. Available from: https://eprints.qut.edu.au/29751/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

19. Lassoued, Rafika. Contributions aux équations d'évolution frac-différentielles : Contributions to frac-differential evolution equations.

Degree: Docteur es, Mathématiques, 2016, La Rochelle; Université de Monastir (Tunisie)

URL: http://www.theses.fr/2016LAROS001

►

Dans cette thèse, nous nous sommes intéressés aux équations différentielles fractionnaires. Nous avons commencé par l'étude d'une équation différentielle fractionnaire en temps. Ensuite, nous avons… (more)

▼

Dans cette thèse, nous nous sommes intéressés aux équations différentielles fractionnaires. Nous avons commencé par l'étude d'une équation différentielle fractionnaire en temps. Ensuite, nous avons étudié trois systèmes fractionnaires non linéaires ; le premier avec un Laplacien fractionnaire et les autres avec une dérivée fractionnaire en temps définie au sens de Caputo. Dans le premier chapitre, nous avons établi les propriétés qualitatives de la solution d'une équation différentielle fractionnaire en temps qui modélise l'évolution d'une certaine espèce. Plus précisément, l'existence et l'unicité de la solution globale sont démontrées pour certaines valeurs de la condition initiale. Dans ce cas, nous avons obtenu le comportement asymptotique de la solution en t^α. Sous une autre condition sur la donnée initiale, la solution explose en temps fini. Le profil de la solution et l'estimation du temps d'explosion sont établis et une confirmation numérique de ces résultats est présentée. Les chapitres 4, 5 et 6 sont consacrés à l'étude théorique de trois systèmes fractionnaires : un système de la diffusion anormale qui décrit la propagation d'une épidémie infectieuse de type SIR dans une population confinée, le Brusselator avec une dérivée fractionnaire en temps et un système fractionnaire en temps avec une loi de balance. Pour chaque système, on présente l'existence globale et le comportement asymptotique des solutions. L'existence et l'unicité de la solution locale pour les trois systèmes sont obtenues par le théorème de point fixe de Banach. Cependant, le comportement asymptotique est établi par des techniques différentes : le comportement asymptotique de la solution du premier système est démontré en se basant sur les estimations du semi-groupe et le théorème d'injection de Sobolev. Concernant le Brusselator fractionnaire, la technique utilisée s'appuie sur un argument de feedback. Finalement, un résultat de régularité maximale est utilisé pour l'étude du dernier système.

In this thesis, we are interested in fractional differential equations. We begin by studying a time fractional differential equation. Then we study three fractional nonlinear systems ; the first system contains a fractional Laplacian, while the others contain a time fractional derivative in the sense of Caputo. In the second chapter, we establish the qualitative properties of the solution of a time fractional equation which describes the evolution of certain species. The existence and uniqueness of the global solution are proved for certain values of the initial condition. In this case, the asymptotic behavior of the solution is dominated by t^α. Under another condition, the solution blows-up in a finite time. The solution profile and the blow-up time estimate are established and a numerical confirmation of these results is presented. The chapters 4, 5 and 6 are dedicated to the study of three fractional systems : an anomalous diffusion system which describes the propagation of an infectious disease in a confined population with a SIR type, the…

Advisors/Committee Members: Kellil, Ferdaous (thesis director), Kirane, Mokhtar (thesis director).

Subjects/Keywords: Calcul fractionnaire; Dérivée fractionnaire; Laplacien fractionnaire; Système de réaction-diffusion; Équation différentielle fractionnaire; Solution explosive; Temps d'explosion; Existence globale; Comportement asymptotique; Fractional calculus; Fractional derivative; Fractional Laplacian; Reaction-diffusion system; Fractional differential equation; Blowing-up solution; Blow-up time; Global existence; Asymptotic behavior

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Lassoued, R. (2016). Contributions aux équations d'évolution frac-différentielles : Contributions to frac-differential evolution equations. (Doctoral Dissertation). La Rochelle; Université de Monastir (Tunisie). Retrieved from http://www.theses.fr/2016LAROS001

Chicago Manual of Style (16^th Edition):

Lassoued, Rafika. “Contributions aux équations d'évolution frac-différentielles : Contributions to frac-differential evolution equations.” 2016. Doctoral Dissertation, La Rochelle; Université de Monastir (Tunisie). Accessed March 04, 2021. http://www.theses.fr/2016LAROS001.

MLA Handbook (7^th Edition):

Lassoued, Rafika. “Contributions aux équations d'évolution frac-différentielles : Contributions to frac-differential evolution equations.” 2016. Web. 04 Mar 2021.

Vancouver:

Lassoued R. Contributions aux équations d'évolution frac-différentielles : Contributions to frac-differential evolution equations. [Internet] [Doctoral dissertation]. La Rochelle; Université de Monastir (Tunisie); 2016. [cited 2021 Mar 04]. Available from: http://www.theses.fr/2016LAROS001.

Council of Science Editors:

Lassoued R. Contributions aux équations d'évolution frac-différentielles : Contributions to frac-differential evolution equations. [Doctoral Dissertation]. La Rochelle; Université de Monastir (Tunisie); 2016. Available from: http://www.theses.fr/2016LAROS001

20. Hnaien, Dorsaf. Equations aux dérivées fractionnaires : propriétés et applications : Fractional differential equations : properties and applications.

Degree: Docteur es, Mathématiques, 2015, La Rochelle; Université de Monastir (Tunisie)

URL: http://www.theses.fr/2015LAROS038

►

Notre objectif dans cette thèse est l'étude des équations différentielles non linéaires comportant des dérivées fractionnaires en temps et/ou en espace. Nous nous sommes intéressés… (more)

▼

Notre objectif dans cette thèse est l'étude des équations différentielles non linéaires comportant des dérivées fractionnaires en temps et/ou en espace. Nous nous sommes intéressés dans un premier temps à l'étude de deux systèmes non linéaires d'équations différentielles fractionnaires en temps et/ou en espace, puis à l'étude d'une équation différentielle fractionnaire en temps. Plus exactement pour la première partie, les questions concernant l'existence globale et le comportement asymptotique des solutions d'un système non linéaire d'équations différentielles comportant des dérivées fractionnaires en temps et en espace sont élucidées. Les techniques utilisées reposent sur des estimations obtenues pour les solutions fondamentales et la comparaison de certaines inégalités fractionnaires. Toujours dans la première partie, l'étude d'un système non linéaire d'équations de réaction-diffusion avec des dérivées fractionnaires en espace est abordée. L'existence locale et l'unicité des solutions sont prouvées à l'aide du théorème du point fixe de Banach. Nous montrons que les solutions sont bornées et analysons leur comportement à l'infini. La deuxième partie est consacrée à l'étude d'une équation différentielle fractionnaire non linéaire. Sous certaines conditions sur la donnée initiale, nous montrons que la solution est globale alors que sous d'autres, elle explose en temps fini. Dans ce dernier cas, nous donnons son profil ainsi que des estimations bilatérales du temps d'explosion. Alors que pour la solution globale nous étudions son comportement asymptotique.

Our objective in this thesis is the study of nonlinear differential equations involving fractional derivatives in time and/or in space. First, we are interested in the study of two nonlinear time and/or space fractional systems. Our second interest is devoted to the analysis of a time fractional differential equation. More exactly for the first part, the question concerning the global existence and the asymptotic behavior of a nonlinear system of differential equations involving time and space fractional derivatives is addressed. The used techniques rest on estimates obtained for the fundamental solutions and the comparison of some fractional inequalities. In addition, we study a nonlinear system of reaction-diffusion equations with space fractional derivatives. The local existence and the uniqueness of the solutions are proved using the Banach fixed point theorem. We show that the solutions are bounded and analyze their large time behavior. The second part is dedicated to the study of a nonlinear time fractional differential equation. Under some conditions on the initial data, we show that the solution is global while under others, it blows-up in a finite time. In this case, we give its profile as well as bilateral estimates of the blow-up time. While for the global solution we study its asymptotic behavior.

Advisors/Committee Members: Kirane, Mokhtar (thesis director), Kellil, Ferdaous (thesis director).

Subjects/Keywords: Calcul fractionnaire; Dérivée fractionnaire au sens de Caputo; Laplacien fractionnaire; Existence locale et globale; Comportement asymptotique; Fonctions de Mittag-Leffler; Solutions explosives; Temps d'explosion; Fractional calculus; Caputo's fractional derivative; Fractional Laplacian; Local and global existence; Asymptotic behavior; Mittag-Leffler’s functions; Blowing-up solutions; Blow-up time

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Hnaien, D. (2015). Equations aux dérivées fractionnaires : propriétés et applications : Fractional differential equations : properties and applications. (Doctoral Dissertation). La Rochelle; Université de Monastir (Tunisie). Retrieved from http://www.theses.fr/2015LAROS038

Chicago Manual of Style (16^th Edition):

Hnaien, Dorsaf. “Equations aux dérivées fractionnaires : propriétés et applications : Fractional differential equations : properties and applications.” 2015. Doctoral Dissertation, La Rochelle; Université de Monastir (Tunisie). Accessed March 04, 2021. http://www.theses.fr/2015LAROS038.

MLA Handbook (7^th Edition):

Hnaien, Dorsaf. “Equations aux dérivées fractionnaires : propriétés et applications : Fractional differential equations : properties and applications.” 2015. Web. 04 Mar 2021.

Vancouver:

Hnaien D. Equations aux dérivées fractionnaires : propriétés et applications : Fractional differential equations : properties and applications. [Internet] [Doctoral dissertation]. La Rochelle; Université de Monastir (Tunisie); 2015. [cited 2021 Mar 04]. Available from: http://www.theses.fr/2015LAROS038.

Council of Science Editors:

Hnaien D. Equations aux dérivées fractionnaires : propriétés et applications : Fractional differential equations : properties and applications. [Doctoral Dissertation]. La Rochelle; Université de Monastir (Tunisie); 2015. Available from: http://www.theses.fr/2015LAROS038

21. N. Abatangelo. Large Solutions for Fractional Laplacian Operators.

Degree: 2015, Università degli Studi di Milano

URL: http://hdl.handle.net/2434/320258

► The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The… (more)

Subjects/Keywords: fractional Laplacian; nonlocal operators; large solutions; L1 weak solutions; nonlinear elliptic equations; Dirichlet problem; boundary singularity; nonlocal curvatures; Settore MAT/05 - Analisi Matematica

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Abatangelo, N. (2015). Large Solutions for Fractional Laplacian Operators. (Thesis). Università degli Studi di Milano. Retrieved from http://hdl.handle.net/2434/320258

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Abatangelo, N.. “Large Solutions for Fractional Laplacian Operators.” 2015. Thesis, Università degli Studi di Milano. Accessed March 04, 2021. http://hdl.handle.net/2434/320258.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Abatangelo, N.. “Large Solutions for Fractional Laplacian Operators.” 2015. Web. 04 Mar 2021.

Vancouver:

Abatangelo N. Large Solutions for Fractional Laplacian Operators. [Internet] [Thesis]. Università degli Studi di Milano; 2015. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/2434/320258.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Abatangelo N. Large Solutions for Fractional Laplacian Operators. [Thesis]. Università degli Studi di Milano; 2015. Available from: http://hdl.handle.net/2434/320258

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

22. A. Fiscella. VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS.

Degree: 2014, Università degli Studi di Milano

URL: http://hdl.handle.net/2434/245334

► My thesis deals with nonlinear elliptic problems involving a non-local integrodifferential operator of fractional type. Our main results concern the existence of weak solutions for… (more)

Subjects/Keywords: integrodifferential operators; fractional Laplacian; Saddle Point Theorem; Palais-Smale condition; Stationary Kirchhoff problems; critical nonlinearities; variational methods; Settore MAT/05 - Analisi Matematica

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APA (6^th Edition):

Fiscella, A. (2014). VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS. (Thesis). Università degli Studi di Milano. Retrieved from http://hdl.handle.net/2434/245334

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Fiscella, A.. “VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS.” 2014. Thesis, Università degli Studi di Milano. Accessed March 04, 2021. http://hdl.handle.net/2434/245334.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Fiscella, A.. “VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS.” 2014. Web. 04 Mar 2021.

Vancouver:

Fiscella A. VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS. [Internet] [Thesis]. Università degli Studi di Milano; 2014. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/2434/245334.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Fiscella A. VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS. [Thesis]. Università degli Studi di Milano; 2014. Available from: http://hdl.handle.net/2434/245334

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

23. Lafleche, Laurent. Dynamique de systèmes à grand nombre de particules et systèmes dynamiques : Dynamics of systems with large number of particles and dynamical systems.

Degree: Docteur es, Mathématiques, 2019, Paris Sciences et Lettres (ComUE)

URL: http://www.theses.fr/2019PSLED010

►

On étudie dans cette thèse le comportement en temps long de solutions d’équations aux dérivées partielles. Celles-ci modélisent des systèmes à grand nombre de particules… (more)

▼

On étudie dans cette thèse le comportement en temps long de solutions d’équations aux dérivées partielles. Celles-ci modélisent des systèmes à grand nombre de particules dont la dynamique est due à des forces externes, internes et à l’interaction entre ces particules. Cependant, on considère différentes échelles. On voyage ainsi du niveau quantique des atomes au niveau macroscopique des étoiles, et l’on voit que des différences apparaissent bien que certaines propriétés soient conservées. Dans ce voyage, on croise le chemin de diverses applications telles que l’astrophysique, les plasmas,les semi-conducteurs, la biologie et l’économie. Ce travail est divisé en trois parties.Dans la première, on étudie le comportement semi-classique de l’équation de Hartree en mécanique quantique et sa limite vers l’équation de Vlasov. On quantifie uniformément en la constante de Planck des propriétés telles que la propagation des moments et de normes de Lebesgue à poids et la dispersion. On les utilise ensuite pour établir des estimées de stabilité entre les deux équations au moyen d’un analogue semi-classique des distances de Wasserstein. Dans la deuxième partie, on regarde le comportement en temps long d’équations cinétiques dont l’opérateur de collision est linéaire et a un équilibre local avec peu de moments, tel que l’opérateur de Fokker-Planck, sa version fractionnaire et un opérateur de Boltzmann linéaire. Deux principales techniques sont utilisées, l’une consistant à construire des entropies et la seconde à utiliser la positivité.Enfin, la dernière partie s’intéresse à des modèles macroscopiques inspirés de l’équation de Keller-Segel et l’on regarde les paramètres sous lesquels ce type de système s’effondre sur lui-même, se disperse ou se stabilise. Le premier effet se voit en introduisant des poids appropriés, le deuxième avec des distances de Wasserstein et le troisième au moyen des normes de Lebesgue.

In this thesis, we study the behavior of solutions of partial differential equations that arise from the modeling of systems with a large number of particles. The dynamic of all these systems is driven by interaction between the particles and external and internal forces. However, we will consider different scales and travel from the quantum level of atoms to the macroscopic level of stars. We will see that differences emerge from the associated dynamics even though the main properties are conserved. In this journey, we will cross the path of various applications of these equations such as astrophysics, plasma, semi-conductors, biology, economy. This work is divided in three parts.In the first one, we study the semi classical behavior of the quantum Hartree equation and its limit to the kinetic Vlasov equation. Properties such as the propagation of moments and weighted Lebesgue norms and dispersive estimates are quantified uniformly in the Planck constant and used to establish stability estimates in a semiclassical analogue of the Wasserstein distance between the solutions of these two equations.In the second part, we…

Advisors/Committee Members: Mischler, Stéphane (thesis director), Golse, François (thesis director).

Subjects/Keywords: Systèmes dynamiques; Particules en interaction; Modèles cinétiques; Mécanique quantique; Laplacien fractionnaire; Dynamical systems; Interacting particles; Kinetic models; Quantum mechanics; Fractional Laplacian; 515.7

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Lafleche, L. (2019). Dynamique de systèmes à grand nombre de particules et systèmes dynamiques : Dynamics of systems with large number of particles and dynamical systems. (Doctoral Dissertation). Paris Sciences et Lettres (ComUE). Retrieved from http://www.theses.fr/2019PSLED010

Chicago Manual of Style (16^th Edition):

Lafleche, Laurent. “Dynamique de systèmes à grand nombre de particules et systèmes dynamiques : Dynamics of systems with large number of particles and dynamical systems.” 2019. Doctoral Dissertation, Paris Sciences et Lettres (ComUE). Accessed March 04, 2021. http://www.theses.fr/2019PSLED010.

MLA Handbook (7^th Edition):

Lafleche, Laurent. “Dynamique de systèmes à grand nombre de particules et systèmes dynamiques : Dynamics of systems with large number of particles and dynamical systems.” 2019. Web. 04 Mar 2021.

Vancouver:

Lafleche L. Dynamique de systèmes à grand nombre de particules et systèmes dynamiques : Dynamics of systems with large number of particles and dynamical systems. [Internet] [Doctoral dissertation]. Paris Sciences et Lettres (ComUE); 2019. [cited 2021 Mar 04]. Available from: http://www.theses.fr/2019PSLED010.

Council of Science Editors:

Lafleche L. Dynamique de systèmes à grand nombre de particules et systèmes dynamiques : Dynamics of systems with large number of particles and dynamical systems. [Doctoral Dissertation]. Paris Sciences et Lettres (ComUE); 2019. Available from: http://www.theses.fr/2019PSLED010

Georgia Tech

24. Einav, Amit. Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian.

Degree: PhD, Mathematics, 2011, Georgia Tech

URL: http://hdl.handle.net/1853/42788

► The presented work deals with two distinct problems in the field of Mathematical Physics. The first part is dedicated to an 'almost' solution of Villani's… (more)

Subjects/Keywords: Fractional laplacian; Villani's conjecture; Entropy production; Kac's model; Trace inequality; Mathematical physics; Statistical mechanics; Transport theory; Particle methods (Numerical analysis); Inequalities (Mathematics)

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Einav, A. (2011). Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/42788

Chicago Manual of Style (16^th Edition):

Einav, Amit. “Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian.” 2011. Doctoral Dissertation, Georgia Tech. Accessed March 04, 2021. http://hdl.handle.net/1853/42788.

MLA Handbook (7^th Edition):

Einav, Amit. “Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian.” 2011. Web. 04 Mar 2021.

Vancouver:

Einav A. Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian. [Internet] [Doctoral dissertation]. Georgia Tech; 2011. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/1853/42788.

Council of Science Editors:

Einav A. Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian. [Doctoral Dissertation]. Georgia Tech; 2011. Available from: http://hdl.handle.net/1853/42788

25. Wei, Peng. Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection.

Degree: PhD, Mathematics, 2019, Texas A&M University

URL: http://hdl.handle.net/1969.1/186199

► In this work, we approximate a time-dependent problem with drift involving fractional powers of elliptic operators. The numerical scheme is based on an integral representation… (more)

▼ In this work, we approximate a time-dependent problem with drift involving fractional powers of elliptic operators. The numerical scheme is based on an integral representation of the stationary problem at each time step. The integral representation is further approximated by an exponentially convergent sinc quadrature. This results in multiple independent reaction-diffusion problems approximated using the finite element method. The resulting error between the solution and its approximation in the energy norm is based on a Strang’s lemma for the consistency errors generated by sinc quadrature and finite element approximations. The L 2 error is obtained by a standard duality argument. A forward Euler method is considered for the time stepping. It is stable provided the sinc quadrature stepping and time stepping is taken sufficiently small. Under the same condition, we also deduce its first order convergence in time. We challenge the analyzed numerical scheme in the context of surface quasi-geostrophic (SQG) dynamics and electroconvection system. In each setting, the governing equations are derived from conservations of various physical quantities. In SQG dynamics the drifting velocity involves the solution to another fractional elliptic problem. The simulations consider two scenarios: inviscid (no diffusion) and inviscid-limit (small diffusion). In both scenarios our simulation results are compared with existing results and good agreements are observed. In electroconvection, the liquid is located in between two concentric circular electrodes which are either assumed to be of infinite height or slim. Each configuration results in a different nonlocal electro-magnetic model defined on a two dimensional bounded domain. Our numerical simulations indicate that slim electrodes are favorable for electroconvection to occur and are able to sustain the phenomena over long period of time. Furthermore, we provide a numerical study on the influence of the three main parameters of the system: the Rayleigh number, the Prandtl number and the electrodes aspect ratio. Advisors/Committee Members: Bonito, Andrea (advisor), Guermond, Jean-Luc (committee member), Muliana, Anastasia (committee member), Walton, Jay (committee member).

Subjects/Keywords: Fractional Laplacian; Finite Element; Surface quasi-geostrophic; Electroconvection

…x29; mapping will introduce fractional Laplacian with power 1/2. Consequetly, the numerical… …considered: the spectral fractional elliptic operator and the integral fractional Laplacian. The… …L = −∆, Ls is referred to as the spectral fractional Laplacian. In addition, when Ω is… …fractional operator, the integral fractional Laplacian, is deﬁned through the Fourier transform on… …the whole space Rd . For v in the Schwartz space, the integral fractional Laplacian is a…

10 more images…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Wei, P. (2019). Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/186199

Chicago Manual of Style (16^th Edition):

Wei, Peng. “Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection.” 2019. Doctoral Dissertation, Texas A&M University. Accessed March 04, 2021. http://hdl.handle.net/1969.1/186199.

MLA Handbook (7^th Edition):

Wei, Peng. “Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection.” 2019. Web. 04 Mar 2021.

Vancouver:

Wei P. Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection. [Internet] [Doctoral dissertation]. Texas A&M University; 2019. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/1969.1/186199.

Council of Science Editors:

Wei P. Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection. [Doctoral Dissertation]. Texas A&M University; 2019. Available from: http://hdl.handle.net/1969.1/186199

26. Wei, Peng. Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection.

Degree: PhD, Mathematics, 2019, Texas A&M University

URL: http://hdl.handle.net/1969.1/186239

► In this work, we approximate a time-dependent problem with drift involving fractional powers of elliptic operators. The numerical scheme is based on an integral representation… (more)

▼ In this work, we approximate a time-dependent problem with drift involving fractional powers of elliptic operators. The numerical scheme is based on an integral representation of the stationary problem at each time step. The integral representation is further approximated by an exponentially convergent sinc quadrature. This results in multiple independent reaction-diffusion problems approximated using the finite element method. The resulting error between the solution and its approximation in the energy norm is based on a Strang’s lemma for the consistency errors generated by sinc quadrature and finite element approximations. The L 2 error is obtained by a standard duality argument. A forward Euler method is considered for the time stepping. It is stable provided the sinc quadrature stepping and time stepping is taken sufficiently small. Under the same condition, we also deduce its first order convergence in time. We challenge the analyzed numerical scheme in the context of surface quasi-geostrophic (SQG) dynamics and electroconvection system. In each setting, the governing equations are derived from conservations of various physical quantities. In SQG dynamics the drifting velocity involves the solution to another fractional elliptic problem. The simulations consider two scenarios: inviscid (no diffusion) and inviscid-limit (small diffusion). In both scenarios our simulation results are compared with existing results and good agreements are observed. In electroconvection, the liquid is located in between two concentric circular electrodes which are either assumed to be of infinite height or slim. Each configuration results in a different nonlocal electro-magnetic model defined on a two dimensional bounded domain. Our numerical simulations indicate that slim electrodes are favorable for electroconvection to occur and are able to sustain the phenomena over long period of time. Furthermore, we provide a numerical study on the influence of the three main parameters of the system: the Rayleigh number, the Prandtl number and the electrodes aspect ratio. Advisors/Committee Members: Bonito, Andrea (advisor), Guermond, Jean-Luc (committee member), Muliana, Anastasia (committee member), Walton, Jay (committee member).

Subjects/Keywords: Fractional Laplacian; Finite Element; Surface quasi-geostrophic; Electroconvection

…x29; mapping will introduce fractional Laplacian with power 1/2. Consequetly, the numerical… …considered: the spectral fractional elliptic operator and the integral fractional Laplacian. The… …L = −∆, Ls is referred to as the spectral fractional Laplacian. In addition, when Ω is… …fractional operator, the integral fractional Laplacian, is deﬁned through the Fourier transform on… …the whole space Rd . For v in the Schwartz space, the integral fractional Laplacian is a…

10 more images…

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Wei, P. (2019). Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/186239

Chicago Manual of Style (16^th Edition):

Wei, Peng. “Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection.” 2019. Doctoral Dissertation, Texas A&M University. Accessed March 04, 2021. http://hdl.handle.net/1969.1/186239.

MLA Handbook (7^th Edition):

Wei, Peng. “Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection.” 2019. Web. 04 Mar 2021.

Vancouver:

Wei P. Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection. [Internet] [Doctoral dissertation]. Texas A&M University; 2019. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/1969.1/186239.

Council of Science Editors:

Wei P. Numerical Approximation of Time Dependent Fractional Diffusion With Drift: Numerical Analysis and Applications to Surface Quasi-Geostrophic Dynamics and Electroconvection. [Doctoral Dissertation]. Texas A&M University; 2019. Available from: http://hdl.handle.net/1969.1/186239

University of Texas – Austin

27. Tang, Lan, 1980-. Random homogenization of p-Laplacian with obstacles on perforated domain and related topics.

Degree: PhD, Mathematics, 2011, University of Texas – Austin

URL: http://hdl.handle.net/2152/ETD-UT-2011-05-3455

Abstract not available. Advisors/Committee Members: Caffarelli, Luis A. (advisor).

Subjects/Keywords: p-capacity; Stationary ergodic; Correctors; Obstacle problem; Fractional p-Laplacian

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Tang, Lan, 1. (2011). Random homogenization of p-Laplacian with obstacles on perforated domain and related topics. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/ETD-UT-2011-05-3455

Chicago Manual of Style (16^th Edition):

Tang, Lan, 1980-. “Random homogenization of p-Laplacian with obstacles on perforated domain and related topics.” 2011. Doctoral Dissertation, University of Texas – Austin. Accessed March 04, 2021. http://hdl.handle.net/2152/ETD-UT-2011-05-3455.

MLA Handbook (7^th Edition):

Tang, Lan, 1980-. “Random homogenization of p-Laplacian with obstacles on perforated domain and related topics.” 2011. Web. 04 Mar 2021.

Vancouver:

Tang, Lan 1. Random homogenization of p-Laplacian with obstacles on perforated domain and related topics. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2011. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/2152/ETD-UT-2011-05-3455.

Council of Science Editors:

Tang, Lan 1. Random homogenization of p-Laplacian with obstacles on perforated domain and related topics. [Doctoral Dissertation]. University of Texas – Austin; 2011. Available from: http://hdl.handle.net/2152/ETD-UT-2011-05-3455

Georgia Tech

28. Yildirim Yolcu, Selma. Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian.

Degree: PhD, Mathematics, 2009, Georgia Tech

URL: http://hdl.handle.net/1853/31649

► Some eigenvalue inequalities for Klein-Gordon operators and fractional Laplacians restricted to a bounded domain are proved. Such operators became very popular recently as they arise… (more)

Subjects/Keywords: Fractional Laplacian; Klein-Gordon operator; Eigenvalue; Laplacian operator; Eigenvalues; Klein-Gordon equation; Spectral theory (Mathematics)

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Yildirim Yolcu, S. (2009). Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/31649

Chicago Manual of Style (16^th Edition):

Yildirim Yolcu, Selma. “Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian.” 2009. Doctoral Dissertation, Georgia Tech. Accessed March 04, 2021. http://hdl.handle.net/1853/31649.

MLA Handbook (7^th Edition):

Yildirim Yolcu, Selma. “Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian.” 2009. Web. 04 Mar 2021.

Vancouver:

Yildirim Yolcu S. Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian. [Internet] [Doctoral dissertation]. Georgia Tech; 2009. [cited 2021 Mar 04]. Available from: http://hdl.handle.net/1853/31649.

Council of Science Editors:

Yildirim Yolcu S. Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian. [Doctoral Dissertation]. Georgia Tech; 2009. Available from: http://hdl.handle.net/1853/31649

Missouri University of Science and Technology

29. Duo, Siwei. Numerical investigation on nonlocal problems with the fractional Laplacian.

Degree: PhD, Mathematics, Missouri University of Science and Technology

URL: https://scholarsmine.mst.edu/doctoral_dissertations/2742

► "Nonlocal models have recently become a powerful tool for studying complex systems with long-range interactions or memory effects, which cannot be described properly by… (more)

▼ "Nonlocal models have recently become a powerful tool for studying complex systems with long-range interactions or memory effects, which cannot be described properly by the traditional differential equations. So far, different nonlocal (or fractional differential) models have been proposed, among which models with the fractional Laplacian have been well applied. The fractional Laplacian (-Δ)^α/2 represents the infinitesimal generator of a symmetric α-stable Lévy process. It has been used to describe anomalous diffusion, turbulent flows, stochastic dynamics, finance, and many other phenomena. However, the nonlocality of the fractional Laplacian introduces considerable challenges in its mathematical modeling, numerical simulations, and mathematical analysis. To advance the understanding of the fractional Laplacian, two novel and accurate finite difference methods – the weighted trapezoidal method and the weighted linear interpolation method are developed for discretizing the fractional Laplacian. Numerical analysis is provided for the error estimates, and fast algorithms are developed for their efficient implementation. Compared to the current state-of-the-art methods, these two methods have higher accuracy but less computational complexity. As an application, the solution behaviors of the fractional Schördinger equation are investigated to understand the nonlocal effects of the fractional Laplacian. First, the eigenvalues and eigenfunctions of the fractional Schrödinger equation in an infinite potential well are studied, and the results provide insights into an open problem in the fractional quantum mechanics. Second, three Fourier spectral methods are developed and compared in solving the fractional nonlinear Schördinger equation (NLS), among which the SSFS method is more effective in the study of the plane wave dynamics. Sufficient conditions are provided to avoid the numerical instability of the SSFS method. In contrast to the standard NLS, the plane wave dynamics of the fractional NLS are more chaotic due to the long-range interactions" – Abstract, page iii.

Subjects/Keywords: Fractional Laplacian; Fractional Schrodinger equation; Montgomery's identity; Plane wave dynamics; Weighted interpolation; Weighted trapezoidal; Mathematics

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APA (6^th Edition):

Duo, S. (n.d.). Numerical investigation on nonlocal problems with the fractional Laplacian. (Doctoral Dissertation). Missouri University of Science and Technology. Retrieved from https://scholarsmine.mst.edu/doctoral_dissertations/2742

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No year of publication.

Chicago Manual of Style (16^th Edition):

Duo, Siwei. “Numerical investigation on nonlocal problems with the fractional Laplacian.” Doctoral Dissertation, Missouri University of Science and Technology. Accessed March 04, 2021. https://scholarsmine.mst.edu/doctoral_dissertations/2742.

Note: this citation may be lacking information needed for this citation format:
No year of publication.

MLA Handbook (7^th Edition):

Duo, Siwei. “Numerical investigation on nonlocal problems with the fractional Laplacian.” Web. 04 Mar 2021.

Note: this citation may be lacking information needed for this citation format:
No year of publication.

Vancouver:

Duo S. Numerical investigation on nonlocal problems with the fractional Laplacian. [Internet] [Doctoral dissertation]. Missouri University of Science and Technology; [cited 2021 Mar 04]. Available from: https://scholarsmine.mst.edu/doctoral_dissertations/2742.

Note: this citation may be lacking information needed for this citation format:
No year of publication.

Council of Science Editors:

Duo S. Numerical investigation on nonlocal problems with the fractional Laplacian. [Doctoral Dissertation]. Missouri University of Science and Technology; Available from: https://scholarsmine.mst.edu/doctoral_dissertations/2742

Note: this citation may be lacking information needed for this citation format:
No year of publication.

30. Dannawi, Ihab. Contributions aux équations d'évolutions non locales en espace-temps : Contributions to non local evolution equations in space-time.

Degree: Docteur es, Mathématiques, 2015, La Rochelle

URL: http://www.theses.fr/2015LAROS007

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Dans cette thèse, nous nous intéressons à l'étude de quatre équations d'évolution non-locales. Les solutions de ces quatre équations peuvent exploser en temps fini. Dans… (more)

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Dans cette thèse, nous nous intéressons à l'étude de quatre équations d'évolution non-locales. Les solutions de ces quatre équations peuvent exploser en temps fini. Dans la théorie des équations d'évolution non-linéaires, une solution est qualifiée de globale si elle est définie pour tout temps positif. Au contraire, si une solution existe seulement sur un intervalle de temps [0; T) borné, elle est dite locale. Dans ce dernier cas et quand le temps maximal d'existence est relié à une alternative d'explosion, on dit aussi que la solution explose en temps fini. Dans un premier travail, nous considérons l'équation de Schrödinger non-linéaire avec une puissance fractionnaire du laplacien, et nous obtenons l'explosion de la solution en temps fini Tmax > 0 pour toute condition initiale positive et non-triviale dans le cas d'exposant sous-critique. Ensuite, nous étudions une équation des ondes amorties avec un potentiel d'espace-temps et un terme non-linéaire et non-local en temps. Nous obtenons un résultat d'existence locale d'une solution dans l'espace d'énergie sous des conditions restrictives sur les données initiales, la dimension de l'espace et la croissance du terme non-linéaire. De plus, nous obtenons l'explosion de la solution en temps fini pour toute condition initiale de moyenne strictement positive. De plus, nous étudions un problème de Cauchy pour l'équation d'évolution avec un p- Laplacien avec une non linéarité non-locale en temps. Dans ce cadre, nous nous intéressons à l'étude de l'existence locale d'une solution de cette équation ainsi qu'un résultat de non-existence de solution globale. Finalement, nous étudions l'intervalle maximal d'existence des solutions de l'équation des milieux poreux avec un terme non-linéaire non-local en temps.

In this thesis, we study four non-local evolution equations. The solutions of these four equations can blow up in finite time. In the theory of nonlinear evolution equations, a solution is qualified as global if it isdefined for any time. Otherwise, if a solution exists only on a bounded interval [0; T), it is called local solution. In this case and when the maximum time of existence is related to a blow up alternative, we say that the solution blows up in finite time. First, we consider the nonlinear Schröodinger equation with a fractional power of the Laplacien operator, and we get a blow up result in finite time Tmax > 0 for any non-trivial non-negative initial condition in the case of sub-critical exponent. Next, we study a damped wave equation with a space-time potential and a non-local in time non-linear term. We obtain a result of local existence of a solution in the energy space under some restrictions on the initial data, the dimension of the space and the growth of nonlinear term. Additionally, we get a blow up result of the solution in finite time for any initial condition positive on average. In addition, we study a Cauchy problem for the evolution p-Laplacien equation with nonlinear memory. We study the local existence of a solution of this equation as well…

Advisors/Committee Members: Kirane, Mokhtar (thesis director).

Subjects/Keywords: Equations de Schrödinger; Explosion en temps fini; Laplacien Fractionnaire; Equation d'onde amortie non linéaire; Exposant sous critique; Problème de Cauchy; Equation de p-Laplacien; Existence locale; Non existence globale; Equation hyperpolique; Solutions douces et faibles; Estimation de Strichartz; Les dérivées et les intégrales fractionnaires de Riemann-Liouville; Schrödinger equations; Blow-up; Fractional Laplacian; Nonlinear damped wave equation; Subcritical potential; Cauchy problem; P-Laplacian equation; Local existence; Global nonexistence; Hyperbolic equation; Mild and weak solutions; Strichartz estimate; Riemann- Liouville fractional integrals and derivatives

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Dannawi, I. (2015). Contributions aux équations d'évolutions non locales en espace-temps : Contributions to non local evolution equations in space-time. (Doctoral Dissertation). La Rochelle. Retrieved from http://www.theses.fr/2015LAROS007

Chicago Manual of Style (16^th Edition):

Dannawi, Ihab. “Contributions aux équations d'évolutions non locales en espace-temps : Contributions to non local evolution equations in space-time.” 2015. Doctoral Dissertation, La Rochelle. Accessed March 04, 2021. http://www.theses.fr/2015LAROS007.

MLA Handbook (7^th Edition):

Dannawi, Ihab. “Contributions aux équations d'évolutions non locales en espace-temps : Contributions to non local evolution equations in space-time.” 2015. Web. 04 Mar 2021.

Vancouver:

Dannawi I. Contributions aux équations d'évolutions non locales en espace-temps : Contributions to non local evolution equations in space-time. [Internet] [Doctoral dissertation]. La Rochelle; 2015. [cited 2021 Mar 04]. Available from: http://www.theses.fr/2015LAROS007.

Council of Science Editors:

Dannawi I. Contributions aux équations d'évolutions non locales en espace-temps : Contributions to non local evolution equations in space-time. [Doctoral Dissertation]. La Rochelle; 2015. Available from: http://www.theses.fr/2015LAROS007

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