Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

Nikki Kedia; Anatoly A. Alikhanov; Vineet Kumar Singh

doi:10.1016/j.matcom.2023.12.034

Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

Nikki Kedia
, Anatoly A. Alikhanov
, Vineet Kumar Singh^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor's series. Theoretical stability and convergence analysis is established in the L₂-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.

Original language	English
Pages (from-to)	337-354
Number of pages	18
Journal	Mathematics and Computers in Simulation
Volume	219
DOIs	https://doi.org/10.1016/j.matcom.2023.12.034
State	Published - May 2024
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2023 International Association for Mathematics and Computers in Simulation (IMACS)

Keywords

Convergence and stability
Fractional derivative with generalized memory kernel
Generalized L1 scheme
Non-linear
Non-smooth solution
Weight function

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science
Numerical Analysis
Modeling and Simulation
Applied Mathematics

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10.1016/j.matcom.2023.12.034

Cite this

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title = "Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution",

abstract = "The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor's series. Theoretical stability and convergence analysis is established in the L2-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.",

keywords = "Convergence and stability, Fractional derivative with generalized memory kernel, Generalized L1 scheme, Non-linear, Non-smooth solution, Weight function",

author = "Nikki Kedia and Alikhanov, \{Anatoly A.\} and Singh, \{Vineet Kumar\}",

note = "Publisher Copyright: {\textcopyright} 2023 International Association for Mathematics and Computers in Simulation (IMACS)",

year = "2024",

month = may,

doi = "10.1016/j.matcom.2023.12.034",

language = "English",

volume = "219",

pages = "337--354",

journal = "Mathematics and Computers in Simulation",

issn = "0378-4754",

publisher = "Elsevier BV",

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TY - JOUR

T1 - Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

AU - Kedia, Nikki

AU - Alikhanov, Anatoly A.

AU - Singh, Vineet Kumar

PY - 2024/5

Y1 - 2024/5

N2 - The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor's series. Theoretical stability and convergence analysis is established in the L2-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.

AB - The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor's series. Theoretical stability and convergence analysis is established in the L2-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.

KW - Convergence and stability

KW - Fractional derivative with generalized memory kernel

KW - Generalized L1 scheme

KW - Non-linear

KW - Non-smooth solution

KW - Weight function

UR - https://www.scopus.com/pages/publications/85181247042

U2 - 10.1016/j.matcom.2023.12.034

DO - 10.1016/j.matcom.2023.12.034

M3 - Article

AN - SCOPUS:85181247042

SN - 0378-4754

VL - 219

SP - 337

EP - 354

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

ER -

Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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