A Space-Time Petrov-Galerkin Spectral Method for Time Fractional Fokker-Planck Equation with Nonsmooth Solution

Wei Zeng, Wei Zeng; Aiguo Xiao, Aiguo Xiao; Weiping Bu, Weiping Bu; Junjie Wang, Junjie Wang; Shucun Li, Shucun Li

doi:10.4208/eajam.260119.140319

Cited by 5 publications

(4 citation statements)

References 33 publications

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“…In [28], the authors used the Legendre spectral method for solving the time fractional nonlinear sine‐Gordon equation with smooth and nonsmooth solutions. In [29], the authors used the Petrov–Galerkin spectral method for solving the time fractional Fokker–Planck equation with nonsmooth solution. Furthermore, the Galerkin–Legendre spectral method for the nonlinear time–space fractional diffusion reaction equations with smooth and nonsmooth solutions was developed by [30].…”

Section: Introductionmentioning

confidence: 99%

Novel spectral schemes to fractional problems with nonsmooth solutions

Atta

Abd-Elhameed

Moatimid

et al. 2023

Math Methods in App Sciences

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In this article, we present two numerical methods for treating the fractional initial‐value problem (FIVP) and time‐fractional partial differential problem (FPDP) that caused the error to decay exponentially rapidly. The derivations of the proposed schemes rely on the use of a spectral Galerkin method that reduces each of the FIVP and FPDP into an algebraic system of equations in the unknown expansion coefficients. The class of orthogonal polynomials, namely, Chebyshev polynomials of the fifth kind is utilized. In terms of new basis functions called regular shifted Chebyshev poly‐fractionomials of fifth kind, approximate solutions to the FIVP and FPDP are obtained. Moreover, convergence and error analysis of the two problems are investigated in depth. Some numerical examples are presented with some comparisons. In conclusion, our spectral methods are effective and convenient.

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Section: Introductionmentioning

confidence: 99%

Novel spectral schemes to fractional problems with nonsmooth solutions

Atta

Abd-Elhameed

Moatimid

et al. 2023

Math Methods in App Sciences

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“…In a similar way, Sepehrian and Radpoor (2015) [15] proposed a Finite Difference scheme associated with the Spline Cubic Collocation Method for solving the non-linear FPE. In a fractional context, the FPE was solved by using the Spectral Method [16].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of the Fokker-Planck Equation Using Orthogonal Collocation

Lima,

Lobato

2023

Sci. Plena

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This study aims to propose a numerical methodology to solve the Fokker-Planck Equation using the Orthogonal Collocation Method. In this approach, the original problem, represented by a partial differential equation, is rewritten as a system of initial value equations via discretization of spatial variable. The resulting system is integrated considering the Runge-Kutta-Fehlberg Method. The proposed methodology is applied in three case studies that presented an analytical solution. The obtained results demonstrate that the proposed approach is a good alternative for solving this class of problems. Keywords: Fokker-Planck Equation, Orthogonal Collocation, Numerical Method.

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“…Fractional diffusion equations (FDEs) are an important class of fractional differential equations and have modelled successfully some physics phenomenons. Some numerical methods such as finite difference method, finite element method, spectral method, and finite volume method have been proposed to solve some fractional differential equations 1–3,5–16 . In particular, for FDEs with nonsmooth solutions, several numerical approaches based on nonuniform refined grids, 17‐21 correction convolution quadrature, 22 and nonpolynomial basis functions 23‐26 have been developed.…”

Section: Introductionmentioning

confidence: 99%

Spectral collocation method for a class of fractional diffusion differential equations with nonsmooth solutions

Wang

Xiao

2020

Math Methods in App Sciences

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In this paper, we develop spectral collocation method for a class of fractional diffusion differential equations. Since the solutions of these fractional differential equations usually exhibit singularities at the end‐points, it can not be well approximated by classical polynomial basis functions. We first give nonclassical interpolants based on the associated Jacobi‐Gauss points and obtain the corresponding fractional differentiation matrices. Second, we prove the convergence for the developed spectral collocation method. Finally, several numerical examples are considered to demonstrate the validity and applicability of the developed basis functions.

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A Space-Time Petrov-Galerkin Spectral Method for Time Fractional Fokker-Planck Equation with Nonsmooth Solution

Cited by 5 publications

References 33 publications

Novel spectral schemes to fractional problems with nonsmooth solutions

Novel spectral schemes to fractional problems with nonsmooth solutions

Numerical Solution of the Fokker-Planck Equation Using Orthogonal Collocation

Spectral collocation method for a class of fractional diffusion differential equations with nonsmooth solutions

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