10.1007/s40096-021-00430-4

A Legendre spectral-finite difference method for Caputo–Fabrizio time-fractional distributed-order diffusion equation

  • M. Fardi*1,
  • J. Alidousti1,
  1. Department of Applied Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord, IR

Published in Issue 2021-09-08

How to Cite

Fardi, M., & Alidousti, J. (2021). A Legendre spectral-finite difference method for Caputo–Fabrizio time-fractional distributed-order diffusion equation. Mathematical Sciences, 16(4 (December 2022). https://doi.org/10.1007/s40096-021-00430-4
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Abstract

Abstract In this paper, we introduce a hybrid method based on a finite difference method and a spectral method for solving the multi-term time-fractional diffusion equations (TFDEs) based on Caputo–Fabrizio fractional operator. We apply a finite difference scheme for discretizing the time derivatives and consider a Legendre-spectral approximation in space discretization to semi-discrete problem. It is known that the spectral method has been an efficient tool for computing numerical solutions of differential equations because of its high-order accuracy. We discuss the convergence of the proposed method in discrete L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document} -norm. Furthermore, we extend the multi-term TFDE to the distributed order and analyze the method for the considered equation. In the end, we confirm the proven theoretical results with the help of some numerical examples.

Keywords

  • Multi-term,
  • Distributed order,
  • Time-fractional,
  • Diffusion equation,
  • Caputo–Fabrizio derivative,
  • Error analysis

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