Chebyshev Polynomials of Sixth Kind for Solving Nonlinear Fractional PDEs with Proportional Delay and Its Convergence Analysis

Khadijeh Sadri,Hossein Aminikhah,Youssri Hassan Youssri

doi:10.1155/2022/9512048

Khadijeh Sadri, Hossein Aminikhah + Show 1 more

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https://doi.org/10.1155/2022/9512048

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Abstract

This work devotes to solving a class of delay fractional partial differential equations that arises in physical, biological, medical, and climate models. For this, a numerical scheme is implemented that applies operational matrices to convert the main problem into a system of algebraic equations; then, solving the resultant system leads to an approximate solution. The two-variable Chebyshev polynomials of the sixth kind, as basis functions in the proposed method, are constructed by the one-variable ones, and their operational matrices are derived. Error bounds of approximate solutions and their fractional and classical derivatives are computed. With the aid of these bounds, a bound for the residual function is estimated. Three illustrative examples demonstrate the simplicity and efficiency of the proposed method.

Highlights

Mathematical modeling of some physical and biological phenomena leads to delay fractional differential equations (DFDEs) [1,2,3]
Few methods exist for solving delay partial differential equations
This paper deals with numerically solving a class of fractional partial differential equations with proportional delays on the domain Ω = 1⁄20, 1 × 1⁄20, 1

Summary

Introduction

Mathematical modeling of some physical and biological phenomena leads to delay fractional differential equations (DFDEs) [1,2,3]. Obtaining exact solutions to these equations necessitates mathematicians to construct some vigorous numerical and semianalytical schemes to handle solving these problems. Few methods exist for solving delay partial differential equations. The interest of scientists and mathematicians in DFDEs has resulted in the presentation of efficient schemes to solve this category of equations. Pimenov and Hendy presented a difference scheme for a class of fractional diffusion equations with fixed time delay [4]. A compact difference scheme was constructed in [5] for the numerical solution of onedimensional fractional parabolic differential equations with delay. Hendy et al [6] introduced a Crank–Nicolson difference approximation for solving multiterm time-fractional diffusion equations with delay. Nandal and Pandey constructed a linearized compact difference scheme for fourthorder nonlinear fractional subdiffusion with time delay [7]

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