Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

Khadijeh Sadri,Dumitru Baleanu,Kamyar Hosseini,Ali Ahmadian,Soheil Salahshour

doi:10.1186/s13662-021-03507-5

Abstract

The shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

Highlights

Partial integro-differential equations (PIDEs) with weakly singular kernels emerge in some physical and chemical phenomena, such as the radiation of heat from semi-infinite solids, stereology, hydrodynamics, heat conduction, and theory of elasticity [1, 2]
The rest of the paper is structured as follows: definitions of the fractional derivative and integral operators, one- and two-variable Chebyshev polynomials of the fifth kind are introduced in Sect
7 Conclusion The fifth-kind Chebyshev polynomials were proposed to deal with the numerical solution of a class of partial integro-differential equations with weakly singular kernels

Summary

Introduction

Partial integro-differential equations (PIDEs) with weakly singular kernels emerge in some physical and chemical phenomena, such as the radiation of heat from semi-infinite solids, stereology, hydrodynamics, heat conduction, and theory of elasticity [1, 2]. The rest of the paper is structured as follows: definitions of the fractional derivative and integral operators, one- and two-variable Chebyshev polynomials of the fifth kind are introduced in Sect. Theorem 3.4 Assume that X(t) is the basis vector in (9) and R0LItσ(t) is the variable-order fractional integral operator in the Riemann–Liouville sense as follows: Itσ (t)

Results

Conclusion