High-order orthogonal spline collocation ADI scheme for a new complex two-dimensional distributed-order fractional integro-differential equation with two weakly singular kernels

Abstract

We present an alternating direction implicit scheme for the numerical solution of the distributed-order fractional integro-differential equation with two weakly singular kernels. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. The weighted and shifted Grünwald difference formula is employed to discretize the distributed-order time-fractional derivative combined with the second-order quadrature convolution rule introduced by Lubich for the Riemann–Liouville fractional integral. We prove the stability of the proposed scheme and derive error estimates. We also describe an efficient implementation of the scheme and show the numerical results demonstrating the accuracy and convergence rates in $L^2$ norm.

Keywords:

• Distributed-order time-fractional derivative
• fractional integral with two weakly singular kernels
• alternating direction implicit scheme
• orthogonal spline collocation
• optimal order convergence

2020 AMS SUBJECT CLASSIFICATIONS:

• 35R11
• 65M12
• 65M70
• 65M15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [12126308,12026263,12101080,12126303] Hubei Superior and Distinctive Discipline Group of ‘Mechatronics and Automobiles’ [XKQ2021018] Scientific Research Foundation of Hunan Provincial Education Department[21C0188] Hunan Provincial Natural Science Foundation of China [2022JJ40463] Excellent young and middle-aged science and Technology Innovation team project of the Education Department of Hubei Province [T2022029] Research ability  cultivation  fund  of Hubei University of Arts and Science [2020kypytd006] the Project of Hubei University of Arts and Science [XK2021022] the Humanities and Social Science Youth Foundation of Ministry of Education of China [ 17YJC630084] Hubei Superior and Distinctive Discipline Group of “New Energy Vehicle and Smart Transportation”[XKTD052023].