Novel and accurate Gegenbauer spectral tau algorithms for distributed order nonlinear time-fractional telegraph models in multi-dimensions

Abstract

This paper sheds light on the numerical solutions of the multi-dimensional distributed-order (DO) nonlinear time-fractional telegraph equations (TFTEs) models. We present novel and accurate shifted Gegenbauer spectral tau schemes for solving the aforementioned models. In the proposed schemes, the known and unknown functions are approximated in terms of the shifted Gegenbauer polynomials (SGPs) by using the Kronecker product structure in time and space. The matrix representation of the DO fractional derivative of SGPs in the Caputo sense has been derived. Also, novel operational matrices of the multiplication of shifted Gegenbauer functions in multi-dimension have been constructed to remove the obstacle that facing the use of the tau method in the presence of nonlinear term or variable coefficients or both of them. The proposed method takes full advantage of the nonlocal nature of DO fractional differential operators. The presented method is applied on six miscellaneous test examples to illustrate its robustness and effectiveness for smooth and non smooth solutions.