A novel method based on fractional order Gegenbauer wavelet operational matrix for the solutions of the multi-term time-fractional telegraph equation of distributed order

In this article, we propose an effective scheme based on a combination of the Tau method and fractional -order Gegenbauer wavelets for solving the multi -term time -fractional differential equations of distributed order. First, we define fractional order Gegenbauer wavelets and then obtain operational matrices of these orthogonal functions. Applying the Legendre-Gauss quadrature for the integral term, we use function approximations obtained by the presented wavelets and the Tau method for the solution of the distributed -order multi -term time -fractional telegraph equation. The proposed method reduces the numerical solution of multi order timefractional equations to a system of algebraic equations. Then, the convergence analysis and error bounds of the proposed scheme are studied. Three illustrative examples are solved to justify the effectiveness of the proposed method compared with some previously published results.