A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations

• An efficient numerical scheme for time fractional diffusion-wave equations is proposed.
• A time–space Jacobi tau approximation is developed for such equations.
• Several tau-spectral methods can be achieved as special cases.
• The validity and applicability of the proposed method are demonstrated.
• Accurate numerical results are obtained by selecting limited collocation nodes.

In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.