Fractional-Order Orthogonal Jacobi Function-Based Operational Approach for Multi-Term Diffusion-Wave Equations of Fractional Order

Solving fractional differential equations using spectral collocation methods based on classical orthogonal polynomials often leads to a reduced convergence rate due to the limited regularity of the solutions. Therefore, spectral collocation methods that employ non-smooth orthogonal functions are frequently preferred for solving various fractional differential equations. This study focuses on solving one- and two-dimensional time-fractional diffusion-wave equations (DWEs). A spectral collocation technique is developed based on fractional-order orthogonal Jacobi functions to approximate the time-fractional derivatives and orthogonal Jacobi polynomials in the spatial directions. For the first time, a fractional-order orthogonal Jacobi functions-based operational matrix is derived and combined with an orthogonal Jacobi polynomials-based operational matrix of second-order derivatives to solve one- and two-dimensional time-fractional DWEs. Three test problems are conducted to evaluate the efficiency of the proposed numerical technique.