Numerical Approximation Scheme for Solving Fractional Laplacian and Time Distributed Order Fractional Differential Equation

ABSTRACT

In this manuscript, our aim is to solve a newly formulated problem equation involving a distributed‐order time fractional derivative and a space‐regional fractional Laplacian operator. This work presents the first application of the proposed numerical technique to such a model. Utilizing 2D shifted Legendre polynomials, we construct novel operational matrices for both the regional fractional Laplacian and distributed‐order fractional derivatives, employing the Legendre–Gauss quadrature rule. The original complex problem is transformed into a system of linear algebraic equations using the standard Tau method combined with collocation points to handle initial and boundary conditions. The approximate solutions obtained demonstrate the effectiveness of the proposed approach. Furthermore, we establish error bounds, perform convergence analysis, and verify the numerical stability of the method. Several illustrative examples are provided to confirm the accuracy and computational efficiency of the proposed technique. We further compare our numerical results in the limiting case with existing results reported in the literature for time distributed‐order models involving the classical spatial Laplacian operator.