Fully spectral‐Galerkin method for the one‐ and two‐dimensional fourth‐order time‐fractional partial integro‐differential equations with a weakly singular kernel
Farhad Fakhar–Izadi
Abstract
Abstract In the current paper, a space–time spectral‐Galerkin method is presented for the one‐ and two‐dimensional (1D & 2D) fourth‐order time‐fractional partial integro‐differential equation (TFPIDE) with a weakly singular kernel. In temporal direction, a Petrov‐Galerkin approach is used for discretization. Indeed, eigenfunctions of the first and second kind fractional Sturm‐Liouville problem (FSLP), which called Jacobi polyfractonomials, are used as temporal basis for the trial and test spaces, respectively. Also, spatial discretization is based on the Galerkin approximation with a combination of Legendre polynomials as basis. Fully discrete scheme, give rises to obtaining approximate solution of desired problem via solving a system of linear algebraic equations. Spectral accuracy and efficiency of the considered method are numerically demonstrated by some test problems with smooth and non‐smooth exact solutions in one‐ and two‐dimensional cases.