A unified spectral method for FPDEs with two-sided derivatives; Part II: Stability, and error analysis

We present the stability and error analysis of the unified Petrov–Galerkin spectral method, developed in [1], for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any (1+d)-dimensional space-time hypercube, d=1,2,3,⋯, subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method. Finally, we perform several numerical simulations to compare the theoretical and computational rates of convergence. •We perform the rigorous well-posedness study of problem in any (1+d)-dimensions.•We carry out the corresponding discrete stability of Petrov–Galerkin method.•We complete the theoretical study by performing the error analysis of the spectral method.•We run several numerical simulations to show the agreement between our computational experiments and the developed theory.