A convergent collocation approach for one- and two-dimensional fractional diffusion equations with generalized kernel

Purpose This study aims to develop and analyze a numerical method to solve fractional diffusion equations (FDEs) of one dimension (1D) and two dimensions (2D) that incorporate the Caputo derivative with a generalized kernel (CDGK). Design/methodology/approach The collocation method is applied to compute the numerical solution, with a detailed discussion of the error and convergence properties. Findings The proposed CDGK is efficient, accurate and effective for solving 1D and 2D FDEs. The study demonstrates that the provided approach successfully handles both smooth and nonsmooth solutions. Furthermore, varying the scale function within the CDGK framework significantly influences, providing flexibility in modeling complex diffusion processes. Originality/value The novelty stems from applying the collocation method within the CDGK framework, providing a comprehensive investigation of error and convergence properties. Unlike existing methods, this study systematically explores the impact of varying the scale function on numerical solutions. It addresses both smooth and nonsmooth solutions under homogeneous and nonhomogeneous boundary conditions.