Finite Difference Methods for Caputo–Hadamard Fractional Differential Equations

Abstract

In this paper, we study finite difference methods for fractional differential equations (FDEs) with Caputo–Hadamard derivatives. First, smoothness properties of the solution are investigated. The fractional rectangular, $${L}_{\mathrm{log},1}$$ interpolation, and modified predictor–corrector methods for Caputo–Hadamard fractional ordinary differential equations (FODEs) are proposed through approximating the corresponding equivalent Volterra integral equations. The stability and error estimate of the derived methods are proved as well. Then, we investigate finite difference methods for fractional partial differential equations (FPDEs) with Caputo–Hadamard derivative. By applying the constructed L1 scheme for approximating the time fractional derivative, a semi-discrete difference scheme is derived. The stability and convergence analysis are shown too in detail. Furthermore, a fully discrete scheme is established by the standard second-order difference scheme in spacial direction. Stability and error estimate are also presented. The numerical experiments are displayed to verify the theoretical results.