High order stable numerical algorithms for generalized time-fractional deterministic and stochastic telegraph models

Abstract

The aim of this manuscript is to design and analyze a hybrid stable numerical algorithm for generalized fractional derivative (GFD) defined in Caputo sense \(\mathscr {D}^{\alpha }_{0, Z,\omega }\) on non-uniform grid points in the temporal direction. An efficient and hybrid high order discretization is proposed for GFD by incorporating a \((3 - \alpha )\)-th order approximation using the moving refinement grid method for the initial interval in the temporal direction. The physical applications of the developed high order approximation are employed to design a hybrid numerical algorithm to determine the solution of the generalized time-fractional telegraph equation (GTFTE) and the generalized time-fractional stochastic telegraph equation (GTFSTE). The proposed numerical techniques are subjected to rigorous error analysis and a thorough investigation of theoretical results i.e. solvability, unconditional stability, convergence analysis, and comparative study are conducted with the existing scheme (Kumar et al. in Numer Methods Partial Differ Equ 35(3):1164–1183, 2019). Several test functions are utilized to verify that second-order convergence is attained in time which is higher than the order of convergence produced by the existing scheme (Kumar et al. 2019). In spatial direction, fourth-order convergence is obtained utilising the compact finite difference methods in spatial approximation on uniform meshes. A reduced first-order convergence in the temporal direction is reported for the GTFSTE model. Further, certain scaling and weight functions are used to show cast the impact of scaling and weight functions in the GFD.