High order approximation on non-uniform meshes for generalized time-fractional telegraph equation

Abstract

This paper presents a high order approximation scheme to solve the generalized fractional telegraph equation (GFTE) involving the generalized fractional derivative (GFD). The GFD is characterized by a scale function σ(t) and a weight function ω(t). Thus, we study the solution behavior of the GFTE for different σ(t) and ω(t). The scale function either stretches or contracts the solution while the weight function dramatically shifts the numerical solution of the GFTE. The time fractional GFTE is approximated using quadratic scheme in the temporal direction and the compact finite difference scheme in the spatial direction. To improve the numerical scheme’s accuracy, we use the non-uniform mesh. The convergence order of the whole discretized scheme is, O(τ2α−3,h4), where τ and h are the temporal and spatial step sizes respectively. The outcomes of the work are as follows: •The error estimate for approximation of the GFD on non-uniform meshes is established.•The numerical scheme’s stability and convergence are examined.•Numerical results for four examples are compared with those obtained using other method. The study shows that the developed scheme achieves higher accuracy than the scheme discussed in literature.