A high-accuracy Vieta-Fibonacci collocation scheme to solve linear time-fractional telegraph equations

Abstract

The vital target of the current work is to construct two-variable Vieta-Fibonacci polynomials which are coupled with a matrix collocation method to solve the time-fractional telegraph equations. The emerged fractional derivative operators in these equations are in the Caputo sense. Telegraph equations arise in the fields of thermodynamics, hydrology, signal analysis, and diffusion process of chemicals. The orthogonality of derivatives of shifted Vieta-Fibonacci polynomials is proved. A bound of the approximation error is ascertained in a Vieta-Fibonacci-weighted Sobolev space that admits increasing the number of terms of the series solution leads to the decrease of the approximation error. The proposed scheme is implemented on four illustrated examples and obtained numerical results are compared with those reported in some existing research works.