Extended cubic B-splines in the numerical solution of time fractional telegraph equation

Abstract

A finite difference scheme based on extended cubic B-spline method for the solution of time fractional telegraph equation is presented and discussed. The Caputo fractional formula is used in the discretization of the time fractional derivative. A combination of the Caputo fractional derivative together with an extended cubic B-spline is utilized to obtain the computed solutions. The proposed scheme is shown to possess the unconditional stability property with second order convergence. Numerical results demonstrate the applicability, simplicity and the strength of the scheme in solving the time fractional telegraph equation with accuracies very close to the exact solutions.and ∂ α u(x,t) ∂t α denote the Caputo fractional derivative of order 2α and α, respectively, which will be defined in Sect. 2.This model has been developed to overcome the shortcomings of the classical telegraph equation which might not adequately model the abnormal diffusion phenomena during a long transmission process in a transmission line [2].

Numerical analysis of two new finite difference methods for time-fractional telegraph equation

“…I-E difference method for time fractional telegraph. Similar to the method of constructing the E-I scheme, we give the I-E scheme of the time fractional telegraph equation (1): The odd number layers is calculated by the implicit scheme:…”

“…. (16) this is Richardson scheme of the time fractional telegraph equation (1). However, it should be pointed out that the calculation process using the I-E scheme (15) is also an unknown number, the process of joint use (15) is equivalent to using the implicit scheme (13) to calculate u 2n+1 i from u 2n i , and then using the explicit scheme (14) to calculate…”

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

“…Akram et al [31,32] solved the time-fractional diffusion problems using ECBS in Caputo and Riemann-Liouville sense. Various numerical techniques based on ECBS functions have been used to approximate fractional partial differential models, such as linear and non-linear time-fractional telegraph models [33,34], fractional Fisher's model [35], time fractional Burger's model [36], fractional Klein-Gordon model [37], time-fractional diffusion wave model [38], fractional advection-diffusion model [39].…”

A numerical study on time fractional Fisher equation using an extended cubic B-spline approximation

“…Mohyud-din [25] obtained the solution of fractional advection diffusion equation by ECuBS approach. Akram et al [8,9] presented numerical solutions of linear FPDEs via ECuBS method and Caputo fractional derivative (CFD). Akram et al [7,11] derived ECuBS method for the solution of time fractional Burgers and time fractional Klein-Gordon equations.…”