A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

Abstract

In this paper, we investigate a fully implicit finite difference scheme for solving the time fractional advection-diffusion equation. The time fractional derivative is estimated using Caputo's formulation, and the spatial derivatives are discretized using extended cubic B-spline functions. The convergence and stability of the fully implicit scheme are analyzed. Numerical experiments conducted indicate that the scheme is feasible and accurate.

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

“…Various numerical methods based on spline functions have also been employed by researchers in pursue of reliable solutions for fractional-order differential equations [29][30][31]. The B-spline functions provide decent approximations in contrast with rest of numerical schemes due to the nominal, compact support and C 2 continuity [32]. The approximate solution of a time-fractional DWE via cubic trigonometric B-splines was explored in [33].…”

Difference numerical solutions for time-space fractional advection diffusion equation

“…The fundamental solutions were obtained by the Laplace transform and Fourier transform, and the numerical results were also obtained. A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection diffusion equation was obtained in [19]. This study [20] was concerned with the mixed initial boundary value problem for a dipolar body in the context of the thermoelastic theory, and the Hölder-type stability was also discussed.…”

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

“…Yaseen et al [28] presented a scheme for the numerical solution of fractional diffusion equation using a finite difference method based on cubic trigonometric B-spline basis functions. Mohyud-Din et al [29] constructed a fully implicit finite difference scheme for solving a time fractional diffusion equation by incorporating an extended cubic Bspline (ExCuBs) approach in its formulation. Because of the promising results obtained by this scheme, efforts are now being made in this work extend the formulation to solve the more complicated telegraph equation with fractional order derivatives.…”

“…In Table 2, we calculate the L 2 -norm for different spatial and temporal step size h = 5, τ = 1 M , (M = 20, 40, 80). In Table 3, we determine the order of convergence [16,29] from the computed data and present maximum absolute errors at different space-time step sizes. We give L 2 -norm and maximum errors for α = 0.6, 0.7, 0.8, 0.9.…”