Abstract
In this paper, we propose an efficient numerical scheme for the approximate solution of a time fractional diffusion-wave equation with reaction term based on cubic trigonometric basis functions. The time fractional derivative is approximated by the usual finite difference formulation, and the derivative in space is discretized using cubic trigonometric B-spline functions. A stability analysis of the scheme is conducted to confirm that the scheme does not amplify errors. Computational experiments are also performed to further establish the accuracy and validity of the proposed scheme. The results obtained are compared with finite difference schemes based on the Hermite formula and radial basis functions. It is found that our numerical approach performs superior to the existing methods due to its simple implementation, straightforward interpolation and very low computational cost. A convergence analysis of the scheme is also discussed.
Highlights
1.1 Problem description For T > and = [a, b], we consider the following model of the time fractional diffusionwave equation with reaction term: ∂γ ∂∂tγ u(x, t) + αu(x, t) = ∂x u(x, t) + f (x, t), < γ ≤, x ∈, ≤t ≤T ( )with initial conditions ⎧⎨u(x, ) = φ (x), ⎩ut(x, ) = φ (x), x∈, and the following boundary conditions:⎨u(a, t) = ψ (t), ⎩u(b, t) = ψ (t),≤ t ≤ T, Yaseen et al Advances in Difference Equations (2017) 2017:274 where a, b, φ (x), φ (x), ψ (t) and ψ (t) are given, α > is the reaction coefficient and
Numerical solutions to the fractional diffusion-wave equation under Dirichlet and Neumann boundary conditions were obtained by Povstenko
An efficient numerical scheme based on trigonometric cubic B spline functions is presented in this paper to find the approximate solutions of a time fractional diffusion-wave equation with reaction term
Summary
Zeng [ ] proposed two second order stable and one conditionally stable finite difference schemes for the time fractional diffusion-wave model. Numerical solutions to the fractional diffusion-wave equation under Dirichlet and Neumann boundary conditions were obtained by Povstenko. Ren and Sun [ ] obtained efficient numerical solutions of the multi-term time fractional diffusion-wave equation by using a compact finite difference scheme with fourth-order accuracy. Jianfei et al [ ] presented two efficient finite difference schemes to approximate solutions of time fractional diffusion equations. An efficient numerical scheme based on trigonometric cubic B spline functions is presented in this paper to find the approximate solutions of a time fractional diffusion-wave equation with reaction term. In Section , we present the derivation of the scheme for the fractional diffusion-wave equation using the trigonometric cubic B-spline functions. The last section is devoted to the concluding remarks of the study
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