A numerical algorithm based on extended cubic B-spline functions for solving time-fractional convection-diffusion-reaction equation with variable coefficients

Authors

Keywords:

Time fractional convection-diffusion-reaction equation, Extended cubic B-spline basis functions, Caputo fractional derivative, Stability and convergence, Crank–Nicolson finite difference formulation

Abstract

In this research, we develop an innovative and efficient numerical method for solving a nonlinear one-dimensional time-fractional convection-diffusion-reaction equation (TFCDRE) with a variable coefficient. This method integrates the Crank-Nicolson finite difference scheme with extended cubic B-spline (ExCBS) basis functions. The Caputo fractional derivative is applied for temporal discretization, while the ExCBS functions are utilized for spatial discretization. The stability of the method was discussed by the von Neumann method, which shows unconditional stability; and the convergence analysis secure an order of $O(h^2+\triangle t^{2-\alpha})$. We demonstrated the efficiency and simplicity of our method through three numerical experiments and verified its accuracy using the absolute error $(L_2)$ and maximum error $(L_\infty)$ norms in temporal and spatial dimensions. The results are also graphically represented and show substantial agreement with the approximated and exact solution. We confirmed the effectiveness and accuracy of our approach over the local discontinuous Galerkin finite element and the compact finite difference methods, respectively, from the literature.

Dimensions

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Comparison plot of exact and approximate solutions with N = 32, ? = 0.45, ?t = 0.01 for Example 1 at different time levels.

Published

2025-02-01

How to Cite

A numerical algorithm based on extended cubic B-spline functions for solving time-fractional convection-diffusion-reaction equation with variable coefficients. (2025). Journal of the Nigerian Society of Physical Sciences, 7(1), 2323. https://doi.org/10.46481/jnsps.2025.2323

Issue

Volume 7, Issue 1, February 2025

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2024 Anthony Anya Okeke, Nur Nadiah Abd Hamid, Wen Eng Ong, Muhammad Abbas

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

A numerical algorithm based on extended cubic B-spline functions for solving time-fractional convection-diffusion-reaction equation with variable coefficients. (2025). Journal of the Nigerian Society of Physical Sciences, 7(1), 2323. https://doi.org/10.46481/jnsps.2025.2323

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