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

Anthony Anya Okeke; Nur Nadiah Abd Hamid; Wen Eng Ong; Muhammad Abbas

doi:10.46481/jnsps.2025.2323

Authors

Anthony Anya Okeke
[email protected]

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia
https://orcid.org/0000-0002-5931-3437
Nur Nadiah Abd Hamid
Nur Nadiah Academic Services, 13800 Butterworth, Penang, Malaysia
https://orcid.org/0000-0001-9674-6082
Wen Eng Ong
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia
https://orcid.org/0000-0001-8073-8774
Muhammad Abbas
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
https://orcid.org/0000-0002-0491-1528

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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