Universal Journal of Applied Mathematics Vol. 14(2), pp. 23 - 38
DOI: 10.13189/ujam.2026.140201
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A Comparative Study of Cubic B-Spline-Based Numerical Techniques for Solving Nonlinear Reaction-Diffusion Equations

Rajni Rohila ¹, R. C. Mittal ^2,*
1 Department of Applied Sciences, The NorthCap University, Gurugram, Haryana, India
² Department of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand, India

ABSTRACT

Nonlinear time-dependent reaction–diffusion equations arise in several scientific and engineering applications, including chemical kinetics, biological pattern formation, heat transfer, and population dynamics. Due to their nonlinear behavior and time dependence, obtaining accurate and computationally efficient numerical solutions remains a significant challenge. In this work, three B-spline-based numerical methods are proposed for the numerical solution of nonlinear reaction–diffusion equations and are analyzed for their computational effectiveness. The developed approaches are referred to as Method I, Method II, and Method III. Method I is based on the modified cubic B-spline differential quadrature method, which provides an efficient approximation of spatial derivatives. Method II employs cubic B-spline collocation for spatial discretization in combination with the Crank-Nicolson scheme for time integration to enhance stability and numerical accuracy. Method III uses modified cubic B-spline functions for spatial approximation, reducing the governing equation to a system of ordinary differential equations that is solved using the SSP-RK43 scheme, a low-storage method that reduces computational cost and minimizes error accumulation. The proposed methods are validated through numerical experiments on nonlinear reaction– diffusion models, demonstrating satisfactory accuracy, stability, and computational efficiency in approximating nonlinear dynamics. Comparative observations indicate that the proposed spline-based methods provide reliable numerical solutions while preserving computational precision. The study contributes to the advancement of efficient B-spline numerical frameworks for solving nonlinear partial differential equations and offers practical relevance for modeling diffusion–reaction phenomena in applied sciences and engineering. A limitation of the present work is its focus on one-dimensional problems; however, the proposed methodology can be extended to multidimensional systems in future studies.

KEYWORDS
Reaction-Diffusion Equations, Differential Quadrature Method, Collocation Method, Finite Difference, Stability Analysis, Result Analysis

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Rajni Rohila , R. C. Mittal , "A Comparative Study of Cubic B-Spline-Based Numerical Techniques for Solving Nonlinear Reaction-Diffusion Equations," Universal Journal of Applied Mathematics, Vol. 14, No. 2, pp. 23 - 38, 2026. DOI: 10.13189/ujam.2026.140201.

(b). APA Format:
Rajni Rohila , R. C. Mittal (2026). A Comparative Study of Cubic B-Spline-Based Numerical Techniques for Solving Nonlinear Reaction-Diffusion Equations. Universal Journal of Applied Mathematics, 14(2), 23 - 38. DOI: 10.13189/ujam.2026.140201.