Numerical techniques demonstrate uniformly convergent solutions for nonlinear reaction–diffusion equations, highlighting stability and convergence rates.
This study presents a higher-order uniformly convergent numerical method for a singularly perturbed nonlinear reaction–diffusion equation. The quasilinearization technique is used to transform a nonlinear term into a linear boundary value problem. The equivalent singularly perturbed reaction–diffusion differential equation is discretized using a central finite-difference method on the Shishkin and Bakhvalov meshes. The stability and parameter-uniform convergence are rigorously analyzed and established. Theoretically, the proposed method is of second order. The application of Richardson extrapolation demonstrates that the Bakhvalov mesh achieves a pure CM−4 rate, whereas the Shishkin mesh attains an ε-uniform convergence rate of CM−4ln4M. Three numerical examples have been solved to corroborate the theoretical findings.
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Gelu et al. (2026) studied this question.
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