An Efficient Numerical Method for Reaction-Diffusion Problems Based on Particle Swarm Optimization Algorithm

In this paper, based on the standard particle swarm optimization algorithm, an efficient numerical method for solving a singularly perturbed reaction-diffusion problem on a Shishkin mesh. An upwind finite difference scheme on a Shishkin mesh is developed to approximate the integro-differential equation transformed from the singularly perturbed reaction-diffusion equation. In order to obtain the best the Shishkin mesh parameters and first-derivative of the exact solution at point \(x=0\), we construct a nonlinear unconstrained optimization problem, which is solved by using the PSO algorithm. Compared with the other algorithms, the PSO algorithm can obtain more accurate numerical results, which demonstrate the feasibility and effectiveness of the presented method.

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