A Rational Spectrum Allocation and Particle Swarm Optimization for Nonlinear Singularly Perturbed Problems

For a class of parametric perturbed problems, we use a rational spectral method of barycenter with sinh transformation. By using sinh transform, more Chebyshev points are mapped to the singularity of the solution interval. Then, in order to calculate the mesh parameters of boundary layer width after sinh transformation and the parameters in the problem, this paper designs a nonlinear optimization with the objective function as the minimum error. In order to solve the optimization problem, we choose particle swarm optimization algorithm to solve the problem. Finally, through numerical experiments, it is found that particle swarm optimization (PSO) algorithm has significant advantages in solving such problems.