Sinc collocation method and sinh transform for solving singular perturbed problems with a transition layer

High-order Sinc collocation method was used to solve a class of singular perturbed problems with a transition layer. Combined with the double exponential transform, Sinc collocation method defined naturally on the unbounded domain as available to the two-point boundary value problem with homogenous Dirichlet boundary conditions. For non-homogenous Dirichlet boundary conditions, a linear function was designed to reconstruct the unknown function. That is, it transformed the original problem into the problem with homogenous Dirichlet boundary conditions. The sinh transform was utilized to make the Sinc nodes dense in the location of the transition layer, which leaded to good accuracy with less computational cost. Based on the double exponential transform, a Sinc-barycentric form interpolation formulae defined on the bounded domain was proposed to improve the performance of the numerical approximation of non-node function values. Numerical results verified the computational efficiency of the Sinc collocation method, which combined with the double exponential transform and the sinh transform, for solving singular perturbed problems with a transition layer and the non-stationary Burgers’ equation with a shock wave.