Singularly Perturbed Two-point Boundary Value Problem Based On Chebyshev-Gauss Grid

Singular perturbation theory and methods cover a very wide range of disciplines,it is a important way that is used to solve the equations of mathematical physics ofnon-linear, high-order or variable coefficients for approximate solution, and thecurrent study is very active and constantly expanded. The goal is to solve differentialequations containing a small parameter, the approximate solution is obtained bysolving some relatively simple equation that is related with the original equation, theapproximate solution is called the analytic approximate solution. It has been graduallyestablished a number of effective singular perturbation numerical methods, such asfinite difference method, finite element method, spectral method, etc. And they havebeen widely used in various fields of natural sciences, playing an important role insolving practical problems. The spectral method is a numerical method which is usedto construct approximate solutions by orthogonal functions or characteristic functions.Because of its superior convergent rate, there has been extensive attention on it. It willbe introduced and then used in the paper. The spectral method derived fromRitz-Galerkin method, which based on orthogonal polynomials (such as Chebyshevpolynomials, Legendre polynomials, Trigonometric polynomials) as basis functions.The paper solves the approximate solution of singularly perturbed problems based onChebyshev-Gauss grid, the main research contents are as follows:1. Background and introduction of spectral methods is presented.2. A class of singularly perturbed two-point boundary value problem based onChebyshev-Gauss grid is studied, and the error estimates of the approximate solutionis given.3. A class of singularly perturbed boundary value problem containing the delayparameters based on Chebyshev-Gauss grid is given, and the error estimates of theapproximate solution is given.