Solving Singularly Perturbed Problems In The Reproducing Kernel Space

Singularly perturbed differential equations(SPDEs) appear in several branches ofapplied mathematics. The principal characteristic of the problem is that the solution ofsingularly perturbed problem has a multiscale character. That is, there are thin layerswhere the solution varies very rapidly, while away from the layer the solution behavesregularly and varies slowly. The numerical treatment of singularly perturbed prob-lems presents some major computational difficulties. Classical numerical methodsfail to produce good approximation and in recent years a larger number of nonclassi-cal numerical methods have been proposed to provide accurate numerical solutions.Moreover, two-parameter singular perturbation problems have so many applications inlubrication theory, reactor theory and dc motor analysis. Higher-order singularly per-turbed ordinary differential equations have important applications in Fluid Dynamics.Therefore, in this paper, we discuss how to solve the singularly perturbed two-pointboundary problems with two-parameter and the singularly perturbed fourth order or-dinary differential equations in the reproducing kernel space. For singularly perturbedfourth order ordinary differential equation, we transform the equation into a systemof two ordinary differential equations, then we solve the system by the method forsolving the differential system in the reproducing kernel space. Their exact solutionsare constructed analytically and are represented in the form of series in the reproduc-ing kernel space. In the mean time, the n-term approximate to the exact solutions areobtained by truncating the series. Some numerical examples are studied to demon-strate the accuracy of the present method. Results obtained by the method indicate themethod is simple and effective.