In this paper, an asymptotic semi-analytical method for solving a class of third-order singularly perturbed boundary value problems in which the highest order derivative is multiplied by a small parameter is presented. The method is distinguished by the following facts: first, the given problem is transformed into an equivalent system of two ODEs and a zeroth-order asymptotic expansion for the solution of the transformed system is constructed. Then, the reduced terminal value system is solved analytically using Differential Transform Method. The method results in approximate analytical solution for the considered problems. Some illustrating examples are given to demonstrate the accuracy and efficiency of the method. Numerical results obtained by the method are compared with the exact solution and its derivatives and are found to be in good agreement with each other not only in the boundary layer, but also away from the layer.

In this paper, a new A-stable explicit one-step integration method is developed for numerically solving stiff differential systems which characterize several kinds of linear reactions and diffusion from biochemistry, physiology, etc. The method is based on deriving a nonlinear relation between the dependent variable and its derivatives from the well known Taylor expansion. The method can be classified as a rational method. The accuracy and stability properties of the method are investigated and shown to yield at least fourth-order and A-stable. Some differential systems arising in chemical reactions will be solved to illustrate the performance and accuracy of the method. AMS Subject Classification: 65L05