Solving singular perturbation with one boundary layer problem of second order ODE using the method of matched asymptotic expansion (MMAE) / Firdawati Mohamed and Masnira Ramli

Any physical problems when modeled into equations, may become linear or nonlinear equations with known and unknown boundaries. The exact solution for those equations is not easy to obtain. Hence, an analytical approximation solution in terms of asymptotic expansion is sought. This study seeks to solve a singular perturbation in second order ordinary differential equations. Solutions to several perturbed ordinary differential equations are obtained in terms of asymptotic expansion. The main focus of this study is to find an approximate analytical solution for perturbation problems (linear and nonlinear equations) that occur at one boundary layer using the classical method of matched asymptotic expansion (MMAE). Besides, the underlying concepts and principles of MMAE will also be clarified. Mathematica computer algebra system is used to perform the detail algebraic computations. The results of approximation analytical solution are illustrated by graphs using selected parameters which show the outer, inner and composite solutions separately. Hence, the exact solution and composite solution obtained using MMAE are compared by plotting the graph to show their accuracy. From the comparison, MMAE is one of the best methods to solve singular perturbation problems in second order ordinary differential equation since the results obtained are very close to the exact solution.