Higher-Order Asymptotic Numerical Solutions for Singularly Perturbed Problems with Variable Coefficients

For the purpose of solving a second-order singularly perturbed problem (SPP) with variable coefficients, a m th-order asymptotic-numerical method was developed, which decomposes the solutions into two independent sub-problems: a reduced first-order linear problem with a left-end boundary condition; and a linear second-order problem with the boundary conditions given at two ends. These are coupled through a left-end boundary condition. Traditionally, the asymptotic solution within the boundary layer is carried out in the stretched coordinates by either analytic or numerical method. The present paper executes the m th-order asymptotic series solution in terms of the original coordinates. After introducing 2 ( m + 1 ) new variables, the outer and inner problems are transformed together to a set of 3 ( m + 1 ) first-order initial value problems with the given zero initial conditions; then, the Runge–Kutta method is applied to integrate the differential equations to determine the 2 ( m + 1 ) unknown terminal values of the new variables until they are convergent. The asymptotic-numerical solution exactly satisfies the boundary conditions, which are different from the conventional asymptotic solution. Several examples demonstrated that the newly proposed method can achieve a better asymptotic solution. For all values of the perturbing parameter, the method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy of the solution in the entire domain. We derive the sufficient conditions, which terminate the series of asymptotic solutions for inner and outer problems of the SPP without having the spring term. For a specific case, we can derive a closed-form asymptotic solution, which is also the exact solution of the considered SPP.