Abstract: Based on the precise integration method (PIM), a coupling technique of the high order multiplication perturbation method (HOMPM) and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value problems (TPBVPs) with one boundary layer. First, the inhomogeneous ordinary differential equations (ODEs) are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer matrix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.

Key words: variable coefficient singular perturbation problem, two-point boundary value problem, modified Laguerre orthogonal approximation and interpolation, multiple dimensions, spectral and pseudospectral methods, nonlinear partial differential equations, high order multiplication perturbation method (HOMPM), reduction method