An asymptotic numerical method for singularly perturbed third-order ordinary differential equations of convection-diffusion type

Abstract

Singularly perturbed two-point boundary value problems (SPBVPs) for third-order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. Then, the computational method, presented in this paper, is applied to this system. In this method, we reduce the weakly coupled system into a decoupled system. Then, to solve this decoupled system numerically, we apply a ‘boundary value technique (BVT)’, in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions. Then, we solve the decoupled system over these regions as two point boundary value problems. An exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme, in the outer region. The boundary conditions at the transition point are obtained using the zero-order asymptotic expansion approximation of the solution of the problem. This computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing. This method can be extended to a system of two ordinary differential equations, of which, one is of first order and the other is of second order. Numerical examples are given to illustrate the method.