An asymptotic numerical fitted mesh method for singularly perturbed third order ordinary differential equations of reaction–diffusion type

摘要

Singularly perturbed boundary value problems (SPBVPs) for third order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative of the form:−εy‴(x)+b(x)y′(x)+c(x)y(x)=f(x),y(0)=p,y′(0)=q,y′(1)=r,where b(x),c(x) and f(x) are sufficiently smooth functions satisfying certain conditions, are considered. Firstly, this third order singularly perturbed boundary value problem (SPBVP) is transformed into equivalent problem of weakly coupled system of one first order and one second order ODE, with a small parameter multiplying the highest derivative of the second order ODE, subject to initial and boundary conditions, respectively. A computational method is suggested in this paper to solve this system. In this method, we first find the zero order asymptotic expansion approximation of the solution of the weakly coupled system. Then this system is decoupled by approximating the first component of the solution by its zero order asymptotic expansion approximation in the second equation. Finally the second equation is solved by the fitted mesh method (J.J.H. Miller, E. O'Riordan, G.I. Shishkin, in: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore, 1996). Numerical experiments are conducted.