A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations

Singularly perturbed two-point boundary value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive 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 system of two ODEs subject to suitable boundary conditions. Then, the domain of definition of the differential equation (a closed interval) is divided into two nonoverlapping subintervals, which we call “inner region” (boundary layer) and “outer region”. Then, the DE is solved in these intervals separately. The solutions obtained in these regions are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval) we use mostly zero-order asymptotic expansion of the solution of the BVP. First, linear equations are considered and then nonlinear equations. To solve nonlinear equations, Newtonʹs method of quasilinearization is applied. The present method is demonstrated by providing examples. The method is easy to implement.