A computational method for solving boundary value problems for third-order singularly perturbed ordinary differential equations

• Published in: Applied mathematics and computation Vol. 129; no. 2; pp. 345 - 373
• Main Authors: Valarmathi, S., Ramanujam, N.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 10.07.2002; Elsevier
• Subjects: Asymptotic approximation; Boundary layer; Exact sciences and technology; Exponentially fitted finite difference scheme; Mathematical analysis; Mathematics; Non-self-adjoint boundary value problem; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use; Singular perturbation; Third-order differential equation; Exponentially fitted finite difference scheme; Non-self-adjoint boundary value problem; Asymptotic approximation; Singular perturbation; Third-order differential equation; Boundary layer; Third order equation; Boundary value problem; Differential equation; Quasilinearization; Numerical method; Boundary condition; Newton method
• DOI: 10.1016/S0096-3003(01)00044-3
• ISSN: 0096-3003

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 system of two ODEs subject to suitable initial and boundary conditions. Then the domain of definition of the differential equation (a closed interval) is divided into two sub-intervals, which we call the inner region (boundary layer) and the outer region. Then the DE is solved in these intervals separately. The solutions obtained in these intervals are combined to give the solution in the whole interval. To obtain boundary conditions at the transition points (boundary values inside this interval) we use mostly the zeroth-order asymptotic expansion of the solution of the BVP or a suitable asymptotic expansion solution. First, the linear equations are considered and then the semi-linear equations. To solve semi-linear equations Newton's method of quasi-linearisation is applied. Examples are provided to illustrate the method. The method is easy to implement and suitable for parallel computing.