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

• Published in: Applied mathematics and computation Vol. 129; no. 2; pp. 269 - 294
• Main Authors: Shanthi, V., Ramanujam, N.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 10.07.2002; Elsevier
• Subjects: Asymptotic expansion; Boundary layer; Exact sciences and technology; Exponentially fitted difference scheme; Finite difference scheme; Fourth-order ordinary differential equation; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; Self-adjoint boundary value problem; Singularly perturbed problems; Finite difference scheme; Singularly perturbed problems; Asymptotic expansion; Fourth-order ordinary differential equation; Exponentially fitted difference scheme; Self-adjoint boundary value problem; Boundary layer; Difference scheme; Boundary value problem; Singular perturbation; Fourth order equation; Partial differential equation; Self adjoint operator; Finite difference method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(01)00040-6

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 three non-overlapping sub-intervals, which we call them inner regions (boundary layers) 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 non-linear equations. To solve non-linear equations, Newton's method of quasi-linearization is applied. The present method is demonstrated by providing examples. The method is easy to implement and suitable for parallel computing.