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

• Published in: Applied mathematics and computation Vol. 132; no. 1; pp. 87 - 104
• Main Authors: Valarmathi, S., Ramanujam, N.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 2002; Elsevier
• Subjects: Asymptotic expansion approximation; Boundary layers; Exact sciences and technology; Fitted mesh method; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; Self-adjoint boundary value problem; Singular perturbation problems; Third order differential equation; Asia; Singapore; Fitted mesh method; Self-adjoint boundary value problem; Asymptotic expansion approximation; Boundary layers; Singular perturbation problems; Third order differential equation; Initial condition; Differential equation; Singular perturbation; Numerical method; Boundary condition; Reaction diffusion equation; Partial differential equation; First order equation; Third order equation; Boundary value problem; Smooth function; Two-dimensional calculations; Asymptotic approximation; Mesh method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(01)00179-5

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.