An asymptotic numerical method for singularly perturbed third-order ordinary differential equations of convection-diffusion type
S. Valarmathi,N. Ramanujam +1 more
TLDR
A numerical method to solve 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.Abstract:
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 weakly coupled system of two ODEs subject to suitable initial and boundary conditions. Then, the computational method, presented in this paper, is applied to this system. In this method, we reduce the weakly coupled system into a decoupled system. Then, to solve this decoupled system numerically, we apply a ‘boundary value technique (BVT)’, in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions. Then, we solve the decoupled system over these regions as two point boundary value problems. An exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme, in the outer region. The boundary conditions at the transition point are obtained using the zero-order asymptotic expansion approximation of the solution of the problem. This computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing. This method can be extended to a system of two ordinary differential equations, of which, one is of first order and the other is of second order. Numerical examples are given to illustrate the method.read more
Citations
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Journal ArticleDOI
A brief survey on numerical methods for solving singularly perturbed problems
Mohan K. Kadalbajoo,Vikas Gupta +1 more
TL;DR: This survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems.
Journal ArticleDOI
Singular perturbations for third-order nonlinear multi-point boundary value problem
Zengji Du,Weigao Ge,Mingru Zhou +2 more
TL;DR: In this article, the existence, uniqueness and asymptotic estimates of solutions of the boundary value problem are given by using priori estimates, differential inequalities technique and Leray-Schauder degree theory.
Journal ArticleDOI
A recent survey on computational techniques for solving singularly perturbed boundary value problems
TL;DR: A survey of singular perturbation methods for boundary value problems can be found in this paper, where a summary of the results of some recent methods is presented and this leads to conclusions and recommendations regarding methods to use.
Journal ArticleDOI
Approximate analytical solutions of singularly perturbed fourth order boundary value problems using differential transform method
TL;DR: In this paper, a reliable algorithm is presented to develop approximate analytical solutions of fourth order singularly perturbed two-point boundary value problems in which the highest order derivative is multiplied by a small parameter.
References
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Book
Maximum principles in differential equations
TL;DR: The One-Dimensional Maximum Principle (MDP) as mentioned in this paper is a generalization of the one-dimensional maximum principle (OMP) for the construction of hyperbolic equations.
Book
Introduction to Perturbation Methods
TL;DR: The WKB and Related Methods are described and the method of Homogenization is explained, followed by a discussion of the properties of Transition Layer Equations and asymptotic approximations.
Book
Singular perturbation methods for ordinary differential equations
TL;DR: In this paper, a monograph on applications of mathematics is intended for students of mathematics, engineering and the sciences, which is intended to provide the basis for the reader to go on to solve new problems.
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