A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations
V. Shanthi,N. Ramanujam +1 more
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In this article, a numerical method is suggested to solve singularly perturbed two-point boundary value problems (BVPs) for fourth-order ODEs with a small positive parameter multiplying the highest derivative.Abstract:
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, Newton's method of quasilinearization is applied. The present method is demonstrated by providing examples. The method is easy to implement.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
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
Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations
TL;DR: It is shown that the proposed technique provides first-order accuracy independent of the perturbation parameter and the classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function.
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.
Journal ArticleDOI
Asymptotic numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer
V. Shanthi,N. Ramanujam +1 more
TL;DR: A computational method for solving Singularly perturbed two-point boundary value problems (SPBVPs) of convection–diffusion type for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is presented.
References
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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.
Journal ArticleDOI
Uniform numerical methods for problems with initial and boundary layers
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Numerical methods for singularly perturbed differential equations : convection-diffusion and flow problems
TL;DR: In this article, the analytical behavior of solutions for second-order boundary value problems and higher-order problems was analyzed. But the analytical behaviour of solutions was not analyzed for the first order boundary value problem.
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