Mohan K. Kadalbajoo

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory. Starting from Prandtls work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984-2000. The work done in this area during the period 1905-1984 has already been surveyed by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223] for details. Due to the space constraints we have covered only singularly perturbed one-dimensional problems.

Numerical analysis of singularly perturbed delay differential equations with layer behavior

In this paper, we present a numerical method to solve boundary-value problems for singularly perturbed differential-difference equations with negative shift. In recent papers, the term negative shift has been used for delay [SIAM J. Appl. Math. 54 (1994) 249; SIAM J. Appl. Math. 42 (1982) 502; SIAM J. Appl. Math. 45 (1985) 687; SIAM J. Appl. Math. 45 (1985) 708; SIAM J. Appl. Math. 54 (1994) 273]. Similar BVPs are associated with expected first-exit time problem of the membrane potential in models for neuron and in variational problem in control theory. The stability and convergence analysis of the method is discussed. Also the effect of small shift on the boundary layer solution in both the cases, i.e., boundary layer on the left side as well as the right side is discussed by considering numerical experiments. Several numerical examples are solved using the presented method, compared the computed result with exact solution and plotted the graphs of the solution of the problems.

A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations

Abstract A boundary value problem for second order singularly perturbed delay differential equation is considered. When the delay argument is sufficiently small, to tackle the delay term, the researchers [M.K. Kadalbajoo, K.K. Sharma, Numerical analysis of singularly perturbed delay differential equations with layer behavior, Appl. Math. Comput. 157 (2004) 11–28, R.E. O’Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991] used Taylor’s series expansion and presented an asymptotic as well as numerical approach to solve such type boundary value problem. But the existing methods in the literature fail in the case when the delay argument is bigger one because in this case, the use of Taylor’s series expansion for the term containing delay may lead to a bad approximation. In this paper to short out this problem, we present a numerical scheme for solving such type of boundary value problems, which works nicely in both the cases, i.e., when delay argument is bigger one as well as smaller one. To handle the delay argument, we construct a special type of mesh so that the term containing delay lies on nodal points after discretization. The proposed method is analyzed for stability and convergence. To demonstrate the efficiency of the method and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution several test examples are considered.

Asymptotic and numerical analysis of singular perturbation problems: A survey

This paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems. The purpose is to find out what problems are treated and what numerical/asymptotic methods are employed, with an eye toward the goal of developing general methods to solve such problems. A summary of the results of some recent methods is presented, and this leads to conclusions and recommendations about what methods to use on singular perturbation problems. Finally, some areas of current research are indicated. A bibliography of about 130 items is provided.

A brief survey on numerical methods for solving singularly perturbed problems

In the present paper, a brief survey on computational techniques for the different classes of singularly perturbed problems is given. This survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000–2009. However some older important relevant papers are also included in this survey. We also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223–259, 130 (2002) 457–510, 134 (2003) 371–429] for details. Thus 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.

Numerical analysis of boundary-value problems for singularly-perturbed differential-difference equations with small shifts of mixed type

In this paper, we use a numerical method to solve boundary-value problems for a singularly-perturbed differential-difference equation of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift. Similar boundary-value problems are associated with expected first exit time problems of the membrane potential in models for the neuron. The stability and convergence analysis of the method is given. The effect of a small shift on the boundary-layer solution is shown via numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method.

Initial-value technique for a class of nonlinear singular perturbation problems

An initial-value technique, which is simple to use and easy to implement, is presented for a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. It is distinguished by the following fact: The original second-order problem is replaced by an asymptotically equivalent first-order problem and is solved as an initial-value problem. Numerical experience with several examples is described.

Numerical treatment of singularly perturbed two point boundary value problems

We propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This is a practical method and can be easily implemented on a computer. The original problem is divided into inner and outer region differential equation systems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem (TPBVP). In turn, the outer region problem is also solved as a TPBVP. Both these TPBVPs are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions. The stability of some recurrence relations involved in the algorithm is investigated. The proposed method is iterative on the terminal point. Some numerical examples are included, and the computational results are compared with exact solutions. It is observed that the accuracy predicted can always be achieved with very little computational effort.

Singularly perturbed problems in partial differential equations: a survey

This survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory. Starting from Prandtls work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980-2000. In this review we have covered singularly perturbed partial differential equations. About ODEs the survey has already been done by us [see M.K. Kadalbajoo, K.C. Patidar, Appl. Math. Comput. 130 (2002) 457-510].

Fitted mesh B-spline collocation method for singularly perturbed differential–difference equations with small delay

Abstract This paper deals with the singularly perturbed boundary value problem for a linear second order differential–difference equation of the convection–diffusion type with small delay parameter δ of o ( e ) whose solution has a boundary layer. The fitted mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layers. B-spline collocation method is used with fitted mesh. Parameter-uniform convergence analysis of the method is discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small delay δ on boundary layer has also been discussed. Several examples are considered to demonstrate the performance of the proposed scheme and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution.