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In the past ** Mohan K. Kadalbajoo** has collaborated on articles with

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AbstractThe object of this paper is to present a new approach based on the method of inner boundary condition for solving singular perturbation problems. The original problem is partitioned into inner and outer region differential equation systems. Asymptotic expansion is used to obtain the terminal boundary condition. Using an appropriate transformation, a new inner region problem is obtained and solved as a two point boundary value problem. The derivative boundary condition at the terminal point is then derived from the solution of the inner region problem. Using this condition, the outer region problem is efficiently solved by employing the classical finite difference scheme. The proposed method is iterative on the terminal point. Some numerical examples have been solved to demonstrate the efficiency of the method.

AbstractWe have developed two methods for the nonseparable elliptic equations of the type ▿[middot](a▿u)=f, based on the extension of the symmetric marching technique developed in [2]. In method I, symmetric marching has been applied to carry out the iterations in the iterative scheme developed by Concus and Golub [3]. Method II is a noniterative method based on the direct extension of symmetric marching to the above problem. Numerical results are given to illustrate the behavior of the two methods.

AbstractQuadrature methods for solving singular integral equations, especially with logarithmic kernels, have been known to break down completely; even if they succeed, the results are often inaccurate and the accuracy difficult to estimate. Product integration methods (to eliminate the singularity in the kernel function) can tackle the problem effectively if the required solution is a sufficiently smooth function, but at a prohibitive cost of computational time. The author proposes a different type of method for solving singular integral equations by reducing them into equivalent Fredholm integral equations (whenever possible) and then imbedding the latter in a class of equivalent initial value problems, which tend to be stable and can be solved in a routine manner. The validity of the method is also discussed.

AbstractWe 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.

AbstractThis 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.

AbstractThird-order variable-mesh methods based on cubic spline approximation for nonlinear singularly perturbed boundary-value problems of the form εy″ = f(x,y), y(a) = α, y(b) = β are presented. The convergence analysis is given and the method is shown to have third-order convergence. Several tet examples are solved to demonstrate the efficiency of the method.

AbstractThis 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 Prandtl's 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.

AbstractWe consider some problems arising from singularly perturbed general differential difference equations. First we construct (in a new way) and analyze a “fitted operator finite difference method (FOFDM)” which is first order ε-uniformly convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the “fitted mesh finite difference method (FMFDM)”. This FMFDM is a direct method and ε-uniformly convergent with the nodal error as O(n-2ln2n) which is an improvement over the existing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson’s extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as O(n-1lnn) where n denotes the total number of sub-intervals of [0, 1]. Comparative numerical results are presented in support of the theory.

AbstractA numerical study is made for a class of boundary value problems of second-order differential equations in which the highest order derivative is multiplied by a small parameter ϵ and both the differentiated(convection) and undifferentiated(reaction) terms are with negative shift δ. We analyze three difference operators Lkh, k = 1, 2, 3 a simple upwind scheme, midpoint upwind scheme and a hybrid scheme, respectively, on a Shishkin mesh to approximate the solution of the problem. Theoretical error bounds are established. The accuracy by one of the grid adaptation strategies, namely grid redistribution is examined by solving the considered problem using hybrid method for some values of ϵ and N. As a result the accuracy gets improved by this grid adaptation strategy with almost the same computational cost. A few numerical results exhibiting the performance of these three schemes are presented.

AbstractIn this paper, a method based on initial value technique is proposed for solving non-linear two-point singularly perturbed boundary value problems for second order ordinary differential equations (ODEs) with a boundary layer at one (either left or right) end. The original singularly perturbed boundary value problem is reduced to an initial value problem approximated by its outer solution (asymptotic approximation). The new initial value problem is solved by proposed non-linear single step explicit scheme followed the idea given in [F.D. Van Niekerk, Non-linear one-step methods for initial value problems, Comput. Math. Appl. 13 (1987) 367–371]. The proposed scheme has been shown to be of order two. To demonstrate the applicability of the proposed scheme several (linear and non-linear) problems have been solved. It is observed that the present scheme approximate the exact solution very well.

AbstractThe hyperbolic partial differential equation, which contains integral condition in place of classical boundary condition arises in many application of modern physics and technologies. In this article, we propose a numerical method to solve the hyperbolic equation with nonlocal boundary condition using radial basis function based finite difference method. Several numerical experiments are presented and compared with some existing method to demonstrate the efficiency of the proposed method.

AbstractThis paper is devoted to the numerical study of the boundary value problems for nonlinear singularly perturbed differential-difference equations with small delay. Quasilinearization process is used to linearize the nonlinear differential equation. After applying the quasilinearization process to the nonlinear problem, a sequence of linearized problems is obtained. To obtain parameter-uniform convergence, a piecewise-uniform mesh is used, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method has shown to have almost second-order parameter-uniform convergence. The effect of small shift on the boundary layer(s) has also been discussed. To demonstrate the performance of the proposed scheme two examples have been carried out. The maximum absolute errors and uniform rates of convergence have been presented in the form of the tables.

AbstractThis paper deals with a numerical method for solving one-dimensional unsteady Burgers–Huxley equation with the viscosity coefficient ε. The parameter ε takes any values from the half open interval (0, 1]. At small values of the parameter ε, an outflow boundary layer is produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a non-linear singularly perturbed problem with a singular perturbation parameter ε. Using singular perturbation analysis, asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. We construct a numerical scheme that comprises of implicit-Euler method to discretize in temporal direction on uniform mesh and a monotone hybrid finite difference operator to discretize the spatial variable with piecewise uniform Shishkin mesh. To obtain better accuracy, we use central finite difference scheme in the boundary layer region. Shishkin meshes are refined in the boundary layer region, therefore stability constraint is satisfied by proposed scheme. Quasilinearization process is used to tackle the non-linearity and it is shown that quasilinearization process converges quadratically. The method has been shown to be first order uniformly accurate in the temporal variable, and in the spatial direction it is first order parameter uniform convergent in the outside region of boundary layer, and almost second order parameter uniform convergent in the boundary layer region. Accuracy and uniform convergence of the proposed method is demonstrated by numerical examples and comparison of numerical results made with the other existing methods.

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