Kapil K. Sharma

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.

ε-Uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay

Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-uniformly convergent.

A parameter-uniform implicit difference scheme for solving time-dependent Burgers’ equations

Abstract A numerical study is made for solving one dimensional time dependent Burgers’ equation with small coefficient of viscosity. Burgers’ equation is one of the fundamental model equations in the fluid dynamics to describe the shock waves and traffic flows. For high coefficient of viscosity a number of solution methodology exist in the literature [6] , [7] , [8] , [9] and [14] but for the sufficiently low coefficient of viscosity, the exist solution methodology fail and a discrepancy occurs in the literature. In this paper, we present a numerical method based on finite difference which works nicely for both the cases, i.e., low as well as high viscosity coefficient. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on uniform mesh and a standard upwind finite difference scheme to discretize in spacial direction on piecewise uniform mesh. The quasilinearzation process is used to tackle the non-linearity. An extensive amount of analysis has been carried out to obtain the parameter uniform error estimates which show that the resulting method is uniformly convergent with respect to the parameter. To illustrate the method, numerical examples are solved using the presented method and compare with exact solution for high value of coefficient of viscosity.

Numerical analysis of singularly perturbed delay differential turning point problem

Abstract In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper.

ϵ-Uniform fitted mesh method for singularly perturbed differential-difference equations: Mixed type of shifts with layer behavior

An ϵ-uniform fitted mesh method is presented to solve boundary-value problems for singularly perturbed differential-difference equations containing negative as well as positive shifts with layer behavior. Such types of BVPs arise at various places in the literature such as the variational problem in control theory and in the determination of the expected time for the generation of action potentials in nerve cells. The method consists of the standard upwind finite difference operator on a special type of mesh. Here, we consider a piecewise uniform fitted mesh, which turns out to be sufficient for the construction of ϵ-uniform method. One may use some more complicated meshes, but the simplicity of the piecewise uniform mesh is supposed to be one of their major attractions. The error estimate is established which shows that the method is ϵ-uniform. Several numerical examples are solved to show the effect of small shifts on the boundary layer solution. Numerical results in terms of maximum errors are tabulated and graphs of the solution are drawn to demonstrate the efficiency of the method. †E-mail: kapilks@iitk.ac.in

Optimal equi-scaled families of Jarratt's method

In this paper, we present many new fourth-order optimal families of Jarratts method and Ostrowskis method for computing simple roots of nonlinear equations numerically. The proposed families of Jarratts method having the same scaling factor of functions as that of Jarratts method (i.e. quadratic scaling factor of functions in the numerator and denominator of the correction factor) are the main finding of this paper. It is observed that the body structures of our proposed families of Jarratts method are simpler than those of the original families of Jarratts method. The efficiency of these methods is tested on a number of relevant numerical problems. Furthermore, numerical examples suggest that each member of the proposed families can be competitive to other similar robust methods available in the literature.

Simply constructed family of a Ostrowski's method with optimal order of convergence

In this paper, we propose a simple modification over Chuns method for constructing iterative methods with at least cubic convergence [5]. Using iteration formulas of order two, we now obtain several new interesting families of cubically or quartically convergent iterative methods. The fourth-order family of Ostrowskis method is the main finding of the present work. Per iteration, this family of Ostrowskis method requires two evaluations of the function and one evaluation of its first-order derivative. Therefore, the efficiency index of this Ostrowskis family is E=43~1.587, which is better than those of most third-order iterative methods E=33~1.442 and Newtons method E=2~1.414. The performance of Ostrowskis family is compared with its closest competitors, namely Ostrowskis method, Jarratts method and Kings family in a series of numerical experiments.

Hyperbolic partial differential-difference equation in the mathematical modeling of neuronal firing and its numerical solution

Abstract First we assume a simple neuronal model based on Stein’s Model [Richard B. Stein, A theoretical analysis of neuronal variability, Biophys. J. 5 (1965) 173–194] in which, after a refractory period, excitatory and inhibitory exponentially decaying inputs of constant size occur at random intervals and sum until a threshold is reached. We briefly discuss the distribution of time intervals between successive neuronal firings, the firing rate as a function of input frequency, the strength–duration curve and the role of inhibition. Then a first-order partial differential-difference equation for the distribution of neuronal firing intervals is derived and a numerical scheme based on finite difference is constructed for solving such type of initial and boundary value problems. The proposed method is analyzed for stability and convergence. Finally, some test examples are given to validate convergence and the computational efficiency of the present scheme.

A solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted operator for solving singularly perturbed differential equations

Abstract The solution of the boundary value problems for singularly perturbed differential equations, i.e., where the highest order derivative is multiplied by a small parameter, exhibits layer behavior. The classical numerical schemes to solve such types of the boundary value problems do not give satisfactory result when the perturbation parameter is sufficiently small. To resolve this difficulty, there are mainly two approaches, namely, fitted operator and fitted mesh. Both the scheme are uniformly convergent, i.e., their convergence is independent of the small perturbation parameter. It is justified to adopt the two approach rather than the classical numerical schemes to solve the boundary value problems for singularly perturbed differential equations. Now if we compare the two approaches, one thing is common that both the approaches give the parameter-uniform schemes which is the primary requirement in construction of the numerical scheme to solve such type of problem. Secondly, one desire a higher order numerical scheme to approximate the solution of a problem. As far as order of convergence is concerned, the numerical scheme based on fitted operator approach is better than the numerical scheme constructed using fitted mesh approach. The researchers who adopted the fitted mesh rather than fitted operator approach to solve a singularly perturbed problem faced the question due to the loss of order of convergence, most of them justified it by quoting the simplicity of the method and there are some non-linear problems for which a parameter uniform scheme cannot be constructed based on fitted operator approach while for the same problem, a parameter uniform scheme is constructed based on fitted mesh method. Now question remains unanswered in the case of linear problem. In this article, we replied to this question by giving an example of linear problem for which one cannot construct a parameter uniform scheme based on fitted operator approach while for the same problem a parameter uniform numerical scheme based fitted mesh approach has been constructed [M.K. Kadalbajoo, K.K. Sharma, e uniform fitted mesh method for singularly perturbed differential difference equations with mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004) 49–62]. A theoretical reason behind the inability in construction of a parameter uniform scheme using fitted operator approach is revealed. In support of the predicted theory, a number of numerical experiments are carried out.

Parameter uniform numerical method for singularly perturbed differential–difference equations with interior layers

The present study is devoted to the numerical study of boundary value problems for singularly perturbed linear second-order differential–difference equations with a turning point. The points of the domain where the coefficient of the convection term in the singularly perturbed differential equation vanishes are known as the turning points. The solution of such type of differential equations exhibits boundary layer(s) or interior layer(s) behaviour depending upon the nature of the coefficient of convection term and the reaction term. In particular, this paper focuses on problems whose solution exhibits interior layers. In the development of numerical schemes for singularly perturbed differential–difference equations with a turning point, we use El-Mistikawy–Werle exponential finite difference scheme with some modifications. Some priori estimates have been established and parameter uniform convergence analysis of the proposed scheme is also discussed. Several examples are considered to demonstrate the performance of the proposed scheme and effect of the size of the delay/advance arguments and coefficients of the delay/advance term on the layer behaviour of the solution.