Vellore Institute of Technology (VIT) is a private university located in&nbsp;Tamil Nadu, India. Founded in 1984, as Vellore Engineering College, the institution offers 20 undergraduate, 34 postgraduate, four integrated and four research programs. It has campuses in Vellore, Amravati, Bhopal and Chennai.

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VIT University

In this paper, we presented exponentially fitted finite difference methods for a class of time dependent singularly perturbed one-dimensional convection diffusion problems with small shifts. Similar boundary value problems arise in computational neuroscience in determination of the behavior of a neuron to random synaptic inputs. When the shift parameters are smaller than the perturbation parameter, the shifted terms are expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed. The proposed finite difference scheme is unconditionally stable and is convergent with order O(Δ t+ h 2 ) where Δ t and h respectively the time and space step-sizes. When the shift parameters are larger than the perturbation parameter a special type of mesh is used for the space variable so that the shifts lie on the nodal points and an exponentially fitted scheme is developed. This scheme is also unconditionally stable. By means of two examples, it is shown that the proposed methods provide uniformly convergent solutions with respect to the perturbation parameter. On the basis of the numerical results, it is concluded that the present methods offer significant advantage for the linear singularly perturbed partial differential difference equations. © 2017, Foundation for Scientific Research and Technological Innovation.

Department of Mathematics

School of Advanced Sciences

Vellore Campus

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Differential Equations and Dynamical Systems

Fitted Numerical Methods for Singularly Perturbed One-Dimensional Parabolic Partial Differential Equations with Small Shifts Arising in the Modelling of Neuronal Variability

In this paper, an adaptive mesh has been generated using the concept of entropy function for solving convection-diffusion singularly perturbed parabolic partial differential equations with a small delay. Similar problems are associated with a furnace used to process a metal sheet in control theory. The beauty of the method is, unlike the popular adaptive meshes (Bakhvalov and Shishkin), prior information of the width and position of the layers are not required. The method is independent of perturbation parameter ε and gives us an oscillation-free solution, without any user-introduced parameters. The applicability of the proposed method is illustrated by means of two examples. © Springer Nature Switzerland AG 2019.

Trends in Mathematics

An adaptive mesh selection strategy for solving singularly perturbed parabolic partial differential equations with a small delay

In this paper, an adaptive mesh strategy is presented for solving singularly perturbed delay differential equation of convection-diffusion type using second order central finite difference scheme. Layer adaptive meshes are generated via an entropy production operator. The details of the location and width of the layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

VIT Unitversity

Fulltext

Mathematical Modelling and Analysis

A NUMERICAL SCHEME FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS OF CONVECTION-DIFFUSION TYPE ON AN ADAPTIVE GRID

This paper deals with singularly perturbed boundary value problem for a linear second order delay differential equation. It is known that the classical numerical methods are not satisfactory when applied to solve singularly perturbed problems in delay differential equations. In this paper we present an exponentially fitted finite difference scheme to overcome the drawbacks of the corresponding classical counter parts. The stability of the scheme is investigated. The proposed scheme is analyzed for convergence. Several linear singularly perturbed delay differential equations have been solved and the numerical results are presented to support the theory. © 2015 Ain Shams University

Ain Shams Engineering Journal

An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays

This paper deals with the singularly perturbed boundary value problem for a second order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. A difference scheme on a uniform mesh is accomplished by the method based on cubic spline in tension. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results. © 2016, Springer India Pvt. Ltd.

International Journal of Applied and Computational Mathematics

Numerical Solution of Second Order Singularly Perturbed Delay Differential Equations via Cubic Spline in Tension

This paper deals with the singularly perturbed boundary value problem for the second-order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. A fitted numerical scheme has been developed to solve the boundary value problem. The difference scheme which is shown to converge to the continuous solution uniformly with respect to the perturbation parameter is illustrated with numerical results. © Shiraz University 2017.

Iranian Journal of Science and Technology, Transactions A: Science

An Exponentially Fitted Spline Method for Second-Order Singularly Perturbed Delay Differential Equations

This paper deals with the singularly perturbed boundary value problem for the second order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. An exponentially fitted difference scheme on a uniform mesh is accomplished by the method based on cubic spline in compression. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results. © 2015, Pramod Chakravarthy et al.

Advances in Difference Equations

A fitted numerical scheme for second order singularly perturbed delay differential equations via cubic spline in compression

This paper deals with the singularly perturbed boundary value problem for a linear second order differential-difference equation of convection-diffusion type. In the numerical treatment of such type of problems, first we use Taylor's approximation to tackle the term containing the small shift. A fitting parameter has been introduced in a tridiagonal finite difference method and is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the tridiagonal system. The method is analysed for convergence. Several numerical examples are solved to demonstrate the applicability of the method. Graphs are plotted for the solutions of these problems to illustrate the effect of small shift on the boundary layer solution. © 2014 Production and hosting by Elsevier B.V.

An exponentially fitted tridiagonal finite difference method for singularly perturbed differential-difference equations with small shift

In this paper, we present a finite difference method for singularly perturbed linear second order differential-difference equations of convection-diffusion type with a small shift, i.e., where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. Here, the study focuses on the effect of shift on the boundary layer behavior or oscillatory behavior of the solution via finite difference approach. An extensive amount of computational work has been carried out to demonstrate the proposed method and to show the effect of shift parameter on the boundary layer behavior and oscillatory behavior of the solution of the problem. © 2012 Elsevier Inc.

Applied Mathematical Modelling

A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior

In this paper, we present a domain decomposition method for solving singularly perturbed differential-difference equation with a small negative shift and the boundary layer at left end of the under lying interval. The method is distinguished by the following fact: The original singularly perturbed boundary value problem is divided into two problems, namely inner region problem and outer region problem. The boundary condition at the cutting point is obtained from the theory of singular perturbations. Using stretching transformation, a modified inner region problem is constructed and is solved by using the upwind finite difference scheme. The outer region problem is solved by a Taylor polynomial approach. We combine the solutions of both the problems to obtain an approximate solution to the original problem. The proposed method is iterative on the cutting point. The process is to be repeated for various choices of the cutting point, until the solution profiles stabilize. A lower bound for the cutting point of the boundary layer region is obtained in terms of the perturbation parameter and delay parameter. The method is analyzed for stability and convergence. Some numerical examples have been solved to demonstrate the applicability of the method. © Euro Journals Publishing, Inc. 2012.

European Journal of Scientific Research

A domain decomposition method for singularly perturbed differential- difference equations with a small negative shift

In this paper a modified fourth order Numerov method is presented for singularly perturbed differential-difference equation of mixed type, i.e., containing both terms having a negative shift and terms having positive shift. Similar boundary value problems are associated with expected first exit time problems of the membrane potential in the models for the neuron. To handle the negative and positive shift terms, we construct a special type of mesh, so that the terms containing shift lie on nodal points after discretization. The proposed finite difference method works nicely when the shift parameters are smaller or bigger to perturbation parameter. An extensive amount of computational work has been carried out to demonstrate the proposed method and to show the effect of shift parameters on the boundary layer behavior or oscillatory behavior of the solution of the problem. © 2012 Elsevier B.V.