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.

VIT is one of the top ranked private universities in India according to NIRF, THE and QS Rankings.&nbsp;Govt. of India has recognized&nbsp;VIT, Vellore as an&nbsp;Institution of Eminence. This has allowed VIT to take independent quality initiatives and move up in world ranking.

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This article presents a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction–diffusion initial-boundary-value problem. For the discretization of the time derivative, we use the implicit Euler scheme on the uniform mesh and for the spatial discretization, we use the central difference scheme on the Shishkin mesh, which provides a second-order convergence rate. To enhance the order of convergence, we apply the Richardson extrapolation technique. We prove that the proposed method converges uniformly with respect to the perturbation parameter and also attains almost fourth-order convergence rate. Finally, to support the theoretical results, we present some numerical experiments by using the proposed method. © 2020, Korean Society for Informatics and Computational Applied Mathematics.

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Journal of Applied Mathematics and Computing

A fourth-order numerical scheme for singularly perturbed delay parabolic problem arising in population dynamics

In this article, we study the numerical solution of a singularly perturbed 2D delay parabolic convection–diffusion problem. First, we discretize the domain with a uniform mesh in the temporal direction and a special mesh in the spatial directions. The numerical scheme used to discretize the continuous problem, consists of the implicit-Euler scheme for the time derivative and the classical upwind scheme for the spatial derivatives. Stability analysis is carried out, and parameter-uniform error estimates are derived. The proposed scheme is of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are carried out to verify the theoretical results. © 2018, Korean Society for Computational and Applied Mathematics.

Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection–diffusion problems on Shishkin mesh

In this article, we propose a second-order uniformly convergent numerical method for a singularly perturbed 2D parabolic convection–diffusion initial–boundary-value problem. First, we use a fractional-step method to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction, which gives a set of two 1D problems. Then, we use the classical finite difference scheme to discretize those 1D problems on a special mesh, which results almost first-order convergence, i.e., O(N−1+βlnN+Δt). To enhance the order of convergence to O(N−2+βln2N+Δt2), we use the Richardson extrapolation technique. In support of the theoretical results, numerical experiments are performed by employing the proposed technique. © 2017 Elsevier Ltd

Computers and Mathematics with Applications

Higher-order convergence with fractional-step method for singularly perturbed 2D parabolic convection–diffusion problems on Shishkin mesh

In this article, we are interested to approximate the solution of a singularly perturbed 2D delay parabolic convection–diffusion initial-boundary-value problem. To discretize the continuous problem in the temporal direction, we use a fractional step method which results a set of two 1D problems. Next, we apply classical finite difference scheme on a special mesh to discretize those 1D problems in the spatial directions. Fractional step method for the time variable permits the computational cost reduction and the special mesh is used to capture the boundary layers. We derive the truncation errors for the scheme to obtain the error estimates, which shows that the scheme is uniformly convergent of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are presented to support the theoretical results. © 2018, Springer (India) Private Ltd., part of Springer Nature.

International Journal of Applied and Computational Mathematics

Fractional Step Method for Singularly Perturbed 2D Delay Parabolic Convection Diffusion Problems on Shishkin Mesh

In this paper, we consider a class of singularly perturbed 2D delay parabolic convection-diffusion initial-boundary-value problems. To solve this problem numerically, we use a modified Shishkin mesh (Bakhvalov-Shishkin mesh) for the discretisation of the domain in the spatial directions and uniform mesh in the temporal direction. The time derivative is discretised by the implicit- Euler scheme and the spatial derivatives are discretised by the upwind finite difference scheme.We derive some conditions on the mesh-generating functions which are useful for the convergence of the method, uniformly with respect to the perturbation parameter. We prove that the proposed scheme on the Bakhvalov- Shishkin mesh is first-order convergent in the discrete supremum norm, which is optimal and does not require any extra computational effort compared to the standard Shishkin mesh. Numerical experiments verify the theoretical results. © 2018 Inderscience Enterprises Ltd.

International Journal of Mathematical Modelling and Numerical Optimisation

Uniformly convergent numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Bakhvalov-Shishkin mesh

This article proposes a second-order uniformly convergent numerical method for singularly perturbed delay parabolic convection-diffusion equation having a regular boundary layer. To handle this layer phenomenon, the problem is solved on a priori special mesh by using the implicit-Euler scheme for the discretization of the time derivative and the upwind scheme for the spatial derivatives which results almost first-order convergence, that is, O(N-1 In N + Δt). It is shown that, the implementation of Richardson extrapolation technique enhanced the order of convergence to O(N-2 In-2 N + Δt-2). To support the theoretical results, numerical experiments are carried out by applying the proposed technique on two test examples. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

International Journal of Computer Mathematics

Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equations

This article studies the numerical solution of singularly perturbed delay parabolic convection-diffusion initial-boundary-value problems. Since the solution of these problems exhibit regular boundary layers in the spatial variable, we use the piecewise-uniform Shishkin mesh for the discretization of the domain in the spatial direction, and uniform mesh in the temporal direction. The time derivative is discretized by the implicit-Euler scheme and the spatial derivatives are discretized by the hybrid scheme. For the proposed scheme, the stability analysis is carried out, and parameter-uniform error estimates are derived. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme. © 2015 Elsevier Inc.

Applied Mathematics and Computation

Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh

Present study intends to propose identification methodologies for multistorey shear buildings using powerful technique of artificial neural network models which can handle interval data. Identification with crisp data is known and neural network method has already been used by various researchers for this case. Here, the input and output data are handled in interval form. This is because, in general, we may not get the corresponding input and output values exactly (in crisp form), but we may have only the uncertain information of the data. These uncertain data are assumed in terms of interval and the corresponding problem of system identification is investigated. © 2014 Taylor & Francis.