Author

# R. Nageshwar Rao

Other affiliations: Visvesvaraya National Institute of Technology

Bio: R. Nageshwar Rao is an academic researcher from VIT University. The author has contributed to research in topics: Boundary value problem & Finite difference method. The author has an hindex of 7, co-authored 15 publications receiving 94 citations. Previous affiliations of R. Nageshwar Rao include Visvesvaraya National Institute of Technology.

##### Papers

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TL;DR: In this article, a finite difference method for singularly perturbed linear second order differential-difference equations of convection-diffusion type with a small shift is presented, where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter.

23 citations

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TL;DR: In this article, the authors presented exponentially fitted finite difference methods for a class of time dependent singularly perturbed one-dimensional convection diffusion problems with small shifts, where the shifted terms are expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed.

Abstract: 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 $$O(\Delta t+h^{2})$$
where $$\Delta t$$
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.

19 citations

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TL;DR: In this article, 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.

Abstract: 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.

15 citations

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TL;DR: In this article, the singularly perturbed boundary value problem for a linear second order delay differential equation is solved by an exponentially fitted finite difference scheme. But the stability of the scheme is investigated.

15 citations

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TL;DR: In this article, a finite difference method is presented for singularly perturbed differential-difference equations with small shifts of mixed type (i.e., terms containing both negative shift and positive shift).

12 citations

##### Cited by

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2,084 citations

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1,242 citations

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TL;DR: In this article, the numerical treatment of a class of singularly perturbed delay boundary value problems with a left layer based on the reproducing kernel theory has been studied and the error estimate of the present method is established.

73 citations

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TL;DR: In this paper, a method to solve singularly perturbed differential-difference equations of mixed type, i.e., containing both terms having a negative shift and terms having positive shift in terms of Fibonacci polynomials, is introduced.

Abstract: In this paper, we introduce a method to solve singularly perturbed differential-difference equations of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift in terms of Fibonacci polynomials. Similar boundary value problems are associated with expected first exit time problems of the membrane potential in the models for the neuron. First, we present some preliminaries about polynomial interpolation and properties of Fibonacci polynomials then a new approach implementing a collocation method in combination with matrices of Fibonacci polynomials is introduced to approximate the solution of these equations with variable coefficients under the boundary conditions. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving these equations.

30 citations

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Tanta University

^{1}TL;DR: In this paper, a singularly perturbed semi-linear boundary value problem with two-parameters is considered and the problem is solved using exponential spline on a Shishkin mesh.

30 citations