R. Nageshwar Rao

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.

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

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.

A modified Numerov method for solving singularly perturbed differential-difference equations arising in science and engineering

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.

Solving singularly perturbed differential-difference equations arising in science and engineering with Fibonacci polynomials

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.