(R1511) Numerical Solution of Differential Difference Equations Having Boundary Layers at Both the Ends

In this paper, numerical solution of differential-difference equation having boundary layers at both ends is discussed. Using Taylor’s series, the given second order differential-difference equation is replaced by an asymptotically equivalent first order differential equation and solved by suitable choice of integrating factor and finite differences. The numerical results for several test examples are presented to demonstrate the applicability of the method

An Initial Value Technique using Exponentially Fitted Non Standard Finite Difference Method for Singularly Perturbed Differential-Difference Equations

In this paper, an exponentially fitted non standard finite difference method is proposed to solve singularly perturbed differential-difference equations with boundary layer on left and right sides of the interval. In this method, the original second order differential difference equation is replaced by an asymptotically equivalent singularly perturbed problem and in turn the problem is replaced by an asymptotically equivalent first order problem. This initial value problem is solve by using exponential fitting with non standard finite differences. To validate the applicability of the method, several model examples have been solved by taking different values for the delay parameter δ , advanced parameter η and the perturbation parameter ε . Comparison of the results is shown to justify the method. The effect of the small shifts on the boundary layer solutions has been investigated and presented in figures. The convergence of the scheme has also been investigated

Quartic B-Spline Method for Non-Linear Second Order Singularly Perturbed Delay Differential Equations

This paper introduces a novel computational approach utilizing the quartic B-spline method on a uniform mesh for the numerical solution of non-linear singularly perturbed delay differential equations (NSP-DDE) of second-order with a small negative shift. These types of equations are encountered in various scientific and engineering disciplines, including biology, physics, and control theory. We are using quartic B-spline methods to solve NSP-DDE without linearizing the equation. Thus, the set of equations generated by the quartic B-spline technique is non-linear and the obtained equations are solved by Newton-Raphson method. The success of the approach is assessed by applying it to a numerical example for different values of perturbation and delay parameter parameters, the maximum absolute error (MAE) is obtained via the double mesh principle. The convergence rate of the proposed method is four. Obtained numerical results are compared with existing numerical techniques in literature and observe that the proposed method is superior with other numerical techniques. The quartic B-spline method provides the numerical solution at any point of the given interval. It is easy to implement on a computer and more efficient for handling second-order NSP-DDE