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This paper aims to establish the finite difference scheme on Bakhvalov-type mesh for the singularly perturbed pseudo-parabolic problems with time-delay. Using the energy estimates, the bounds of the solution are investigated. An error analysis is derived in the discrete energy norm and the almost second order convergence is obtained. The reliability of the suggested method is displayed by means of a numerical example. [ABSTRACT FROM AUTHOR]

In the present work, we consider a class of singularly perturbed turning point problems with delay for their numerical solution. Due to the presence of a delay, there occurs an interior layer in the solution along with twin boundary layers (SIAM J. Appl. Math. 42(3): 502–530). Some a priori estimates are given on the exact solution and its derivatives. A numerical technique composed of hybrid finite difference scheme on a layer-adapted Shishkin mesh is proposed. We establish the consistency, stability and convergence of the proposed technique. To validate the theoretical predictions, we present some numerical illustrations. The proposed scheme is proved to have an almost second-order convergence rate, uniform with respect to the perturbation parameter ε. [ABSTRACT FROM AUTHOR]

This paper considers the Runge-Kutta methods used on one-dimensional delay differential equations (DDEs), which undergo Hopf bifurcation from the origin. It not only shows the numerical approximations undergo Neimark-Sacker bifurcation, but also these Neimark-Sacker bifurcation points close the Hopf bifurcation point of the DDEs with the order of methods. Meanwhile, compared with another interpolation method, we conclude that the given interpolation method is better in the convergence of Neimark-Sacker bifurcation points. Finally, by using 2-stage Gauss method to delay Logistic equation, the theoretical analysis is supported. [ABSTRACT FROM AUTHOR]

In this paper a class of delay differential equations with a perturbation parameter ε is examined. A hybrid finite difference scheme on an appropriate piecewise uniform mesh of Shishkin-type is derived. We show that the scheme is almost second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical study of singularly perturbed boundary value problems for delay differential equations with a turning point. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighbourhood of the boundary layers. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Some numerical experiments are carried out to illustrate, in practice, the result of convergence proved theoretically and demonstrate the effect of the delay argument and the coefficient of the delay term on the layer behaviour of the solution. [ABSTRACT FROM AUTHOR]

A numerical method based on the Adomian decomposition method which has been developed by Adomian [Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method . Kluwer Academic Publishers, Boston, MA] is introduced in this paper for the approximate solution of delay differential equations. The algorithm is illustrated by studying an initial value problem. The results obtained are presented and show that only a few terms are required to obtain an approximate solution which is found to be accurate and efficient. [ABSTRACT FROM AUTHOR]

In this article, we aim to introduce a high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer. The domain is discretized by a uniform mesh in the time direction and a piecewise-uniform Shishkin mesh for the spatial direction. We use the Crank–Nicolson method for the time derivative and we develop a fourth-order compact difference method to solve the set of ordinary differential equations at each time level. The stability analysis and the truncation error are discussed. Parameter-uniform error estimates are derived and it is shown that the method is-uniformly convergent of second-order accurate in time, and in the spatial direction it is of second-order outside region of boundary layer, and of almost fourth-order inside the layer region. Numerical examples are presented to verify the theoretical results and to confirm the efficiency and high accuracy of the proposed method. [ABSTRACT FROM PUBLISHER]

Differential transform method (DTM) is extended for delay differential equations. By using DTM, we manage to obtain the numerical, analytical, and exact solutions of both linear and nonlinear equations. In comparison with the existing techniques, the DTM is a reliable method that needs less work and does not require strong assumptions and linearization. [ABSTRACT FROM AUTHOR]