Solving Delay Differential Equations By Elzaki Transform Method

Methods used for ordinary differential equations cannot generally be used to solve delay differential equation, which is reflected in the choice of suitable software. The paper shows possibilities of software solution of delay differential equations. The aim of the paper is to present the possibilities of current software packages and programme systems (e.g. Matlab, Maple, R). The paper further shows how the aforementioned equations can be used in solutions of dynamical models. The conclusion of the papers demonstrates a solution of a specific dynamical model -- the Phillips curve applied to the Czech Republic. Setting up the model required the use of analytic and synthetic methods, dynamical modelling and solving the system of two delay differential equations. In the conclusion the authors claim that although the quality of the available software, suitable for solving delay differential equations, is not as good as that of software used for solving ordinary differential equations, such software that meets basic requirements can be found. Therefore such software can be used as a supportive tool for exact modelling methods in practice.

A technique which is known as Sumudu Transform Method (STM) is studied for the construction of solutions of a most general form of delay differential equations of pantograph type. This is a pioneer study on using the STM to construct the solutions of delay differential equations of pantograph type with variable coefficients. We obtain the exact and approximate solutions of nonlinear problems with multiproportional delays and variable coefficients. The strength of STM is illustrated in reducing the complex computational work as compared to the well-known methods. This paper shows how to succinctly identify the Lagrange multipliers for nonlinear delay differential equations with variable coefficients, using the STM. The potency and suitability of the STM are exhibited by giving expository examples. The solutions of nonlinear Volterra integro-differential equations of pantograph type are also obtained.

Periodic solutions of delay differential equations (DDEs) splitting into stable and unstable branches are examined in an infinite-dimensional space for fixed and multiple time delays. The center manifold theorem and the classical Hopf bifurcation theorem for the study of periodic solutions of ordinary differential equations (ODEs) are employed to reduce the infinite-dimensional character of the DDEs to finite-dimensional ODEs. Using integral averaging method, the vector field of the ODEs is converted and averaged into amplitude a and phase φ relations. From these relations bifurcation equations of the form ℑ(a, φ) = 0 for the solution branches are derived.

In this paper, the relation between the oscillatory behavior of the solutions of second order differential equation and the oscillatory behaviour of the solutions of the corresponding delay differential equation is established. Also a necessary and sufficient condition for every bounded solution of the delay differential equations to have nonooscillatory solutions is given. A.M.S.C.: 34C10, 34C11, 34K15. Key words: Zeros, oscillatory and nonoscillatory solutions.

Preservation of nonoscillatory behavior of solutions of second-order delay differential equations under impulsive perturbations

Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments

The paper presents a study on the validity of perturbation methods, suchas the method of multiple scales, the Lindstedt–Poincare method and soon, in seeking for the periodic motions of the delayed dynamic systemsthrough an example of a Duffing oscillator with delayed velocityfeedback. An important observation in the paper is that the method ofmultiple scales, which has been widely used in nonlinear dynamics, worksonly for the approximate solutions of the first two orders, and givesrise to a paradox for the third-order approximate solutions of delaydifferential equations. The same problem appears when theLindstedt–Poincare method is implemented to find the third-orderapproximation of periodic solutions for delay differential equations,though it is effective in seeking for any order approximation ofperiodic solutions for nonlinear ordinary differential equations. Apossible explanation to the paradox is given by the results obtained byusing the method of harmonic balance. The paper also indicates thatthese perturbation methods, despite of some shortcomings, are stilleffective in analyzing the dynamics of a delayed dynamic system sincethe approximate solutions of the first two orders already enable one togain an insight into the primary dynamics of the system.

General Lagrange-hybrid functions and numerical solution of differential equations containing piecewise constant delays with bibliometric analysis

In 1999 Ivanov and Losson [A.F. Ivanov and J. Losson, Stable rapidly oscillating solutions in delay differential equations with negative feedback, Differ. Int. Eqns 12 (1999), pp. 811–832] presented a computer assisted proof that a particular delay differential equation (with negative feedback) admits a stable rapidly oscillating periodic solution (ROPS). In this article the delay equation of Ivanov and Losson is embedded in a five-parametric class of differential equations. Conditions on the parameters are given such that the delay equation admits a stable ROPS. Moreover, it is shown that for odd n > 1 the delay equation admits a stable ROPS with n humps per unit time if the parameters satisfy some explicitly given conditions. The delay equation of Ivanov and Losson satisfies all conditions on the five parameters. This gives an analytic proof and a considerable generalization of the result of Ivanov and Losson. The conditions on the parameters are believed to be sharp in a certain sense. The second result proves part of a conjecture in Stoffer [D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Differ. Eqns 20(1) (2008), pp. 201–238]. For a class of stiff delay differential equations with piecewise constant nonlinearity (positive or negative feedback) and for every n the following holds: if the stiffness parameter is sufficiently large then there are 2a(n) essentially different stable ROPSs with n humps per time unit. a(n) is the number of essentially different binary n-stage shift register sequences.

PurposeIn this paper, Picard–S hybrid iterative process is defined, which is a hybrid of Picard and S-iterative process. This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid and Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.Design/methodology/approachThis new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings.FindingsShowed the fastest convergence of this new iteration and then other iteration defined in this paper. The author finds the solution of delay differential equation using this hybrid iteration. For new iteration, the author also proved a theorem for nonexpansive mapping.Originality/valueThis new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.

Characterising small solutions in delay differential equations through numerical approximations

This paper considers the numerical solution of delay differential equations. The predictor–corrector scheme based on generalized multistep methods are implemented in variable order variable stepsize techniques. The formulae are represented in divided difference form where the integration coefficients are computed by a simple recurrence relation. This representation produces simpler calculation as compared with the modified divided difference form, but no sacrifice is made in efficiency and accuracy of the method. Numerical results prove that the method is reliable, efficient and accurate. The P- and Q-stability regions for a fixed stepsize of the predictor–corrector scheme are illustrated for various orders.

Quasi-periodic solutions for perturbed autonomous delay differential equations

Construction of quasi-periodic solutions of delay differential equations via KAM techniques

In this article an introduction to the wide field of retarded or delay differential equations with state-dependent delays is given. Hereby, the most important features of this type of differential equations which are profoundly different from ordinary differential equations are discussed. The presence of discontinuities in higher derivatives of the solution of delay differential equations require suitable integration methods. Thus, an efficient numerical code for the integration of delay differential equations is presented. This code is based on a fourth order RK one step algorithm. Furthermore, the presence of discontinuities may lead to severe problems when parameters are to be estimated using higher order optimization techniques. These problems are pointed out and a systematic way for the analysis of the smoothness of a least squares objective function is presented.