Numerical Algorithm for Nonlinear Delayed Differential Systems of nnth Order

The purpose of this paper is to propose a semi-analytical technique convenient for numerical approximation of solutions of the initial value problem for $p$-dimensional delayed and neutral differential systems with constant, proportional and time varying delays. The algorithm is based on combination of the method of steps and the differential transformation. Convergence analysis of the presented method is given as well. Applicability of the presented approach is demonstrated in two examples: A system of pantograph type differential equations and a system of neutral functional differential equations with all three types of delays considered. Accuracy of the results is compared to results obtained by the Laplace decomposition algorithm, the residual power series method and Matlab package DDENSD. Comparison of computing time is done too, showing reliability and efficiency of the proposed technique.Comment: arXiv admin note: text overlap with arXiv:1501.00411 Author's reply: the text overlap may be caused by the fact that this article is concerning systems of equations, while the other paper was about single equation

Zhou Method for the Solutions of System of Proportional Delay Differential Equations

In this paper, we consider a viable semi-analytical approach for the approximate-analytical solutions of certain system of functional differential equations (SFDEs) engendered by proportional delays. The proposed semi-analytical technique is built on the basis of the classical Differential Transform Method (DTM). The effectiveness and robustness of the proposed technique is illustratively demonstrated and the results are compared with their exact forms. We note also that using this method, the SFDEs with proportional delays need not be converted to SFDEs with constant delays before obtaining their solutions, and no symbolic calculation or initial guesstimates are required

A new modified homotopy perturbation method for fractional partial differential equations with proportional delay

In this paper, we suggest and analyze a technique by combining the Shehu transform method and the homotopy perturbation method. This method is called the Shehu transform homotopy method (STHM). This method is used to solve the time-fractional partial differential equations (TFPDEs) with proportional delay. The fractional derivative is described in Caputo's sense. The solutions proposed in the series converge rapidly to the exact solution. Some examples are solved to show the STHM is easy to apply

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

Exp-function Method for Wick-type Stochastic Combined KdV-mKdV Equations

Exp-function method is proposed to present soliton and periodic wave solutions for variable coefficients combined KdV- mKdV equation. By means of Hermite transform and white noise analysis, we consider the variable coefficients and Wick-type stochastic combined KdV-mKdV equations. As a result, we can construct new and more general formal solutions. These solutions include exact stochastic soliton and periodic wave solutions.Keywords: combined KdV-mKdV equation, Exp-function method, Wick product, Hermite transform, White noise

Analytical Approximate Solutions for a General Class of Nonlinear Delay Differential Equations

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods

Geometric nonlinear vibration analysis for pretensioned rectangular orthotropic membrane

The geometric nonlinear vibrations of pretensioned orthotropic membrane with four edges fixed, which is commonly applied in building membrane structure, are studied. The nonlinear partial differential governing equations are derived by von Kármán’s large deflection theory and D’Alembert’s principle. Because of the strong nonlinearity of governing equations, the homotopy perturbation method (HPM) to solve them is applied. The approximate analytical solution of the vibration frequency and displacement function is obtained. In the computational example, the frequency, vibration mode and displacement as well as the time curve of each feature point are analyzed. It is proved that HPM is an effective, simple and high-precision method to solve the geometric nonlinear vibration problem of membrane structures. These results provide some valuable computational basis for the vibration control and dynamic design of building and other analogous membrane structures.Вивчено геометрично нелінійні коливання попередньо напруженої ортотропної мембрани з чотирма фіксованими краями, яка звичайно використовується в будівельних мембранних конструкціях. Нелінійні рівняння динаміки в частинних похідних отримано на базі теорії фон Кармана про великі прогини і принципу Д‘Алямбера. Застосовано метод гомотопічного збурення для розв’язування отриманих сильно нелінійних рівнянь. Отримано наближений аналітичний розвязок для частоти коливань і функції зміщень. У числовому прикладі проаналізовано частоти, форми коливань, зміщення і залежні від часу криві у кожній характерній точці. Доведено, що цей метод є ефективним, простим і високоточним для розвязування задач про геометрично нелінійні коливання мембранних конструкцій. Ці результати створюють певну корисну базу для обчислення задач про управління коливаннями і динамічне конструювання будівельних та інших аналогічних мембранних конструкцій

Optimal Homotopy Asymptotic Method for Solving Delay Differential Equations

We extend for the first time the applicability of the optimal homotopy asymptotic method (OHAM) to find the algorithm of approximate analytic solution of delay differential equations (DDEs). The analytical solutions for various examples of linear and nonlinear and system of initial value problems of DDEs are obtained successfully by this method. However, this approach does not depend on small or large parameters in comparison to other perturbation methods. This method provides us with a convenient way to control the convergence of approximation series. The results which are obtained revealed that the proposed method is explicit, effective, and easy to use

Stability Analysis of a Strongly Displacement Time-Delayed Duffing Oscillator Using Multiple Scales Homotopy Perturbation Method

In the present study, some perturbation methods are applied to Duffing equations having a displacement time-delayed variable to study the stability of such systems. Two approaches are considered to analyze Duffing oscillator having a strong delayed variable. The homotopy perturbation method is applied through the frequency analysis and nonlinear frequency is formulated as a function of all the problem’s parameters. Based on the multiple scales homotopy perturbation method, a uniform second-order periodic solution having a damping part is formulated. Comparing these two approaches reveals the accuracy of using the second approach and further allows studying the stability behavior. Numerical simulations are carried out to validate the analytical finding