Search

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 perturbationmethods. 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. [ABSTRACT FROM AUTHOR]

The (G'/G)-expansion method is used for the first time to find traveling wave solutions for thin film equations, where it is found that the related balance numbers are not the usual positive integers. The closed-form solution obtained via this method is in good agreement with the previously obtained solutions of other researchers. It is also noted that, for appropriate parameters, new solitary waves solutions are found. [ABSTRACT FROM AUTHOR]

A new analytic method is applied to singular initial-value Lane-Emden-type problems, and the effectiveness and performance of the method is studied. The proposed method obtains a Taylor expansion of the solution, and when the solution is polynomial, our method reproduces the exact solution. It is observed that the method is easy to implement, valuable for handling singular phenomena, yields excellent results at a minimum computational cost, and requires less time. Computational results of several test problems are presented to demonstrate the viability and practical usefulness of the method. The results reveal that the method is very effective, straight forward, and simple. [ABSTRACT FROM AUTHOR]

We consider the higher order nonlinear rational difference equation xn+1 = (α+βxn + γxn-k)/(A+Bxn +Cxn-k), n = 0, 1, 2, . . ., where the parameters α, β, γ, A, B, C are positive real numbers and the initial conditions x-k, . . . , x-1, x0 are nonnegative real numbers, κ ∊ {1, 2, . . .}. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable. [ABSTRACT FROM AUTHOR]

Direct solution of a class of nth-order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge-Kutta method (RK78). [ABSTRACT FROM AUTHOR]

The time evolution of the multispecies Lotka-Volterra system is investigated by the homotopy analysis method (HAM). The continuous solution for the nonlinear system is given, which provides a convenient and straightforward approach to calculate the dynamics of the system. The HAM continuous solution generated by polynomial base functions is of comparable accuracy to the purely numerical fourth-order Runge-Kutta method. The convergence theorem for the three-dimensional case is also given. [ABSTRACT FROM AUTHOR]

We consider the higher order nonlinear rational difference equation xn+1 = (α+βxn + γxn-k)/(A+Bxn +Cxn-k), n = 0, 1, 2, . . ., where the parameters α, β, γ, A, B, C are positive real numbers and the initial conditions x-k, . . . , x-1, x0 are nonnegative real numbers, κ ∊ {1, 2, . . .}. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable. [ABSTRACT FROM AUTHOR]