In this paper, we suggest and analyze some iterative methods for solving nonlinear equations using the homotopy perturbation technique coupled with system of equations. In addition, we show that the homotopy polynomial is exactly the the Adomian polynomial and this establishes the equivalence between the homotopy perturbation and decomposition techniques. Our method of establishing the equivale...

The Adomian’s decomposition method (ADM) is a solution method with a wide range of applications including the solution of algebraic, differential, integral and integro-differential equations or system of equations. This method was first introduced by Adomian [1, 2] in the beginning of the 1980’s. In this method the solution is considered as an rapidly converging, infinite series. The convergenc...

In this paper, we have discussed the integrability of a nonlinear partial differential equation, with a focus on the Painlevé property, the compatibility condition and the Bäcklund transformation. Afterwards, the Adomian decomposition method, which accurately computes the series solution, has been used to obtain an approximate solution. The convergence analysis based on the wave number and nonl...

Adomian s decomposition method (ADM) is proposed to approximate the numerical and analytical solutions of system two-dimensional Burgers equations (STDBE) with initial conditions. The advantages of this work are the decomposition method reduces the computational work and improvement with regard to its accuracy and rapid convergence. Some examples are given to illustrate the performance of the m...

In this paper, we use the optimal homotopy analysis method (OHAM) for approximate solutions of the fractional order Logistic equation. The numerical results obtained are compared with the results obtained by using variational iteration method (VIM) and Adomian decomposition method (ADM). The fractional derivatives are described by Caputo's sense. Exact and/or approximate analytical solutions of...

In this work, the homotopy perturbation method (HPM), variational iteration method (VIM) and Adomian decomposition method (ADM) are applied to solve the Rosenau–Hyman equation. We used the initial condition ( u(x, 0) = −83 cos 2 ( x 4 )) for aforementioned methods. Numerical solutions obtained by these methods are compared with the exact solutions, revealing that the obtained solutions are of h...

where 3 2 ( , , ) , ( , ) x y z a b ∈ ∈ R R . It is a quadratic and Lorenz-like model.The integrability [9], the limit cycles [10], the chaos synchronization [11] and the feedback controller [12] et al., have been discussed very recently. Many researchers have made much effort to analytical and numerical methods for solving the chaotic systems. For example, the variational iteration method (VIM...

Adomian decomposition method (ADM) is applied to approximately solve stochastic fractional integrodifferential equations involving nonlocal initial condition. The convergence of the ADM for the considered problem is proved. The mean square error between approximate solution and accurate solution is also given. [Mahmoud M. El-Borai, M.A.Abdou, Mohamed Ibrahim M. Youssef. On Adomian’s Decompositi...

In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integrodifferential equation by using the following powerful, efficient but simple methods: (i) Modified Adomian decomposition method (MADM), (ii) Variational iteration method (VIM), (iii) Homotopy analysis method (HAM) and (iv) Modified homotopy perturbation method (MHPM). The uniqueness of the solution and the conve...

By considering the Adomian decomposition scheme, we solve a generalized Boussinesq equation. The method does not need linearization or weak nonlinearly assumptions. By using this scheme, the solutions are calculated in the form of a convergent power series with easily computable components. The decomposition series analytic solution of the problem is quickly obtained by observing the existence ...