An approximation of the analytical solution of the Jeffery-Hamel flow by decomposition method

Many researchers have been interested in application of mathematical methods to find analytical solutions of nonlinear equations and for this purpose, new methods have been developed. Since most of fluid mechanics problems due to boundary layer are strongly nonlinear, so analytical solution of them is confronted with some difficulty. In this Letter, the Jeffery-Hamel flow—a nonlinear equation of 3rd order—is studied by Adomian decomposition method. After introducing Adomian decomposition method and the way of obtaining Adomian's polynomial, we solved the problem for divergent and convergent channels. Finally, velocity distribution and shear stress constant is depicted at various Reynolds numbers and comparing our results with some earlier works illustrated their excellent accuracy.     

Closed form solution to a second order boundary value problem and its application in fluid mechanics

The Adomian decomposition method is used by many researchers to investigate several scientific models. In this Letter, the modified Adomian decomposition method is applied to construct a closed form solution for a second order boundary value problem with singularity.     

Hall Effect on Falkner-Skan Boundary Layer Flow of FENE-P Fluid over a Stretching Sheet

The Falkner-Skan boundary layer steady flow over a flat stretching sheet is investigated in this paper. The mathematical model consists of continuity and the momentum equations, while a new model is proposed for MHD Finitely Extensible Nonlinear Elastic Peterlin (FENE-P) fluid. The effects of Hall current with the variation of intensity of non-zero pressure gradient are taken into account. The governing partial differential equations are first transformed to ordinary differential equations using appropriate similarity transformation and then solved by Adomian decomposition method (ADM). The obtained results are validated by generalized collocation method (GCM) and found to be in good agreement. Effects of pertinent parameters are discussed through graphs and tables. Comparison with the existing studies is made as a limiting case of the considered problem at the end.     

A numerical solution of the vibration equation using modified decomposition method

In the present paper the well-known vibration equation for very large membrane with the help of powerful modification of Adomian decomposition method proposed by Wazwaz [A reliable modification of Adomian decomposition method, Applied Mathematics and Computation 102 (1999) 77-86] has been solved. By using initial value, the explicit solutions of the equation for different cases have been derived, which accelerate the rapid convergence of the series solution. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented graphically.     

A new numerical solution for the MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method

In this Letter, we considered a numerical treatment for the solution of the hydromagnetic peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube using Adomian decomposition method and a modified form of this method. The axial velocity is obtained in a closed form. Comparison is made between the results obtained by only three terms of Adomian series with those obtained previously by perturbation technique. It is observed that only few terms of the series expansion are required to obtain the numerical solution with good accuracy.     

MHD Boundary Layer Flow of a Non-Newtonian Fluid on a Moving Surface with a Power-Law Velocity

A theoretical analysis for MHD boundary layer flow on a moving surface with the power-law velocity is presented. An accurate expression of the skin friction coefficient is derived. The analytical approximate solution is obtained by means of Adomian decomposition methods. The reliability and efficiency of the approximate solutions are verified by numerical ones in the literature.     

An approximate analytic solution of the heat conduction equation at nanoscale

The Adomian decomposition method (ADM) and the Adomian double decomposition method (ADDM) for solving the 3D non-Fourier heat conduction equation at nanoscale based on the dual-phase-lag framework are proposed. We show that the noise terms that appear in ADM solution can be removed, if the ADDM is employed.     

Free vibration analysis of elastically connected multiple-beams by using the Adomian modified decomposition method

The Adomian modified decomposition method (AMDM) is employed in this paper to investigate the free vibrations of N elastically connected parallel Euler–Bernoulli beams, which are continuously joined by a Winkler-type elastic layer. The proposed AMDM method can be used to analyze the vibration of beam system consisting of an arbitrary number of beams. By using boundary conditions the natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The numerical results for different boundary conditions, beam numbers and the stiffness of the Winkler-type elastic layer are presented. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of multiple-connected beams with arbitrary boundary conditions.     

Numerical study of magneto-convective heat and mass transfer from inclined surface with Soret diffusion and heat generation effects: A model for ocean magnetic energy generator fluid dynamics

A mathematical model is developed for steady state magnetohydrodynamic (MHD) heat and mass transfer flow along an inclined surface in an ocean MHD energy generator device with heat generation and thermo-diffusive (Soret) effects. The governing equations are transformed into nonlinear ordinary differential equations with appropriate similarity variables. The emerging two-point boundary value problem is shown to depend on six dimensionless thermophysical parameters - magnetic parameter, Grashof number, Prandtl number, modified Prandtl number, heat source parameter and Soret number in addition to plate inclination. Numerical solutions are obtained for the nonlinear coupled ordinary differential equations for momentum, energy and salinity (species) conservation, numerically, using the Nachtsheim–Swigert shooting iteration technique in conjunction with the Runge–Kutta sixth order iteration scheme. Validation is achieved with Nakamura's implicit finite difference method. Further verification is obtained via the semi-numerical Homotopy analysis method (HAM). With an increase in magnetic parameter, skin friction is depressed whereas it generally increases with heat source parameter. Salinity magnitudes are significantly reduced with increasing heat source parameter. Temperature gradient is decreased with Prandtl number and salinity gradient (mass transfer rate) is also reduced with modified Prandtl number. Furthermore, the flow is decelerated with increasing plate inclinations and temperature also depressed with increasing thermal Grashof number.     

Dark and kink soliton solutions of the generalized ZK–BBM equation by iterative scheme

The generalized ZK–BBM equation is solved using iterative scheme of the Adomian decomposition method (ADM) and variational iteration method (VIM). A dark and a kink soliton solutions of the generalized ZK–BBM equation are obtained under initial conditions. The convergence analysis of the ADM and VIM solution shows that these solutions are convergent. The comparison of the ADM and VIM solutions with the exact solution shows that the solutions of the generalized ZK–BBM equation by the iterative methods are almost exact. The absolute errors show that the accuracy and efficiency of the ADM and VIM depend on the problem and its domain. It is found that the iterative scheme of Adomian decomposition method and variational iteration method are quite efficient for the soliton solution of the generalized ZK–BBM equation.     

Adomian Decomposition Method and Exact Solutions of the Perturbed KdV Equation

The Adomian decomposition method is used to solve the Cauchy problem of the perturbed KdV equation.Three types of exact solitary wave solutions are reobtained via the A domian‘s approach by selecting the initial conditionsappropriately.     

Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations

In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

He's variational iteration method applied to the solution of the prey and predator problem with variable coefficients

In this Letter, a mathematical model of the problem of prey and predator is presented and He's variational iteration method is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The results are compared with the results obtained by Adomian decomposition method and homotopy perturbation method. Comparison of the methods show that He's variational iteration method is a powerful method for obtaining approximate solutions to nonlinear equations and their systems.     

Newly modified method and its application to the coupled Boussinesq equation in ocean engineering with its linear stability analysis

Investigating the dynamic characteristics of nonlinear models that appear in ocean science plays an important role in our lifetime. In this research, we study some features of the paired Boussinesq equation that appears for two-layered fluid flow in the shallow water waves. We extend the modified expansion function method (MEFM) to obtain abundant solutions, as well as to find new solutions. By using this newly modified method one can obtain novel and more analytic solutions comparing to MEFM. Also, numerical solutions via the Adomian decomposition scheme are discussed and favorable comparisons with analytical solutions have been done with an outstanding agreement. Besides, the instability modulation of the governing equations are explored through the linear stability analysis function. All new solutions satisfy the main coupled equation after they have been put into the governing equations.     

Adomian Decomposition Method and Pade Approximants for Nonlinear Differential-Difference Equations

Combining Adomian decomposition method （ADM） with Pade approximants, we solve two differentiaidifference equations （DDEs）： the relativistic Toda lattice equation and the modified Volterra lattice equation. With the help of symbolic computation Maple, the results obtained by ADM-Pade technique are compared with those obtained by using ADM alone. The numerical results demonstrate that ADM-Pade technique give the approximate solution with faster convergence rate and higher accuracy and relative in larger domain of convergence than using ADM.     

A NOTE ON USING ADOMIAN DECOMPOSITION METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS

In this paper we outline a reliable strategy to use Adomian decomposition method properly for solving nonlinear partial differential equations with boundary conditions. Our fundamental goal in this paper has two features: (i) it introduces an efficient way for using Adomian decomposition method for boundary value problems, and (ii) it also would present the framework in a general way so that it may be used in BVPs of the same type. A numerical example is included to dwell upon the importance of the analysis presented.     

Series solution of flow over nonlinearly stretching sheet with chemical reaction and magnetic field

In this Letter, the boundary-layer equation of flow over a nonlinearly stretching sheet in the presence of a chemical reaction and a magnetic field is investigated by employing the Adomian decomposition method (ADM). The series solution of the governing nonlinear problem is developed. The present solution is shown to agree very well with the existing solution.     

计算应力强度因子的离散分离变量法

提出一种用于数值求解带有一条边界裂纹的多角形区域上的Navier's方程组边值问题的半离散方法。做一个适当的坐标变换后,将原边值问题化为半无限长条上的不连续系数问题。将其半离散化以后,等价于一个常系数常微分方程组的边值问题。进一步,用直接法来求解这个边值问题,便得到原问题的半离散近似解。值得指出的是,这个用分离变量形式给出的半离散近似解自然地具有原问题的奇性。数值例子显示,用该方法可以很方便地计算出在裂纹顶端的应力强度因子的近似值。     

Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

In this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method. Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation, nonlinear fractional discrete Schrodinger equation, fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger's equation. The obtained solution is verified by comparison with exact solution when $\alpha=1$.     

Numerical Solutions of Fractional Boussinesq Equation

Based upon the Adomian decomposition method, a scheme is developed to obtain numerical solutions of a fractional Boussinesq equation with initial condition, which is introduced by replacing some order time and space derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the traditional Adomian decomposition method for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional differential equations. The solutions of our model equation are calculated in the form of convergent series with easily computable components.