Convergence of Discrete Adomian Method for Solving a Class of Nonlinear Fredholm Integral Equations

In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In this paper a discretized version of the ADM is introduced and the proposed version will be called discrete Adomian decomposition method (DADM). An accelerated formula of Adomian polynomials is used in calculations. Based on this formula, a new convergence approach of ADM is introduced. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian’s series solution. Also, we prove that the solution of nonlinear Fredholm integral equation by DADM converges to ADM solution. Finally, some numerical examples were introduced.

Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method

In this study, the effects of magnetic field and nanoparticle on the Jeffery- Hamel flow are studied using a powerful analytical method called the Adomian decomposition method （ADM）. The traditional Navier-Stokes equation of fluid mechanics and Maxwell＇s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The obtained results are well agreed with that of the Runge-Kutta method. The present plots confirm that the method has high accuracy for different a, Ha, and Re numbers. The flow field inside the divergent channel is studied for various values of Hartmann ：number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.

SOLUTION OF SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD

The aim of this paper is to apply the relatively new Adomian decomposition method to solving the system of linear fractional, in the sense of Riemann-Liouville and Caputo respectively, differential equations. The solutions are expressed in terms of Mittag-Leffier functions of matric argument. The Adomian decomposition method is straightforward, applicable for broader problems and avoids the difficulties in applying integral transforms. As the order is 1, the result here is simplified to that of first order differential equation.

Adomian decomposition method and Padé approximants for solving the Blaszak-Marciniak lattice

The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.

Analytical solution of fractionally damped beam by Adomian decomposition method

The analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the AdoInian decomposition method. The solution contains arbitrary initial conditions and zero input. For specific analysis, the initial conditions were assumed homogeneous, and the input force was treated as a special process with a particular beam. Two simple cases, step and impulse function responses, were considered respectively. Subsequently, some figures were plotted to show the displacement of the beam under different sets of parameters including different orders of the fractional derivatives.

Nonlinear Problems via a Convergence Accelerated Decomposition Method of Adomian

The present paper is devoted to the convergence control and accelerating the traditional Decomposition Methodof Adomian (ADM). By means of perturbing the initial or early terms of the Adomian iterates by adding aparameterized term, containing an embedded parameter, new modified ADM is constructed. The optimal value ofthis parameter is later determined via squared residual minimizing the error. The failure of the classical ADM is alsoprevented by a suitable value of the embedded parameter, particularly beneficial for the Duan–Rach modification ofthe ADM incorporating all the boundaries into the formulation. With the presented squared residual error analysis,there is no need to check out the results against the numerical ones, as usually has to be done in the traditional ADMstudies to convince the readers that the results are indeed converged to the realistic solutions. Physical examplesselected from the recent application of ADM demonstrate the validity, accuracy and power of the presented novelapproach in this paper. Hence, the highly nonlinear equations arising from engineering applications can be safelytreated by the outlined method for which the classical ADM may fail or be slow to converge.

The Modified Adomian Decomposition Method for Nonlinear Fractional Boundary Value Problems

We use the modified Adomian decomposition method（ADM） for solving the nonlinear fractional boundary value problem {D α0＋u（x）=f（x,u（x）） ,0〈x〈1,3〈α≤4u（0）=α0, u″（0）=α2u（1）=β0,u″（1）β2where Dα 0 ＋u is Caputo fractional derivative and α0, α2, β0, β2 is not zero at all, and f ： [0, 1] × R→R is continuous. The calculated numerical results show reliability and efficiency of the algorithm given. The numerical procedure is tested on lineax and nonlinear problems.

Adomian Decomposition Method and Padé 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.

Solitary Wave Solutions of Discrete Complex Ginzburg-Landau Equation by Modified Adomian Decomposition Method

In this paper, exact and numerical solutions are calculated for discrete complex Ginzburg-Landau equation with initial condition by considering the modified Adomian decomposition method （mADM）, which is an efficient method and does not need linearization, weak nonlinearity assumptions or perturbation theory. The numerical solutions are also compared with their corresponding analytical solutions. It is shown that a very good approximation is achieved with the analytical solutions. Finally, the modulational instability is investigated and the corresponding condition is given.

Adomian decomposition method simulation of von Kármán swrling bioconvection nanofluid flow

The study reveals analytically on the 3-dimensional viscous time-dependent gyrotactic bioconvection in swirling nanofluid flow past from a rotating disk.It is known that the deformation of the disk is along the radial direction.In addition to that Stefan blowing is considered.The Buongiorno nanofluid model is taken care of assuming the fluid to be dilute and we find Brownian motion and thermophoresis have dominant role on nanoscale unit.The primitive mass conservation equation,radial,tangential and axial momentum,heat,nano-particle concentration and micro-organism density function are developed in a cylindrical polar coordinate system with appropriate wall(disk surface)and free stream boundary conditions.This highly nonlinear,strongly coupled system of unsteady partial differential equations is normalized with the classical von Kármán and other transformations to render the boundary value problem into an ordinary differential system.The emerging 11th order system features an extensive range of dimensionless flow parameters,i.e.,disk stretching rate,Brownian motion,thermophoresis,bioconvection Lewis number,unsteadiness parameter,ordinary Lewis number,Prandtl number,mass convective Biot number,Péclet number and Stefan blowing parameter.Solutions of the system are obtained with developed semi-analytical technique,i.e.,Adomian decomposition method.Validation of the said problem is also conducted with earlier literature computed by Runge-Kutta shooting technique.

The Adomian Decomposition Method for Solving Volterra-Fredholm Integral Equation Using Maple

In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables and the amount of error between the exact solution and the numerical solution, it is very small and almost non-existent and is also illustrated through the graph how the exact solution of completely applies to the numerical solution This proves the accuracy of the method, which is the Adomian decomposition method (ADM) for solving the Volterra Fredholm integral equation using Maple 18. And that this method is characterized by ease, speed and great accuracy in obtaining numerical results.

Solving <i>nth</i>-Order Integro-Differential Equations Using the Combined Laplace Transform-Adomian Decomposition Method

In this paper, the Combined Laplace Transform-Adomian Decomposition Method is used to solve nth-order integro-differential equations. The results show that the method is very simple and effective.

The Adomian Decomposition Method for Solving Nonlinear Partial Differential Equation Using Maple

The nonlinear partial differential equation is solved using the Adomian decomposition method (ADM) in this article. A number of examples have been provided to illustrate the numerical results, which is the comparison of the exact and numerical solutions, and it has been discovered through the tables that the amount of error between the exact and numerical solutions is very small and almost non-existent, and the graph also shows how the exact solution of absolutely applies to the numerical solution. This demonstrates the precision of the Adomian decomposition method (ADM) for solving the nonlinear partial differential equation with Maple18. And that in terms of obtaining numerical results, this approach is characterized by ease, speed, and high accuracy.

On Discrete Adomian Decomposition Method with Chebyshev Abscissa for Nonlinear Integral Equations of Hammerstein Type

We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Pade approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.

On the Application of the Adomian’s Decomposition Method to a Generalized Thermoelastic Infinite Medium with a Spherical Cavity in the Framework Three Different Models

A mathematical model is elaborated for a thermoelastic infinite body with a spherical cavity.A generalized set of governing equations is formulated in the context of three different models of thermoelasticity:the Biot model,also known as“coupled thermoelasticity”model;the Lord-Shulman model,also referred to as“generalized thermoelasticity with one-relaxation time”approach;and the Green-Lindsay model,also called“generalized thermoelasticity with two-relaxation times”approach.The Adomian’s decomposition method is used to solve the related mathematical problem.The bounding plane of the cavity is subjected to harmonic thermal loading with zero heat flux and strain.Numerical results for the temperature,radial stress,strain,and displacement are represented graphically.It is shown that the angular thermal load and the relaxation times have significant effects on all the studied fields.

The Adomian Decomposition Method for a Type of Fractional Differential Equations

Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is in Caputo sense. Through the numerical test, we can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations. The numerical results also show the efficiency of this method.

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

A Study of Some Nonlinear Partial Differential Equations by Using Adomian Decomposition Method and Variational Iteration Method

In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Mahony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration Method (VIM). The results reveal that the two methods are very effective, simple and very close to the exact solution.

On the Adomian Decomposition Method for Solving PDEs

In this paper, we explore some issues related to adopting the Adomian decomposition method （ADM） to solve partial differential equations （PDEs）, particularly linear diffusion equations. Through a proposition, we show that extending the ADM from ODEs to PDEs poses some strong requirements on the initial and boundary conditions, which quite often are violated for problems encountered in engineering, physics and applied mathematics. We then propose a modified approach, based on combining the ADM with the Fourier series decomposition, to provide solutions for those problems when these conditions are not met. In passing, we shall also present an argument that would address a long-term standing ＂pitfall＂ of the original ADM and make this powerful approach much more rigorous in its setup. Numerical examples are provided to show that our modified approach can be used to solve any linear diffusion equation （homogeneous or non-homogeneous）, with reasonable smoothness of the initial and boundary data.