Use Of Adomian Decomposition Method Research Articles

REACTION-DIFFUSION FISHER’S EQUATIONS VIA DECOMPOSITION METHOD

The effect of the source, initial or boundary conditions in the use of Adomian decomposition method (ADM) on nonlinear partial differential equation or nonlinear equation in general is enormous. Sometimes the equation in question result to continuous exact solution in series form, other times it result to discrete approximate analytical solutions. In this paper, we show that continuous exact solitons can be obtained on application of ADM to the Fisher's equation with the deployment Taylor theorem to the terms(s) in question. And, the resulting series is split into the integral equations during the solution process. Resulting to multivariate Taylor's series of the exact solitons with the help of Adomian polynomials of the nonlinear reaction term correctly calculated. More physical results are further depicted in 2D, 3D and contour plots.

A DOMIAN DECOMPOSITION METHOD FOR STEADY FREE CONVECTIVE COUETTE FLOW IN A VERTICAL CHANNEL WITH NON-LINEAR THERMAL RADIATION, DYNAMIC VISCOSITY AND DYNAMIC THERMAL CONDUCTIVITY EFFECTS

In this paper, we investigate steady free convective Couette flow in a vertical channel with nonlinear thermal radiation, dynamic viscosity and dynamic thermal conductivity effects. The investigation is motivated by the studies of some researchers which assumed linear thermal radiation and constant fluid properties. However, this is uncalled for; as these assumptions do not reflect true behavior of the flow. For instance; increase in temperature affects fluid viscosity, thermal conductivity thereby changing the transport phenomenon. Here; the investigation considers both the fluid viscosity and thermal conductivity to be dependent on temperature with the thermal radiation adopting nonlinear form. Due to this reasons, the associated flow equations are highly nonlinear and exhibit no analytical solution and therefore require the use of Adomian decomposition method (ADM) of solution. The attained ADM solution is then coded into computer algebra package of mathematica where results under the parameters of interest are presented and discussed. Results of the investigation show that raising the thermal radiation leads to corresponding rise in both the velocity and temperature of the fluid in the channel. Furthermore; lessening the viscosity and thermal conduction of the fluid were identified to escalate both velocity and temperature of the fluid.

Application of Adomian‐type method to solve rectangular laminated thick plates in bending

AbstractThis paper aims the use of Adomian Decomposition Method along with the pb‐2 Rayleigh‐Ritz Method to obtain the response of thick laminated plates. The Mindlin Plate Theory and the equivalent single layer concept are utilised. The differential operator is decomposed accordingly to a hierarchic constitutive scheme, resulting in a superposition of isotropic and anisotropic constitutive tensors at lamina's level. This implies the superposition of isotropic and a remainder plate stiffness with the latter grouping the anisotropic influence of each lamina at plate's level. By the Adomian Decomposition Method, an isotropic plate solution (related to the isotropic stiffness) is recursively enhanced by the laminae influence (remainder stiffness). Convergence criteria and numeric results are presented, discussed and compared with those found in the literature.

Adomian decomposition method used to solve the one-dimensional acoustic equations

In this paper we propose the use of Adomian decomposition method to solve one-dimensional acoustic equations. This recursive method can be calculated easily and the result is an approximation of the exact solution. We use the Maple software to compute the series in the Adomian decomposition. We obtain that the Adomian decomposition method is able to solve the acoustic equations with the physically correct behavior.

Adomian decomposition identifies an approximate analytical solution for a set of coupled strings

The use of Adomian decomposition method (ADM) has been successfully applied in various applications across the applied mechanics and mathematics community. Originally, Adomian developed this method to derive analytical approximate solutions to nonlinear functional equations. It was shown that the solution to the given nonlinear functional equation can be approximated by an infinite series solution of the linear and nonlinear terms, provided the nonlinear terms are represented by a sum of series of Adomian polynomials. Here, ADM is used to derive an approximate analytical solution to a set of partial differential equations (PDEs) describing the motion of two coupled strings that lie orthogonal to each other. The PDEs are derived using Euler-Lagrange equations of motion. The ends of the strings are pinned and the strings are coupled with a nonlinear spring. A finite element model of the system is developed to provide a comparative baseline. Both the finite element model and analytical solution were driven b...

On the solution of characteristic value problems arising in linear stability analysis; semi analytical approach

The linear stability analysis is normally governed by equations that constitute an eigenvalue (characteristic value) problem. In this paper, for the first time, two semi analytical algorithms, (1) Differential Transform Method (DTM) and (2) Adomian Decomposition Method (ADM) are examined for solving a characteristic value problem occurring in linear stability analysis. In this paper, the characteristic value problem of Couette Taylor flow is selected because its simple geometry continues to be a paradigm for theoretical studies of hydrodynamic stability. The results show that DTM handles the solution conveniently and accurately. However, this paper limits the use of ADM for solving characteristic value problems. The results indicate that the present algorithm based on DTM could be used as a promising method for solving characteristic value problems.

Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid

In this paper, Adomian’s decomposition method is used to solve non-linear differential equations which arise in fluid dynamics. We study basic flow problems of a third grade non-Newtonian fluid between two parallel plates separated by a finite distance. The technique of Adomian decomposition is successfully applied to study the problem of a non-Newtonian plane Couette flow, fully developed plane Poiseuille flow and plane Couette–Poiseuille flow. The results obtained show the reliability and efficiency of this analytical method. Numerical solutions are also obtained by solving non-linear ordinary differential equations using Chebyshev spectral method. We present a comparative study between the analytical solutions and numerical solutions. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the Adomian decomposition method.

A note on the use of Adomian decomposition method for high-order and system of nonlinear differential equations

This paper extends an earlier work [Hosseini MM, Nasabzadeh H. Modified Adomian decomposition method for specific second order ordinary differential equations. Appl Math Comput 2007;186:117–23] to high order and system of differential equations. Solution of these problems is considered by proposed modification of Adomian decomposition method. Furthermore, with providing some examples, the aforementioned cases are dealt with numerically.

An adaptation of adomian decomposition for numeric–analytic integration of strongly nonlinear and chaotic oscillators

A novel form of an explicit numeric-analytic technique is developed for solving strongly nonlinear oscillators of engineering interest. The analytic part of this technique makes use of Adomian Decomposition Method (ADM), but unlike other analytical solutions it does not rely on the functional form of the solution over the whole domain of the independent variable. Instead it discretizes the domain and solves multiple IVPs recursively. ADM uses a rearranged Taylor series expansion about a function and finds a series of functions which add up to generate the required solution. The present method discretizes the axis of the independent variable and only collects lower powers of the chosen step size in series solution. Each function constituting the series solution is found analytically. It is next shown that the modified ADM can be used to obtain the analytical solution,in a piecewise form. For nonlinear oscillators such a piecewise solution is valid only within a chosen time step. An attempt has been made to address few issues like the order of local error and convergence of the method. Emphasis has been on the application of the present method to a number of well known oscillators. The method has the advantage of giving a functional form of the solution within each time interval thus one has access to finer details of the solution over the interval. This is not possible in purely numerical techniques like the Runge–Kutta method, which provides solution only at the two ends of a given time interval, provided that the interval is chosen small enough for convergence. It is shown that the present technique successfully overcomes many limitations of the conventional form of ADM. The present method has the versatility and advantages of numerical methods for being applied directly to highly nonlinear problems and also have the elegance and other benefits of analytical techniques.

Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems

In this paper, we will compare the differential transformation method DTM and Adomian decomposition method ADM to solve partial differential equations (PDEs). The definition and operations of differential transform method was introduced by Zhou [Zhou JK. Differential transformation and its application for electrical circuits. Wuuhahn, China: Huarjung University Press; 1986 [in Chinese]]. Adomian decomposition method which is given by Adomian [Adomian G. Convergent series solution of nonlinear equation. J Comput Appl Math 1984;11:113–7; Adomian G. Solutions of nonlinear PDE. Appl Math Lett 1989;11:121–3; Adomian G. Solving Frontier problems of physics. The decomposition method, Boston, 1994] for approximate solution of linear and nonlinear differential equations and to the solutions of various scientific models see [Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons & Fractals 2006;29:313–30; El-Danaf TS, Ramadan MA, Abd Alaal FEI. The use of Adomian decomposition method for solving the regularized long-wave equation. Chaos, Solitons & Fractals 2005;26:747–57; El-Sayed SM. The decomposition method for studying the Klein–Gordon equation. Chaos, Solitions & Fractals 2003;18:1025–30; Helal MA, Mehanna MS. A comparison between two different methods for solving KdV-Burgers equation. Chaos, Solitons & Fractals 2006;28:320–6, Hashim I, Noorani MSM, Ahmad R, Bakar SA, Ismail ES, Zakaria AM. Accuracy of the decomposition method applied to the Lorenz system. Chaos, Solitons & Fractals 2006;28:1149–58; Kaya D, El-Sayed SM. An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons & Fractals 2003;17:869–77; Lesnic D. Blow-up solutions obtained using the decomposition method. Chaos, Solitons & Fractals 2006;28:776–87; Wazwaz AM. Construction of solitary wave solutions and rational solutions for KdV equation by Adomian decomposition method. Chaos, Solitons & Fractals 2001;12:2283–93; Wazwaz AM. Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. Chaos, Solitons & Fractals 2001;12:1549–56]. A distinctive practical feature of the differential transformation method DTM is ability to solve linear or nonlinear differential equations. Higher-order dimensional differential transformation are applied to a few some initial value problems to show that the solutions obtained by the proposed method DTM coincide with the approximate solution ADM and the analytic solutions.

The use of the Adomian decomposition method for solving multipoint boundary value problems

In this paper, a method for solving multipoint boundary value problems is presented. The main idea behind this work is the use of the well-known Adomian decomposition method. In this technique, the solution is found in the form of a rapid convergent series. Using this method, it is possible to obtain the solution of the general form of multipoint boundary value problems. The Adomian decomposition method is not affected by computation round off errors and one is not faced with the necessity of large computer memory and time. To show the efficiency of the developed method, numerical results are presented.

The use of Adomian decomposition method for solving problems in calculus of variations

In this paper, a numerical method is presented for finding the solution of some variational problems. The main objective is to find the solution of an ordinary differential equation which arises from the variational problem. This work is done using Adomian decomposition method which is a powerful tool for solving large amount of problems. In this approach, the solution is found in the form of a convergent power series with easily computed components. To show the efficiency of the method, numerical results are presented.

The use of adomian decomposition method for solving the regularized long-wave equation

Abstract In this paper, an accurate method to obtain an approximate numerical solution for the nonlinear regularized long-wave (in short RLW) equation is considered. The theoretical analysis of the method is investigated. The performance and the accuracy of the algorithm are illustrated by solving two test examples of the problem. The obtained results are presented and compared with the analytical solutions. It is observed that only few terms of the series expansion are required to obtain approximate solutions with good accuracy.

The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications

Over the last 20 years, the Adomian decomposition approach has been applied to obtain formal solutions to a wide class of stochastic and deterministic problems involving algebraic, differential, integro-differential, differential delay, integral and partial differential equations. This method leads to computable, efficient, solutions to linear and nonlinear operator equations. Furthermore in the past, only classical boundary value problems have been considered. The parabolic partial differential equations with non-classical conditions model various physical problems. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the second-order linear parabolic partial differential equation with nonlocal boundary specifications replacing the standard boundary conditions. This scheme is employed for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval. The Adomian decomposition method provides a reliable technique that requires less work when compared with the traditional techniques. This method is used by many researchers to investigate several scientific applications. Some experimental results using the newly proposed procedure are given to confirm our belief of the reliability of the approach.

The use of Adomian decomposition method for solving a specific nonlinear partial differential equations

The use of Adomian decomposition method for solving a specific nonlinear partial differential equations

A study of nonlinear dispersive equations with solitary-wave solutions having compact support

With the use of Adomian decomposition method, the prototypical, genuinely nonlinear K( m, n) equation, u t +( u m ) x +( u n ) xxx =0, which exhibits compactons — solitons with finite wavelength — is solved exactly. Two numerical illustrations, K(2,2) and K(3,3), are investigated to illustrate the pertinent features of the proposed scheme. The technique is presented in a general way so that it can be used in nonlinear dispersive equations.