adomian decomposition method

equations with non-local boundary conditions. If the equations considered have a solution in terms of the series expansion of known function, this powerful method catches the exact solution. Comparison is made between the exact solution and Adomian decomposition method (ADM). The results reveal that the differential transform method is very effective and simple.

The LADM is known for its rapid convergence in solution and also uses only little iteration as successfully applied in Kiymaz [15]. Furthermore, several mod- ifications of the ADM and LADM methods can be seen in [16]-[22] with wide applications ranging from differential equations, partial differential equations, integral equations and integro-differential equations among others. It seems that the LADM method is always open for further modifications especially on discre- tizing the Adomian decomposition. In this paper, we aim at extending the Lap- lace Adomian decomposition method for finding the solution of nonlinear integro-differential equations by firstly discretizing the Adomian decomposition method, followed by coupling some numerical integration schemes or quadra- ture rules. These quadrature rules are used to approximate the definite integrals which are analytically impossible [23] [24] [25] [26] as the solution is given at the nodes. For convenience, the proposed extension is known as Laplace Dis- crete Adomian Decomposition Method (LDADM). The rest of this paper is ar- ranged as follows. Section 2 describes the proposed LDADM. Section 3 presents numerical results with four examples and finally Section 4 concludes.

Abstract: The system of gas dynamic equations governing the motion of one-dimensional unsteady adiabatic flow of a perfect gas in planer, cylindrical and spherical symmetry is solved successfully by applying the Adomian decom- position method under the exponential initial conditions. The solution of the system of equation is computed up to the five components of the decomposition series. The variation of the approximate velocity, density and pressure of the fluid motion with position and time is studied. It is found that there exists discontinuity or shock wave in the distribution of flow variables. The solution of system of gas dynamic equations by Adomian decomposition method is con- vergent for a domain of position and time. The decomposition method provides the variation of flow-variables with position and time separately which was not possible in similarity method.

The SageMath code presented in this paper is a simple one and also easily understandable. The main advantage of the program is its ability to generate Adomian polynomials for nonlinear terms as presented by G. Adomian. In addition to that, we have developed the function “adomiansol” to solve nonlinear ordinary differential equations using Adomian decomposition method for single linear and nonlinear terms. One can extend this concept to multivariable cases and differential equations having more nonlinear terms also. These programs will be useful for the researchers since SageMath is open source software.

The main objective of this work is to obtain a solution for nonlinear fractional partial differential equation. We observe that Adomian Decomposition Method is a powerful method to solve nonlinear fractional partial differential equation. To, show the applicability and efficiency of the proposed method, the method is applied to obtain the solutions of several examples. The obtained results demonstrate the reliability of the algorithm. It is worth mentioning that the proposed technique is capable of reducing the volume of the computational work as compared to the classical methods. Finally, we come to the conclusion that the Adomian Decomposition Method is very powerful and efficient in finding solutions for wide class of nonlinear fractional partial differential equation.

ogeneity of the equation is the major and only reason. The appearance of these terms in the first two compo- nents of the solution series is considered by Adomian [13] as the first condition of demonstrating a fast convergence of the solution. Wazwaz [18,19] stated the inhomogene- ity condition is not sufficient for occurring this phenom- ena and he added another necessary condition (if the ex- act solution explicitly appears in the zeroth component) to interpret this phenomena. Wazwaz [16] modified the Adomian decomposition method by dividing the initial approximation into two parts, and distribute these parts through the calculations. In [20] Wazwaz and Salah ex- pand the same initial approximation using Taylor expan- sion and insert the terms of this expansion through the calculations, they succeed to avoid noise terms for some integral equations.

. Adomian decomposition method has been applied to a large class of linear and nonlinear equations that are et al., 2016; Bulut et al., 2004; Asil et al., 2005). The essential property of this method is the ability to solve these equations appropriately and accurately. Systems of ordinary differential equations have been appea frequently in a wide class of scientific applications in physics, engineering and other fields. So, their importance cannot be undervalued. Adomian decomposition method has been also used efficiently for solving systems of differential equations

Abstract. In this paper, Adomian Decomposition Method(ADM) is applied to Fifth-Order autonomous differential equations. The general concept of ADM to this class of equations was stated in relation to the general concept. Three test problems were used to validate the concept of the decomposition method, and the result in series form of only the first six terms were obtained. The absolute error were also obtained, similarly the plots of both the exact and ADM solutions. The series form solution by ADM gave almost the same result as those obtained by any known closed form method of the continuous function. Thus, justifying the excellent potentials of the decomposition method.

In this paper, linear and nonlinear boundary value problems for fourth-order fractional integro- differential equations are solved by Variational iteration method (VIM) and Adomian decomposition method (ADM). The fractional derivative is considered in the Caputo sense . The solutions of both problems are derived by infinite convergent series . Numerical example are presented to illustrate the efficiency and reliability of two methods.

This research will focus on the algorithm for the methods of variational iteration and Adomian decomposition method. The methods are used to solve fractional diffusion equation. The general equation of fractional diffusion equation is

In this work, we have applied Adomian Decomposition method with Taylor series and Chebyshev polynomials. On applying the method to the examples with inhomoheneous source terms we can conclude from Fig.1, Fig.2, Fig.3 and Fig.4 that Chebyshev polynomial series representation of inhomogeneous source term produces better results than Taylor series expansion.

In this work, we have applied Adomian Decomposition method with Taylor series and Chebyshev polynomials. On applying the method to the examples with inhomoheneous source terms we can conclude from Fig.1, Fig.2, Fig.3 and Fig.4 that Chebyshev polynomial series representation of inhomogeneous source term produces better results than Taylor series expansion.

Many differential equations cannot be solved by standard methods, in such equations, it is sufficient to obtain an approximation solution. Among the approximation method used for solving differential equations, are the Picard iteration method and the Adomian decomposition method. The Picard iteration method named after a French mathematician Charles Emile Picard (1856–1941) is a successive approximation method which is used in giving an approximate solution to an initial-value problem. It is also a constructive method used for initiating the existence and the uniqueness of a solution to a differential equation of the form . The Adomian decomposition method (ADM) is an approximation method used for solving non-linear differential equation [Ado88]. This method involves the decomposition of a solution of a non-linear equation in a series of functions whereby each term of the series is obtained recursively from a polynomial generated from an expansion of an analytic function into a power series [Ado92]. These polynomials are called Adomian polynomials. ADM is most useful in non-linear differential equations and the number of the partial solutions used determines their accuracy. One of the advantages of this method is that it converges fast to the exact solution [Ado92].

The Adomian decomposition method (ADM) can be used to solve a wide range of problems and usually gets the solution in a series form. In this paper, we propose two-step Adomian Decomposition Method (TSAM) for nonlinear integro-differential equations that will facilitate the calculations. In this modification, compared to the standard Adomian decomposition method, the size of calculations was reduced. This modification also avoids computing Adomian polynomials. Numerical results are given to show the efficiency and performance of this method.

In this work, it is shown how the Adomian decomposition method and some of its modification can be adapted in order to be used to the nonlinear Schrodinger. The new method presented in this work has a powerful and easy use. The numerical technique is improved by decomposition of the nonlinear operator. In applying the improved Ado- mian decomposition Method (IADM) to the nonlinear Schrodinger equation, it is found that the method gives accurate results with lesser computational effort as com- pared with other modification.

Adomian decomposition method (ADM) is a powerful tool which enables us to find solutions in case of linear as well as non-linear equations. The method has been successfully applied to a system of fractional differential equation as well as fuzzy differential equation. It is shown that the applicability of Adomian decomposition method to solve the system of fuzzy fractional differential equations of fractional order α (0 < α ≤ 1). The Adomian decomposition method is straightforward and applicable for broader problems. It can avoid the difficulty of finding the inverse of Laplace transform for solving the fuzzy fractional differential equation by Laplace transform method.

On comparison of solution of homogeneous part the equation (2)-(3) with that used by Carroll (1951), this is far much simpler for we do not involve transformations which are not easy to identify. Thus Adomian decomposition method offers quite a simpler means of solving nonlinear differential equations.

Recently, Adomian decomposition method and modified Adomian decomposition method were popular among the researchers who studied integral equations. From the findings this dissertation, it is hoped that the present work can be used as a reference for the future study.

The concept of convergence of the solution obtained by Adomian decomposition method was addressed by [1,18], and extensively by [19,20]. Convergence of the ADM when applied to initial value problem in ordinary differ- ential equation is discussed by many authors. For exam- ple, K. Abbaoui and Y. Cherruault [21,22]. In [23] H. Alzumi et al. discussed the convergence of ADM.

In this paper, the bioconvective nanofluid flow in a horizontal channel was considered. Using the appropriate similarity functions, the partial differential equations of the studied problem resulting from mathematical modeling are reduced to a set of non-linear differential equations. Thereafter, these equations are solved numerically using the fourth order Runge-Kutta method featuring shooting technique and analytically via the Adomian decomposition method (ADM). This study mainly focuses on the effects of several physical parameters such as Reynolds number (Re), thermal parameter (𝛿 𝜃 ), microorganisms density