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.

13 Read more

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.

Show more
20 Read more

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**.

Show more
20 Read more

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.

Show more
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.

Show more
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.

Show more
. **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

Show more
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**.

Show more
10 Read more

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

19 Read more

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].

Show more
12 Read more

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**.

Show more
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.

18 Read more

10 Read more

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.

22 Read more

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

Show more
14 Read more