Applied Numerical Mathematics (original) (raw)
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Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations
2010
We are concerned here with a multi-term nonlinear fractional differential equation (FDE). Two methods are used to solve this type of equations. The first is an analytical method; Adomian decomposition method (ADM). Convergence analysis of this method is discussed. This analysis is used to estimate the maximum absolute truncated error of Adomian's series solution. The second method is a proposed numerical method (PNM). A comparison between the results of the two methods is given. One of the important applications of these equations is Bagley-Torvik equation.
Computers & Mathematics with Applications, 2013
In this paper, we present the Adomian decomposition method and its modifications combined with convergence acceleration techniques, such as the diagonal Padé approximants and the iterated Shanks transforms, to solve nonlinear fractional ordinary differential equations. Two nonlinear numeric examples demonstrate that either the diagonal Padé approximants or the iterated Shanks transforms can efficiently extend the effective convergence region of the decomposition series solution.
Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method
International Journal of Systems Science and Applied Mathematics, 2021
In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differential equations (FDEs) of Caputo sense. These type of equations is very important in engineering applications such as electrical networks, fluid flow, control theory and fractals theory. ADM give analytical solution in form of series solution so the convergence of the series solution and the error analysis will discuss. In addition, existence and uniqueness of the solution will prove. Some numerical examples will solve to test the validity of the method and the given theorems. A comparison of ADM solution with exact and numerical methods are given.
Adomian decomposition method for solving fractional nonlinear differential equations
Applied Mathematics and Computation, 2006
In this article, we have discussed a new application of Adomian decomposition method on time fractional nonlinear fractional differential equations. Three models with fractional-time derivative of order a, 0 < a < 1 are considered and solved by means of Adomian decomposition method. The behaviour of Adomian solutions and the effects of different values of a are investigated. Numerical examples are tested to illustrate the pertinent feature of the proposed algorithm.
Analytical solution of linear and nonlinear fractional differential equations
Nile Journal of Basic Science, 2021
In this paper, we apply the Adomian decomposition method (ADM) for solving linear and nonlinear fractional differential equations (FDEs). The existence and uniqueness of the solution are proved. The convergence of the series solution and the error analysis are discussed. Some applications are solved such as relaxation-oscillation equation, Basset problem and fractional Riccati differential equation.
Delta University Scientific Journal, 2023
In this paper, we apply the Adomian decomposition method (ADM) for solving Fractional Differential Equations (FDEs) with some modifications to the traditional method. The aim of this paper is to make ADM more efficient, rapid in convergence, and easy to use, so we will discuss two modifications. We use the reliable modification to simplify calculations. For difficulties in symbolic integration, we use a numerical implementation method. All these modifications were applied to the integer-order case, so we would apply it to FDEs. Some numerical results are given from solving these cases and comparing the solution with the ADM method.
Adomian Decomposition Method for Solving Nonlinear Fractional PDEs
In this paper, we obtain the approximate analytical solution of fractional differential equations with Caputo-Fabrizio fractional derivative by using the fractional Adomian decomposition method. The approximate solutions of nonlinear differential equations with fractional order are successfully obtained using this method , and the result is compared with the result of the existing methods.
Numerical solutions for systems of fractional differential equations by the decomposition method
Applied Mathematics and Computation, 2005
In this paper we use Adomian decomposition method to solve systems of nonlinear fractional differential equations and a linear multi-term fractional differential equation by reducing it to a system of fractional equations each of order at most unity. We begin by showing how the decomposition method applies to a class of nonlinear fractional differential equations and give two examples to illustrate the efficiency of the method. Moreover, we show how the method can be applied to a general linear multi-term equation and solve several applied problems.
2012
In this article we review the Adomian decomposition method (ADM) and its modifications including different modified and parametrized recursion schemes, the multistage ADM for initial value problems as well as the multistage ADM for boundary value problems, new developments of the method and its applications to linear or nonlinear and ordinary or partial differential equations, including fractional differential equations.