scholarly journals Approximate Solutions of Fractional Riccati Equations Using the Adomian Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Fei Wu ◽  
Lan-Lan Huang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Riccati Equations ◽  
Crucial Step ◽  
Adomian Decomposition

The fractional derivative equation has extensively appeared in various applied nonlinear problems and methods for finding the model become a popular topic. Very recently, a novel way was proposed by Duan (2010) to calculate the Adomian series which is a crucial step of the Adomian decomposition method. In this paper, it was used to solve some fractional nonlinear differential equations.

scholarly journals Application of Natural Transform Method to Fractional Pantograph Delay Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
M. Valizadeh ◽  
Y. Mahmoudi ◽  
F. Dastmalchi Saei
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
New Method ◽  
Transform Method ◽  
Adomian Decomposition ◽  
Delay Differential

In this paper, a new method based on combination of the natural transform method (NTM), Adomian decomposition method (ADM), and coefficient perturbation method (CPM) which is called “perturbed decomposition natural transform method” (PDNTM) is implemented for solving fractional pantograph delay differential equations with nonconstant coefficients. The fractional derivative is regarded in Caputo sense. Numerical evaluations are included to demonstrate the validity and applicability of this technique.

scholarly journals Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Hassan Eltayeb ◽  
Adem Kılıçman
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Adomian Polynomials ◽  
Adomian Decomposition ◽  
Sumudu Transform

We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM).

scholarly journals Variational Approximate Solutions of Fractional Delay Differential Equations with Integral Transform

2021 ◽  
pp. 3679-3689
Author(s):  
Eman Mohmmed Namah
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Integral Transform ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Adomian Decomposition ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

     The idea of the paper is to consolidate Mahgoub transform and variational iteration method (MTVIM) to solve fractional delay differential equations (FDDEs). The fractional derivative was in Caputo sense. The convergences of approximate solutions to exact solution were quick. The MTVIM is characterized by ease of application in various problems and is capable of simplifying the size of computational operations.  Several non-linear (FDDEs) were analytically solved as illustrative examples and the results were compared numerically. The results for accentuating the efficiency, performance, and activity of suggested method were shown by comparisons with Adomian Decomposition Method (ADM), Laplace Adomian Decomposition Method (LADM), Modified Adomian Decomposition Method (MADM) and Homotopy Analysis Method (HAM).

scholarly journals Adomian Decomposition Method for a Class of Nonlinear Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
J. A. Sánchez Cano
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Coupled System ◽  
Nonlinear Ordinary Differential Equations ◽  
Adomian Decomposition

The Adomian decomposition method together with some properties of nested integrals is used to provide a solution to a class of nonlinear ordinary differential equations and a coupled system.

scholarly journals Analytical Solutions for the Mathematical Model Describing the Formation of Liver Zones via Adomian’s Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdelhalim Ebaid
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Applied Mathematics ◽  
Mathematical Formulation ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Stationary Solutions ◽  
Proposed Model ◽  
Adomian Decomposition ◽  
The Mathematical Model

The formation of liver zones is modeled by a system of two integropartial differential equations. In this research, we introduce the mathematical formulation of these integro-partial differential equations obtained by Bass et al. in 1987. For better understanding of this mathematical formulation, we present a medical introduction for the liver in order to make the formulation as clear as possible. In applied mathematics, the Adomian decomposition method is an effective procedure to obtain analytic and approximate solutions for different types of operator equations. This Adomian decomposition method is used in this work to solve the proposed model analytically. The stationary solutions (as time tends to infinity) are also obtained through it, which are in full agreement with those obtained by Bass et al. in 1987.

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Umesh Umesh ◽  
Manoj Kumar
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
The Other ◽  
Step Size ◽  
Adomian Polynomials ◽  
Content Type ◽  
Adomian Decomposition

Purpose The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions. Design/methodology/approach First, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed. Findings The proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples. Originality/value Unlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.

Sign in / Sign up

Export Citation Format

Share Document