Solving Nonlinear Volterra | Fredholm Integro-differential Equations Using the Modified Adomian Decomposition Method

2009 ◽  
Vol 9 (4) ◽  
pp. 321-331 ◽  
Author(s):  
M. A. Fariborzi Araghi ◽  
Sh. Sadigh Behzadi
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Differential Equations ◽  
Error Bound ◽  
Decomposition Method ◽  
Numerical Scheme ◽  
Adomian Decomposition Method ◽  
Integro Differential Equation ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition

AbstractIn this paper, a nonlinear Volterra | Fredholm integro-differential equation is solved by using the modified Adomian decomposition method (MADM). The approximate solution of this equation is calculated in the form of a series in which its components are computed easily. The accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results. The existence, uniqueness and convergence and an error bound of the proposed method are proved. Some examples are presented to illustrate the efficiency and the performance of the modified decomposition method.

scholarly journals Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
S. Narayanamoorthy ◽  
T. L. Yookesh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Approximate Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Delay Differential

We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

scholarly journals Approximate solution of nonlinear ordinary differential equation using ZZ decomposition method

2020 ◽  
Vol 4 (1) ◽  
pp. 448-455
Author(s):  
Mulugeta Andualem ◽  
◽  
Atinafu Asfaw ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Approximate Solution ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Initial Value ◽  
Transform Method ◽  
Adomian Decomposition

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

scholarly journals Numerical solution of Drinfeld–Sokolov–Wilso system by using modified Adomian decomposition method

2021 ◽  
Vol 23 (1) ◽  
pp. 590
Author(s):  
Badran Jasim Salim ◽  
Oday Ahmed Jasim ◽  
Zeiad Yahya Ali
Keyword(s):  
Genetic Algorithm ◽  
Exact Solution ◽  
Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Linear Partial Differential Equations ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Optimal Value ◽  
Mean Square Errors

<p class="Char">In this paper, the modified Adomian decomposition method (MADM) is usedto solve different types of differential equations, one of the numerical analysis methods for solving non linear partial differential equations (Drinfeld–Sokolov–Wilson system) and short (DSWS) that occur in shallow water flows. A Genetic Algorithm was used to find the optimal value for the parameter (a). We numerically solved the system (DSWS) and compared the result to the exact solution. When the value of it is low and close to zero, the MADM provides an excellent approximation to the exact solution. As well as the lower value of leads to the numerical algorithm of (MADM) approaching the real solution.  Finally, found the optimal value when a=-10 by using the Genetic Algorithm (G-MADM). All the computations were carried out with the aid of Maple 18 and Matlab to find the parameter value (a) by using the genetic algorithm as well as to figures drawing. The errors in this paper resulted from cut errors and mean square errors.</p>

scholarly journals Analytic approximation of Volterra’s population model

2017 ◽  
Vol 13 (1) ◽  
pp. 5-17 ◽  
Author(s):  
J. Biazar ◽  
K. Hosseini
Keyword(s):  
Differential Equation ◽  
Population Growth ◽  
Closed System ◽  
Population Model ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Analytic Approximation ◽  
Exact Results ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition

Abstract In this paper, the Volterra’s population model is studied for population growth of a species within a closed system. Modified Adomian decomposition method (MADM) in conjunction with Pade technique is formally proposed to obtain an analytic approximation for the solution of the model, which is a nonlinear intgro-differential equation. The results of the method are compared with the existing exact results, confirming the accuracy and the efficiency of the proposed approach.

scholarly journals Mathematical Modelling in Engineering with Integral Transforms via Modified Adomian Decomposition Method

2021 ◽  
Vol 8 (3) ◽  
pp. 409-417
Author(s):  
Minakshi Mohanty ◽  
Saumya Ranjan Jena ◽  
Satya Kumar Misra
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Integral Transforms ◽  
Physical Problems ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Different Types ◽  
Linear Differential ◽  
Homogeneous Linear

In this work three integral transforms through modified Adomian decomposition method (ADM) are proposed to obtain the approximate analytical solution of different types of mathematical models arising in physical problems. These transformations are applied for both homogeneous and non-homogeneous linear differential equations. The efficiency and accuracy of the proposed methods are implemented through higher order non-homogeneous ordinary differential equations. Numerical tests are reported for applicability of the current scheme based on different transformations and compared with exact solutions.

scholarly journals Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Veyis Turut ◽  
Nuran Güzel
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Fractional Differential

Two tecHniques were implemented, the Adomian decomposition method (ADM) and multivariate Padé approximation (MPA), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM), then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.

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