Numerical Solution of Nonlinear Volterra Integro-Differential Equations of Fractional Order by Using Adomian Decomposition Method

2014 ◽  
Vol 635-637 ◽  
pp. 1582-1585
Author(s):  
Li Feng Wang ◽  
Yun Peng Ma ◽  
Yong Qiang Yang
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Convergence Analysis ◽  
Fractional Order ◽  
Decomposition Method ◽  
Series Solution ◽  
Adomian Decomposition Method ◽  
Computational Method ◽  
Numerical Example ◽  
Adomian Decomposition

In this work we present a computational method for for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on Adomian Decom-position Method. Convergence analysis is dependable enough to estimate the maximum absolute truncated error of the Adomian series solution. Numerical example is included to demonstrate the validity and applicability of the method.

scholarly journals Solving Some Derivative Equations Fractional Order Nonlinear Partials Using the Some Blaise Abbo Method

2021 ◽  
Vol 13 (2) ◽  
pp. 101
Author(s):  
Abdoul wassiha NEBIE ◽  
Frederic BERE ◽  
Bakari ABBO ◽  
Youssouf PARE
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Adomian Decomposition ◽  
Diffusion Convection

In this paper, we propose the solution of some nonlinear partial differential equations of  fractional order that modeled diffusion, convection and reaction problems. For the solution of these equations we will use the SBA method which is a method based on the combination of the Adomian Decomposition Method (ADM), the Picard's principle  and the method of successive approximations.   

scholarly journals A New Computational Approach for Solving Fractional Order Telegraph Equations

2021 ◽  
Vol 13 (3) ◽  
pp. 715-732
Author(s):  
A. Devi ◽  
M. Jakhar
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Series Solution ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Computational Approach ◽  
Order Problem ◽  
Adomian Decomposition ◽  
Telegraph Equations ◽  
Derivatives Of

In this work, a modified decomposition method namely Sumudu-Adomian Decomposition Method (SADM) is implemented to find the exact and approximate solutions of fractional order telegraph equations. The derivatives of fractional-order are expressed in terms of caputo operator. Some numerical examples are illustrated to examine the efficiency of the proposed technique. Solutions of fractional order telegraph equations are obtained in the form of a series solution. It is observed that the solutions of fractional order telegraph equations converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested method.

scholarly journals Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 335 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Muhammad Arif ◽  
Poom Kumam
Keyword(s):  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Third Order ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
In Series

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

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.

scholarly journals Numerical Solution of Coupled System of Nonlinear Partial Differential Equations Using Laplace-Adomian Decomposition Method

2016 ◽  
Vol 12 (8) ◽  
pp. 6530-6544
Author(s):  
Mohamed S M. Bahgat
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Coupled System ◽  
Decomposition Methods ◽  
Partial Differential ◽  
Adomian Decomposition

Aim of the paper is to investigate applications of Laplace Adomian Decomposition Method (LADM) on nonlinear physical problems. Some coupled system of non-linear partial differential equations (NLPDEs) are considered and solved numerically using LADM. The results obtained by LADM are compared with those obtained by standard and modified Adomian Decomposition Methods. The behavior of the numerical solution is shown through graphs. It is observed that LADM is an effective method with high accuracy with less number of components.

scholarly journals An accelerated solution for some classes of nonlinear partial differential equations

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
I. L. El-Kalla ◽  
E. M. Mohamed ◽  
H. A. A. El-Saka
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Series Solution ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Adomian Polynomials ◽  
Numerical Examples ◽  
Adomian Decomposition

AbstractIn this paper, we apply an accelerated version of the Adomian decomposition method for solving a class of nonlinear partial differential equations. This version is a smart recursive technique in which no differentiation for computing the Adomian polynomials is needed. Convergence analysis of this version is discussed, and the error of the series solution is estimated. Some numerical examples were solved, and the numerical results illustrate the effectiveness of this version.

scholarly journals Solutions of Volterra integral and integro-differential equations using modified Laplace Adomian decomposition method

2019 ◽  
Vol 15 (1) ◽  
pp. 5-18
Author(s):  
D. Rani ◽  
V. Mishra
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Convergence Analysis ◽  
Decomposition Method ◽  
Bernstein Polynomials ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Modified Technique ◽  
Adomian Decomposition

Abstract In this paper, an effectual and new modification in Laplace Adomian decomposition method based on Bernstein polynomials is proposed to find the solution of nonlinear Volterra integral and integro-differential equations. The performance and capability of the proposed idea is endorsed by comparing the exact and approximate solutions for three different examples on Volterra integral, integro-differential equations of the first and second kinds. The results shown through tables and figures demonstrate the accuracy of our method. It is concluded here that the non orthogonal polynomials can also be used for Laplace Adomian decomposition method. In addition, convergence analysis of the modified technique is also presented.

HYPERCHAOS IN THE FRACTIONAL-ORDER RÖSSLER SYSTEM WITH LOWEST-ORDER

2009 ◽  
Vol 19 (01) ◽  
pp. 339-347 ◽  
Author(s):  
DONATO CAFAGNA ◽  
GIUSEPPE GRASSI
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Series Solution ◽  
Adomian Decomposition Method ◽  
Point Of View ◽  
Hyperchaotic System ◽  
Adomian Decomposition ◽  
Rossler System ◽  
Fractional Differential ◽  
Rössler System

This Letter analyzes the hyperchaotic dynamics of the fractional-order Rössler system from a time-domain point of view. The approach exploits the Adomian decomposition method (ADM), which generates series solution of the fractional differential equations. A remarkable finding of the Letter is that hyperchaos occurs in the fractional Rössler system with order as low as 3.12. This represents the lowest order reported in literature for any hyperchaotic system studied so far.

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