THE NEW ADM–PADÉ TECHNIQUE FOR THE GENERALIZED EMDEN–FOWLER EQUATIONS

2010 ◽  
Vol 24 (12) ◽  
pp. 1237-1254 ◽  
Author(s):  
HONGMEI CHU ◽  
YINPING LIU
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Padé Approximant ◽  
Convergence Domain ◽  
Adomian Polynomials ◽  
Solution Series ◽  
Pade Approximant ◽  
Adomian Decomposition ◽  
New Type

In this paper, the Emden–Fowler equations are investigated by employing the Adomian decomposition method (ADM) and the Padé approximant. By using the new type of Adomian polynomials proposed by Randolph C. Rach in 2008, our obtained solution series converges much faster than the regular ADM solution of the same order. Meanwhile, we note that the solutions obtained by using the new ADM–Padé technique have much higher accuracy and larger convergence domain than those obtained by using the regular ADM together with the Padé technique. Finally, comparison of our new obtained solutions are given with those existing exact ones graphically to illustrate the validity and the promising potential of the new ADM–Padé technique for solving nonlinear problems.

Adomian Decomposition Method for Approximating the Solutions of the Bidirectional Sawada-Kotera Equation

2010 ◽  
Vol 65 (8-9) ◽  
pp. 658-664 ◽  
Author(s):  
Xian-Jing Lai ◽  
Xiao-Ou Cai
Keyword(s):  
Decomposition Method ◽  
Initial Conditions ◽  
Soliton Solutions ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Absolute Error ◽  
Solitary Wave Solutions ◽  
Adomian Polynomials ◽  
Wave Solutions ◽  
Adomian Decomposition

In this paper, the decomposition method is implemented for solving the bidirectional Sawada- Kotera (bSK) equation with two kinds of initial conditions. As a result, the Adomian polynomials have been calculated and the approximate and exact solutions of the bSK equation are obtained by means of Maple, such as solitary wave solutions, doubly-periodic solutions, two-soliton solutions. Moreover, we compare the approximate solution with the exact solution in a table and analyze the absolute error and the relative error. The results reported in this article provide further evidence of the usefulness of the Adomian decomposition method for obtaining solutions of nonlinear problems

A New Algorithm on the Solutions of Forced Convective Heat Transfer in a Semi-Infinite Flat Plate

2011 ◽  
Vol 27 (1) ◽  
pp. 63-69 ◽  
Author(s):  
P.-Y. Tsai ◽  
C.-K. Chen
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Flat Plate ◽  
Decomposition Method ◽  
Thermal Boundary Layer ◽  
Adomian Decomposition Method ◽  
Padé Approximant ◽  
Temperature Fields ◽  
Pade Approximant ◽  
Adomian Decomposition

ABSTRACTIn this paper, a new algorithm is proposed to solve the velocity and temperature fields in the thermal boundary layer flow over a semi-infinite flat plate. Both the flow and heat transfer solutions are calculated accurately by the Laplace Adomian decomposition method, Padé approximant and the optimal design concept. The Laplace Adomian decomposition method (LADM) is a combination of the numerical Laplace transform algorithm with the Adomian decomposition method (ADM). A hybrid method of the LADM combined with the Padé approximant, named the LADM-Padé approximant technique, is introduced to solve the thermal boundary layer problems directly without any small parameter assumptions, linearizatons or transformations of the boundary value problems to a pair of initial value problems. The LADM-Padé approximant solutions here in are given to show the accuracy in comparison with different method solutions.

scholarly journals Kamal Adomian Decomposition Method for Solving Nonlinear Wave-Like Equation with Variable Coefficients

2019 ◽  
pp. 1-11
Author(s):  
Azhari Ahmad
Keyword(s):  
Nonlinear Wave ◽  
Decomposition Method ◽  
Nonlinear Term ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Variable Coefficients ◽  
The Other ◽  
Transform Method ◽  
Adomian Polynomials ◽  
Adomian Decomposition

In this paper, we applied a new method for solving nonlinear wave-like equation with variable coefficients , when  the exact solution has a closed form. This method is Kamal Adomian De-composition Method (KADM). The Kamal decomposition method is a combined form of the Kamal transform method and the Adomian decomposition method [1,2,3]. The nonlinear term can easily be handled with the help of Adomian polynomials which is considered to be a significant advantage of this technique over the other methods. The results reveal that the Kamal decomposition method is very efficient, simple and can be applied to other nonlinear problems.

scholarly journals Solving nonlinear and non-autonomous ODEs systems by the ADM using a new several-variables Adomian polynomials

2021 ◽  
Author(s):  
Idriss Noureddine Zaouagui ◽  
Toufik Badredine
Keyword(s):  
Taylor Series ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Differential Operators ◽  
Adomian Decomposition Method ◽  
Initial Value ◽  
Adomian Polynomials ◽  
Solution Series ◽  
Several Variables ◽  
Adomian Decomposition

In this work, we adapted another time the Adomian decomposition method for solving nonlinear and non-autonomous ODEs systems. Therefore, our expressions of the Adomian polynomials are determined for a several-variable differential operators. The solution series is shown that it stay coincide with the Taylor series. Thus new conditions of convergence have been established, some systemes has been solved by ADM using Maple 2020. keywords Adomian decomposition method, Adomian polynomials, ODEs systems, initial value problems, several-variables differential operators. Classification 26B12, 34L30, 47E05, 65B10, 65L05, 65L80

scholarly journals Comment on “Adomian Decomposition Method for a Class of Nonlinear Problems”

2013 ◽  
Vol 2013 ◽  
pp. 1-3
Author(s):  
S. Dalvandpour ◽  
A. Motamedinasab
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Adomian Decomposition

Sánchez Cano in his paper “Adomian Decomposition Method for a Class of Nonlinear Problems” in application part pages 8, 9, and 10 had made some mistakes in context; in this paper we correct them.

scholarly journals Approximate Solution of Riccati Differential Equation via Modified Greens Decomposition Method

2020 ◽  
Vol 70 (4) ◽  
pp. 419-424
Author(s):  
Amit Ujlayan ◽  
Mohit Arya
Keyword(s):  
Numerical Solution ◽  
Decomposition Method ◽  
Padé Approximation ◽  
Medical Science ◽  
Adomian Decomposition Method ◽  
Specific Class ◽  
Pade Approximation ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Adomian Decomposition

Riccati differential equations (RDEs) plays important role in the various fields of defence, physics, engineering, medical science, and mathematics. A new approach to find the numerical solution of a class of RDEs with quadratic nonlinearity is presented in this paper. In the process of solving the pre-mentioned class of RDEs, we used an ordered combination of Green’s function, Adomian’s polynomials, and Pade` approximation. This technique is named as green decomposition method with Pade` approximation (GDMP). Since, the most contemporary definition of Adomian polynomials has been used in GDMP. Therefore, a specific class of Adomian polynomials is used to advance GDMP to modified green decomposition method with Pade` approximation (MGDMP). Further, MGDMP is applied to solve some special RDEs, belonging to the considered class of RDEs, absolute error of the obtained solution is compared with Adomian decomposition method (ADM) and Laplace decomposition method with Pade` approximation (LADM-Pade`). As well, the impedance of the method emphasised with the comparative error tables of the exact solution and the associated solutions with respect to ADM, LADM-Pade`, and MGDMP. The observation from this comparative study exhibits that MGDMP provides an improved numerical solution in the given interval. In spite of this, generally, some of the particular RDEs (with variable coefficients) cannot be easily solved by some of the existing methods, such as LADM-Pade` or Homotopy perturbation methods. However, under some limitations, MGDMP can be successfully applied to solve such type of RDEs.

scholarly journals A Modified Adomian Decomposition Method for Solving Higher-Order Singular Boundary Value Problems

2010 ◽  
Vol 65 (12) ◽  
pp. 1093-1100 ◽  
Author(s):  
Weonbae Kim ◽  
Changbum Chun
Keyword(s):  
Boundary Value Problems ◽  
Decomposition Method ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Higher Order ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Adomian Polynomials ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition

In this paper, we present a reliable modification of the Adomian decomposition method for solving higher-order singular boundary value problems. He’s polynomials are also used to overcome the complex and difficult calculation of Adomian polynomials occurring in the application of the Adomian decomposition method. Numerical examples are given to illustrate the accuracy and efficiency of the presented method, revealing its reliability and applicability in handling the problems with singular nature.

EXACT SOLUTION OF VAN DER POL NONLINEAR OSCILLATORS ON FINITE DOMAIN BY PADE APPROXIMANT AND ADOMIAN DECOMPOSITION METHODS

2021 ◽  
Vol 10 (6) ◽  
pp. 2755-2766
Author(s):  
E.U. Agom ◽  
F.O. Ogunfiditimi
Keyword(s):  
Exact Analytical Solution ◽  
Adomian Decomposition Method ◽  
Nonlinear Oscillators ◽  
Decomposition Methods ◽  
Padé Approximant ◽  
Finite Domain ◽  
Van Der Pol ◽  
Pade Approximant ◽  
Adomian Decomposition ◽  
The Given

This paper is concerned with a thorough investigation in achieving exact analytical solution for the Van der Pol (VDP) nonlinear oscillators models via Adomian decomposition method (ADM). The models are nonlinear time dependent second order ordinary differential equations. ADM has already been applied, in existing literatures, to obtain approximate results. But, we adapt the method by adjusting the source term; a procedure that is base on the asymptotic Taylor's series expansion on the term that would have resulted to proliferation of terms during the invertible process. Then, the rational Pade Approximant is applied to clarify and get a better understanding of the uniqueness and convergence of our findings. Two models were used as illustrations and their result pictured to indicate their behaviour in the given domains. And, we found that the adaptation on the models yielded exact results which were further displayed in constructed tables.

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