A Study of Some Nonlinear Partial  Differential Equations by Using Adomian  Decomposition Method and Variational  Iteration Method

In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

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Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations

Journal of Applied Mathematics ◽

10.1155/2013/157956 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 6

Author(s):

Fukang Yin ◽

Junqiang Song ◽

Xiaoqun Cao ◽

Fengshun Lu

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Iteration Method ◽

Nonlinear Partial Differential Equations ◽

Variational Iteration Method ◽

Approximate Solutions ◽

Legendre Wavelets ◽

Partial Differential ◽

Variational Iteration ◽

Block Pulse Functions

This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs). The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.

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A modification of He's variational iteration method by Taylor's series for solving second order nonlinear partial differential equations

Journal of Interpolation and Approximation in Scientific Computing ◽

10.5899/2013/jiasc-00049 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7

Author(s):

Majid Amirfakhrian ◽

Somayeh Keighobadi

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Iteration Method ◽

Nonlinear Partial Differential Equations ◽

Variational Iteration Method ◽

Second Order ◽

Partial Differential ◽

Variational Iteration ◽

He’S Variational Iteration Method ◽

Taylor’S Series

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Solution of Nonlinear Partial Differential Equations by Mixture Adomian Decomposition Method and Sumudu Transform

10.5772/intechopen.94598 ◽

2021 ◽

Author(s):

Tarig M. Elzaki ◽

Shams E. Ahmed

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Decomposition Method ◽

Nonlinear Partial Differential Equations ◽

Adomian Decomposition Method ◽

Convergent Series ◽

Partial Differential ◽

Adomian Decomposition ◽

Sumudu Transform

This chapter is fundamentally centering on the application of the Adomian decomposition method and Sumudu transform for solving the nonlinear partial differential equations. It has instituted some theorems, definitions, and properties of Adomian decomposition and Sumudu transform. This chapter is an elegant combination of the Adomian decomposition method and Sumudu transform. Consequently, it provides the solution in the form of convergent series, then, it is applied to solve nonlinear partial differential equations.

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Solution of Some Systems of Nonlinear Partial Differential Equations by Variational Iteration Method

Journal of Algorithms & Computational Technology ◽

10.1260/1748-3018.4.1.1 ◽

2010 ◽

Vol 4 (1) ◽

pp. 1-14

Author(s):

Mohammad Taghi Darvishi ◽

Farzad Khani ◽

Somayeh Hamedi-Nezhad

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Iteration Method ◽

Nonlinear Partial Differential Equations ◽

Variational Iteration Method ◽

Partial Differential ◽

Variational Iteration

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Solusi Persamaan Diferensial Fraksional Riccati Menggunakan Adomian Decomposition Method dan Variational Iteration Method

Matematika ◽

10.29313/jmtm.v18i1.4931 ◽

2019 ◽

Vol 18 (1) ◽

Author(s):

Muhamad Deni Johansyah ◽

Herlina Napitupulu ◽

Erwin Harahap ◽

Ira Sumiati ◽

Asep K. Supriatna

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Fractional Differential Equations ◽

Iteration Method ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Variational Iteration Method ◽

Variational Iteration ◽

Adomian Decomposition ◽

Fractional Differential

Abstrak. Pada umumnya orde dari persamaan diferensial adalah bilangan asli, namun orde pada persamaan diferensial dapat dibentuk menjadi orde pecahan yang disebut persamaan diferensial fraksional. Paper ini membahas persamaan diferensial fraksional Riccati dengan orde diantara nol dan satu, dan koefisien konstan. Metode numerik yang digunakan untuk mendapatkan solusi dari persamaan diferensial fraksional Riccati adalah Adomian Decomposition Method (ADM) dan Variational Iteration Method (VIM). Tujuan dari paper ini adalah untuk memperluas penerapan ADM dan VIM dalam menyelesaikan persamaan diferensial fraksional Riccati nonlinear dengan turunan Caputo. Perbandingan solusi yang diperoleh menunjukkan bahwa VIM adalah metode yang lebih sederhana untuk mencari solusi persamaan diferensial fraksional Riccati nonlinier dengan orde antara nol dan satu, kemudian hasil yang diperoleh disajikan dalam bentuk grafik.Kata kunci: diferensial, fraksional, riccati, adomian dekomposisiThe solution of Riccati Fractional Differential Equation using Adomian Decomposition methodAbstract. Generally, the order of differential equations is a natural numbers, but this order can be formed into fractional, called as fractional differential equations. In this paper, the Riccati fractional differential equations with order between zero and one, and constant coefficient is discussed. The numerical methods used to obtain solutions from Riccati fractional differential equations are the Adomian Decomposition Method (ADM) and Variational Iteration Method (VIM). The aim of this paper is to expand the application of ADM and VIM in solving nonlinear Riccati fractional differential equations with Caputo derivatives. The comparison of the obtained solutions shows that VIM is simpler method for finding solutions to Riccati nonlinear fractional differential equations with order between zero and one. The obtained results are presented graphically.Keywords: riccati, fractional, differential, adomian, decomposition

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

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v12i8.5075 ◽

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.

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An accelerated solution for some classes of nonlinear partial differential equations

Journal of the Egyptian Mathematical Society ◽

10.1186/s42787-021-00116-9 ◽

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.

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