scholarly journals New Iterative Method: An Application for Solving Fractional Physical Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
A. A. Hemeda
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Fractional Order ◽  
Small Perturbation ◽  
The Other ◽  
Numerical Examples ◽  
Linear And Nonlinear ◽  
The Given ◽  
New Iterative Method

The new iterative method with a powerful algorithm is developed for the solution of linear and nonlinear ordinary and partial differential equations of fractional order as well. The analysis is accompanied by numerical examples where this method, in solving them, is used without linearization or small perturbation which con…firm the power, accuracy, and simplicity of the given method compared with some of the other methods.

scholarly journals New Iterative Method of Solving Nonlinear Equations in Fluid Mechanics

2021 ◽  
Vol 26 (3) ◽  
pp. 163-176
Author(s):  
M. Paliivets ◽  
E. Andreev ◽  
A. Bakshtanin ◽  
D. Benin ◽  
V. Snezhko
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Fluid Mechanics ◽  
Nonlinear Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Linear And Nonlinear ◽  
New Iterative Method

Abstract This paper presents the results of applying a new iterative method to linear and nonlinear fractional partial differential equations in fluid mechanics. A numerical analysis was performed to find an exact solution of the fractional wave equation and fractional Burgers’ equation, as well as an approximate solution of fractional KdV equation and fractional Boussinesq equation. Fractional derivatives of the order α are described using Caputo's definition with 0 < α ≤ 1 or 1 < α ≤ 2. A comparative analysis of the results obtained using a new iterative method with those obtained by the Adomian decomposition method showed the first method to be more efficient and simple, providing accurate results in fewer computational operations. Given its flexibility and ability to solve nonlinear equations, the iterative method can be used to solve more complex linear and nonlinear fractional partial differential equations.

scholarly journals Analytic and numerical solution for duffing equations

2016 ◽  
Vol 5 (2) ◽  
pp. 115 ◽  
Author(s):  
Majeed AL-Jawary ◽  
Sayl Abd- AL- Razaq
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Duffing Equations ◽  
Linear And Nonlinear ◽  
New Iterative Method

<p>Daftardar Gejji and Hossein Jafari have proposed a new iterative method for solving many of the linear and nonlinear equations namely (DJM). This method proved already the effectiveness in solved many of the ordinary differential equations, partial differential equations and integral equations. The main aim from this paper is to propose the Daftardar-Jafari method (DJM) to solve the Duffing equations and to find the exact solution and numerical solutions. The proposed (DJM) is very effective and reliable, and the solution is obtained in the series form with easily computed components. The software used for the calculations in this study was MATHEMATICA<sup>®</sup> 9.0.</p>

A Friendly Iterative Technique for Solving Nonlinear Integro-Differential and Systems of Nonlinear Integro-Differential Equations

2018 ◽  
Vol 15 (03) ◽  
pp. 1850016 ◽  
Author(s):  
A. A. Hemeda
Keyword(s):  
Approximate Solution ◽  
Differential Operator ◽  
Differential Equations ◽  
The Other ◽  
Iterative Technique ◽  
Restrictive Assumption ◽  
Numerical Examples ◽  
Linear And Nonlinear ◽  
Computational Procedures ◽  
Adomian’S Polynomials

In this work, a simple new iterative technique based on the integral operator, the inverse of the differential operator in the problem under consideration, is introduced to solve nonlinear integro-differential and systems of nonlinear integro-differential equations (IDEs). The introduced technique is simpler and shorter in its computational procedures and time than the other methods. In addition, it does not require discretization, linearization or any restrictive assumption of any form in providing analytical or approximate solution to linear and nonlinear equations. Also, this technique does not require calculating Adomian’s polynomials, Lagrange’s multiplier values or equating the terms of equal powers of the impeding parameter which need more computational procedures and time. These advantages make it reliable and its efficiency is demonstrated with numerical examples.

scholarly journals ANALYSIS OF TIME-FRACTIONAL BURGERS AND DIFFUSION EQUATIONS BY USING MODIFIED q-HATM

Fractals ◽  
2021 ◽  
pp. 2240012
Author(s):  
NEHAD ALI SHAH ◽  
PRAVEEN AGARWAL ◽  
JAE DONG CHUNG ◽  
SAAD ALTHOBAITI ◽  
SAMY SAYED ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Nonlinear Partial Differential Equations ◽  
Convergent Sequence ◽  
Exact Result ◽  
Diffusion Equations ◽  
Homotopy Analysis ◽  
Linear And Nonlinear ◽  
And Diffusion

In this paper, the q-homotopy analysis transform technique is implemented to analyze the solution of fractional-order Burgers and diffusion equations with the help of Caputo operator. The results of the proposed method are shown and analyzed with the help of figures. This approach is used to determine the solution in a convergent sequence and illustrate the q-homotopy analysis transform technique solutions convergence to the exact result. Several examples showed the reliability and simplicity of the technique and highlighted the significance of this work. Therefore, the proposed method is successful in investigating other fractional-order linear and nonlinear partial differential equations.

Solution of fractional partial differential equations using iterative method

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Chandradeepa Dhaigude ◽  
Vasant Nikam
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Initial Value Problems ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Fractional Partial Differential Equations ◽  
Initial Value ◽  
Diffusion Wave ◽  
Linear And Nonlinear

AbstractThe purpose of this paper is to obtain solutions for both linear and nonlinear initial value problems (IVPs) for fractional transport equations and fractional diffusion-wave equations using the iterative method.

scholarly journals An efficient new iterative method for finding exact solutions of nonlinear time-fractional partial differential equations

2011 ◽  
Vol 16 (4) ◽  
pp. 403-414 ◽  
Author(s):  
Hüseyin Koçak ◽  
Ahmet Yıldırım
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Convergent Series ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
New Iterative Method

In this paper, a new iterative method (NIM) is used to obtain the exact solutions of some nonlinear time-fractional partial differential equations. The fractional derivatives are described in the Caputo sense. The method provides a convergent series with easily computable components in comparison with other existing methods.

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