scholarly journals New Iterative Method for Fractional Gas Dynamics and Coupled Burger’s Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Mohamed S. Al-luhaibi
Keyword(s):  
Iterative Method ◽  
Analytical Solutions ◽  
Gas Dynamics ◽  
Time Derivative ◽  
Explicit Solutions ◽  
Fractional Partial Differential Equations ◽  
Approximate Analytical Solutions ◽  
Fractional Time Derivative ◽  
Burger's Equations ◽  
New Iterative Method

This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger’s equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterative method is easy to implement and accurate when applied to time-fractional partial differential equations.

scholarly journals Application of He’s Homotopy Perturbation Method to Fractional Diffusion Equations

2010 ◽  
Vol 65 (1-2) ◽  
pp. 53-58 ◽  
Author(s):  
Subir Das ◽  
Praveen Kumar Gupta ◽  
Vinod Sankar Pandey ◽  
Kabindra Nath Rai
Keyword(s):  
Perturbation Method ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Time Derivative ◽  
Fractional Diffusion Equations ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
And Performance ◽  
Fractional Time Derivative

AbstractIn this paper, the approximate analytical solutions of a general diffusion equation with fractional time derivative in the presence of a linear external force are obtained with the help of the homotopy perturbation method (HPM). The explicit solutions of the problem for the initial condition as a function of x have been obtained. It reveals that a few iterations are needed to obtain accurate approximate analytical solutions. The numerical calculations are carried out when the initial conditions are like exponential and periodic functions and the results are depicted through graphs. The examples prove that the method is extremely effective due to its simplistic approach and performance.

Approximate analytical solutions of strongly nonlinear fractional BBM-Burger’s equations with dissipative term

2014 ◽  
Vol 8 ◽  
pp. 7715-7726 ◽  
Author(s):  
Mukiawa Edwin Soh ◽  
Cyril Dennis Enyi ◽  
Olaniyi Samuel Iyiola ◽  
Johnson Daddy Audu
Keyword(s):  
Analytical Solutions ◽  
Dissipative Term ◽  
Strongly Nonlinear ◽  
Approximate Analytical Solutions ◽  
Burger's Equations

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.

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.

Reliable analysis for time-fractional nonlinear differential difference equations

2013 ◽  
Vol 3 (4) ◽  
Author(s):  
Mithilesh Singh ◽  
R. Prajapati
Keyword(s):  
Analytical Solution ◽  
Difference Equations ◽  
Present Method ◽  
Approximate Analytical Solution ◽  
Initial Conditions ◽  
Time Derivative ◽  
Explicit Solutions ◽  
Initial Condition ◽  
Hyperbolic Functions ◽  
Fractional Time Derivative

AbstractIn this study, we used HPM to determine the approximate analytical solution of nonlinear differential difference equations of fractional time derivative. By using initial conditions, the explicit solutions of the coupled nonlinear differential difference equations have been derived which demonstrate the effectiveness, potentiality and validity of the method in reality. The present method is very effective and powerful to determine the solution of system of non-linear DDE. The numerical calculations are carried out when the initial condition in the form of hyperbolic functions and the results are shown through the graphs.

An Approximate Analytical Solution of the Fractional Diffusion Equation with Absorbent Term and External Force by Homotopy Perturbation Method

2010 ◽  
Vol 65 (3) ◽  
pp. 182-190 ◽  
Author(s):  
Subir Das ◽  
Praveen Kumar Gupta
Keyword(s):  
Diffusion Equation ◽  
Perturbation Method ◽  
External Force ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Time Derivative ◽  
Approximate Solutions ◽  
Mathematical Tool ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions

In the present paper, the approximate analytical solutions of a general diffusion equation with fractional time derivative in the presence of an absorbent term and a linear external force are obtained with the help of the powerful homotopy perturbation method (HPM). By using initial values, the approximate analytical solutions of the equation are derived. The results are deduced for different particular cases. The numerical results show that only a few iterations are needed to obtain accurate approximate solutions and these are presented graphically. The presented method is extremely simple, concise, and highly efficient as a mathematical tool in comparison with the other existing techniques.

scholarly journals New Approximate Analytical Solutions of the Falkner-Skan Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Beong In Yun
Keyword(s):  
Iterative Method ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Correction Method ◽  
Approximate Analytical Solutions ◽  
Adomian Decomposition ◽  
Number Of Iterations

We propose an iterative method for solving the Falkner-Skan equation. The method provides approximate analytical solutions which consist of coefficients of the previous iterate solution. By some examples, we show that the presented method with a small number of iterations is competitive with the existing method such as Adomian decomposition method. Furthermore, to improve the accuracy of the proposed method, we suggest an efficient correction method. In practice, for some examples one can observe that the correction method results in highly improved approximate solutions.

The Homotopy Analysis Method for Fractional Cauchy Reaction-Diffusion Problems

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Subir Das ◽  
Rajnesh Kumar ◽  
Praveen Kumar Gupta
Keyword(s):  
Homotopy Analysis Method ◽  
Time Derivative ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Analysis Method ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Fractional Time Derivative ◽  
Linear Interactions ◽  
Diffusion Problems

In this article, homotopy analysis method is successfully applied to obtain the approximate analytical solutions of the characteristic Cauchy reaction-diffusion equation with fractional time derivative. The beauty of the article is the wonderful application of Caputo fractional order time derivative. The linear interactions of the merging populations are examined using perturbation theory and the method of matched asymptotic expansions. The solutions of the problem for different particular cases are presented graphically.

scholarly journals Approximate Analytical Solutions of Fractional Perturbed Diffusion Equation by Reduced Differential Transform Method and the Homotopy Perturbation Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Zhoujin Cui ◽  
Zisen Mao ◽  
Sujuan Yang ◽  
Pinneng Yu
Keyword(s):  
Perturbation Method ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Time Derivative ◽  
Differential Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
Differential Transform ◽  
Reduced Differential Transform Method

The approximate analytical solutions of differential equations with fractional time derivative are obtained with the help of a general framework of the reduced differential transform method (RDTM) and the homotopy perturbation method (HPM). RDTM technique does not require any discretization, linearization, or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology (RDTM) with some known technique (HPM) shows that the present approach is effective and powerful. The numerical calculations are carried out when the initial conditions in the form of periodic functions and the results are depicted through graphs. The two different cases have studied and proved that the method is extremely effective due to its simplistic approach and performance.

scholarly journals Solving Fractional Partial Differential Equations with Corrected Fourier Series Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Nor Hafizah Zainal ◽  
Adem Kılıçman
Keyword(s):  
Fourier Series ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Finite Domain ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fourier Series Method ◽  
Fractional Time Derivative

The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.

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