Solution of Nonlinear Fractional Diffusion Equation Using Similarity Transform and Homotopy Analysis Method

2015 ◽  
Vol 7 (1) ◽  
pp. 22
Author(s):  
S. Das ◽  
P. Kar ◽  
V. Mishra
Keyword(s):  
Diffusion Equation ◽  
Homotopy Analysis Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Analysis Method ◽  
Similarity Transform ◽  
Homotopy Analysis

On the Approximate Solution of Caputo-Riesz-Feller Fractional Diffusion Equation

2020 ◽  
pp. 224-244
Author(s):  
Samir Shamseldeen ◽  
Ahmed Elsaid ◽  
Seham Madkour
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Optimal Homotopy Analysis Method ◽  
Homotopy Analysis ◽  
Fractional Time Derivative ◽  
Complex Exponential ◽  
Fractional Equation

In this work, a space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative is introduced. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. Also, a very useful Riesz-Feller fractional derivative is proved; the property is essential in applying iterative methods specially for complex exponential and/or real trigonometric functions. The analytic series solution of the problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to highlight the effect of changing the fractional derivative parameters on the behavior of the obtained solutions. The results in this work are originally extracted from the author's work.

scholarly journals Analytical solution of non-linear fractional diffusion equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Obaid Alqahtani
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Approximate Analytical Solution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Transform Method ◽  
Numerical Examples ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution

AbstractIn this paper, we obtain an approximate/analytical solution of nonlinear fractional diffusion equation using the q-homotopy analysis transform method. The existence and uniqueness of the solution for this problem are also derived. Further, the applicability of the model is discussed based on graphical results and numerical examples.

scholarly journals Semianalytic Solution of Space-Time Fractional Diffusion Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
A. Elsaid ◽  
S. Shamseldeen ◽  
S. Madkour
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Optimal Homotopy Analysis Method ◽  
Homotopy Analysis ◽  
Time Fractional Diffusion Equation ◽  
Fractional Time Derivative

We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.

A Numerical Study of the Nonlinear Reaction-Diffusion Equation with Different Type of Absorbent Term by Homotopy Analysis Method

2012 ◽  
Vol 67 (10-11) ◽  
pp. 621-627 ◽  
Author(s):  
Praveen Kumar Gupta ◽  
Swati Verma
Keyword(s):  
Diffusion Equation ◽  
Homotopy Analysis Method ◽  
Numerical Study ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Analysis Method ◽  
Nonlinear Reaction ◽  
Nonlinear Reaction Diffusion Equation ◽  
Deformation Equation ◽  
Homotopy Analysis

In this paper, based on the homotopy analysis method (HAM), a new powerful algorithm is used for the solution of the nonlinear reaction-diffusion equation. The algorithm presents the procedure of constructing a set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of the solution series by introducing an auxiliary parameter h. The solutions of the problem of presence and absence of absorbent term and external force for different particular cases are presented graphically.

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