scholarly journals Similarity Solution for Fractional Diffusion Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Ai-Ping Guo ◽  
Wen-Zai Yun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Probability Density Function ◽  
Hausdorff Dimension ◽  
Diffusion Equation ◽  
Similarity Solution ◽  
Mellin Transform ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Fractal Media

Fractional diffusion equation in fractal media is an integropartial differential equation parametrized by fractal Hausdorff dimension and anomalous diffusion exponent. In this paper, the similarity solution of the fractional diffusion equation was considered. Through the invariants of the group of scaling transformations we derived the integro-ordinary differential equation for the similarity variable. Then by virtue of Mellin transform, the probability density functionp(r,t), which is just the fundamental solution of the fractional diffusion equation, was expressed in terms of Fox functions.

scholarly journals Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation

2020 ◽  
Vol 4 (3) ◽  
pp. 39
Author(s):  
Rafał Brociek ◽  
Agata Chmielowska ◽  
Damian Słota
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Parameter Identification ◽  
Fractional Diffusion ◽  
Riesz Space ◽  
Fractional Diffusion Equation ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Space Fractional Diffusion Equation ◽  
Swarm Intelligence Algorithm

This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm’s accuracy.

scholarly journals Similarity Solutions for Multiterm Time-Fractional Diffusion Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
A. Elsaid ◽  
M. S. Abdel Latif ◽  
M. Maneea
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Series Solution ◽  
Error Function ◽  
Fractional Diffusion ◽  
Similarity Solutions ◽  
Fractional Diffusion Equation ◽  
Power Series Solution ◽  
Time Fractional Diffusion Equation ◽  
Similarity Method

Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval(0,1]and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on the obtained results, we propose a definition for a multiterm error function with generalized coefficients.

scholarly journals Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1005
Author(s):  
Andang Sunarto ◽  
Praveen Agarwal ◽  
Jackel Vui Lung Chew ◽  
Jumat Sulaiman
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear System ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Numerical Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential

In this study, the numerical solution of a space-fractional parabolic partial differential equation was considered. The investigation of the solution was made by focusing on the space-fractional diffusion equation (SFDE) problem. Note that the symmetry of an efficient approximation to the SFDE based on a numerical method is related to the compatibility of a discretization scheme and a linear system solver. The application of the one-dimensional, linear, unconditionally stable, and implicit finite difference approximation to SFDE was studied. The general differential equation of the SFDE was discretized using the space-fractional derivative of Caputo with a half-sweep finite difference scheme. The implicit approximation to the SFDE was formulated, and the formation of a linear system with a coefficient matrix, which was large and sparse, is shown. The construction of a general preconditioned system of equation is also presented. This study’s contribution is the introduction of a half-sweep preconditioned successive over relaxation (HSPSOR) method for the solution of the SFDE-based system of equation. This work extended the use of the HSPSOR as an efficient numerical method for the time-fractional diffusion equation, which has been presented in the 5th North American International Conference on industrial engineering and operations management in Detroit, Michigan, USA, 10–14 August 2020. The current work proposed several SFDE examples to validate the performance of the HSPSOR iterative method in solving the fractional diffusion equation. The outcome of the numerical investigation illustrated the competence of the HSPSOR to solve the SFDE and proved that the HSPSOR is superior to the standard approximation, which is the full-sweep preconditioned SOR (FSPSOR), in terms of computational complexity.

scholarly journals Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abderrazak Nabti ◽  
Ahmed Alsaedi ◽  
Mokhtar Kirane ◽  
Bashir Ahmad
Keyword(s):  
Diffusion Equation ◽  
Global Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Laplacian Operator ◽  
Nonlocal Source ◽  
Time Space ◽  
Nonexistence Of Solutions ◽  
Beta 2 ◽  
Fractional Laplacian Operator

Abstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$ u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s for $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$ ( x , t ) ∈ R N × ( 0 , ∞ ) with initial data $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$ u ( x , 0 ) = u 0 ( x ) ∈ L loc 1 ( R N ) , where $p,q,r>1$ p , q , r > 1 , $q(p+r)>q+r$ q ( p + r ) > q + r , $0<\gamma \leq 2 $ 0 < γ ≤ 2 , $0<\alpha <1$ 0 < α < 1 , $0<\beta \leq 2$ 0 < β ≤ 2 , $(-\Delta )^{\frac{\beta }{2}}$ ( − Δ ) β 2 stands for the fractional Laplacian operator of order β, the weight function $\nu (x)$ ν ( x ) is positive and singular at the origin, and $\Vert \cdot \Vert _{q}$ ∥ ⋅ ∥ q is the norm of $L^{q}$ L q space.

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