Convergence Analysis of a High Order Schema for the Ordinary Fractional Diffusion Equation

2014 ◽  
Vol 602-605 ◽  
pp. 3088-3091
Author(s):  
Jun Ying Cao ◽  
Zi Qiang Wang
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Convergence Analysis ◽  
Fractional Differential Equations ◽  
Fractional Diffusion ◽  
High Order ◽  
Fractional Diffusion Equation ◽  
Block Method ◽  
Fractional Differential ◽  
Block Approach

The block-by-block method extended by Kumar and Agrawal to fractional differential equations. Cao et al. proposed a high order schema which is based on an improved block-by-block approach, which consists in finding 4 unknowns simultaneously at each step block through solving a 4 × 4 system. We rigorously analytically prove that this method is convergent with order for , and order 6 for .

NONEXISTENCE OF SOLUTIONS RESULTS FOR CERTAIN FRACTIONAL DIFFERENTIAL EQUATIONS

2011 ◽  
Vol 16 (3) ◽  
pp. 488-497
Author(s):  
Mohamed Berbiche
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Fractional Differential Equations ◽  
Weak Solutions ◽  
Sufficient Conditions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Convective Term ◽  
Nonexistence Of Solutions ◽  
Fractional Differential

This paper is meant to establish sufficient conditions for the nonexistence of weak solutions to nonlinear fractional diffusion equation in space and time with nonlinear convective term. The Fujita exponent is determined.

Application of low-dimensional finite element method to fractional diffusion equation

2014 ◽  
Vol 05 (04) ◽  
pp. 1450022 ◽  
Author(s):  
Jincun Liu ◽  
Hong Li ◽  
Zhichao Fang ◽  
Yang Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Fractional Differential Equations ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Fractional Differential ◽  
Low Dimensional ◽  
Element Method

Classical finite element method (FEM) has been applied to solve some fractional differential equations, but its scheme has too many degrees of freedom. In this paper, a low-dimensional FEM, whose number of basis functions is reduced by the theory of proper orthogonal decomposition (POD) technique, is proposed for the time fractional diffusion equation in two-dimensional space. The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved. Moreover, error estimation of the method is obtained. Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

scholarly journals Symmetry Group Classification and Conservation Laws of the Nonlinear Fractional Diffusion Equation with the Riesz Potential

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 178 ◽  
Author(s):  
Nikita S. Belevtsov ◽  
Stanislav Yu. Lukashchuk
Keyword(s):  
Diffusion Equation ◽  
Conservation Laws ◽  
Symmetry Group ◽  
Fractional Differential Equations ◽  
Riesz Potential ◽  
Fractional Diffusion ◽  
Group Classification ◽  
Fractional Diffusion Equation ◽  
Space Fractional Diffusion Equation ◽  
Fractional Differential

Symmetry properties of a nonlinear two-dimensional space-fractional diffusion equation with the Riesz potential of the order α ∈ ( 0 , 1 ) are studied. Lie point symmetry group classification of this equation is performed with respect to diffusivity function. To construct conservation laws for the considered equation, the concept of nonlinear self-adjointness is adopted to a certain class of space-fractional differential equations with the Riesz potential. It is proved that the equation in question is nonlinearly self-adjoint. An extension of Ibragimov’s constructive algorithm for finding conservation laws is proposed, and the corresponding Noether operators for fractional differential equations with the Riesz potential are presented in an explicit form. To illustrate the proposed approach, conservation laws for the considered nonlinear space-fractional diffusion equation are constructed by using its Lie point symmetries.

scholarly journals Numerical Solution of Fractional Diffusion Equation Model for Freezing in Finite Media

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
R. S. Damor ◽  
Sushil Kumar ◽  
A. K. Shukla
Keyword(s):  
Diffusion Equation ◽  
Interface Velocity ◽  
Biological Tissues ◽  
Fractional Diffusion ◽  
Equation Model ◽  
Fractional Diffusion Equation ◽  
Heat Transfer Analysis ◽  
Engineering Sciences ◽  
Space Fractional Diffusion Equation ◽  
Fractional Differential

Phase change problems play very important role in engineering sciences including casting of nuclear waste materials, vivo freezing of biological tissues, solar collectors and so forth. In present paper, we propose fractional diffusion equation model for alloy solidification. A transient heat transfer analysis is carried out to study the anomalous diffusion. Finite difference method is used to solve the fractional differential equation model. The temperature profiles, the motion of interface, and interface velocity have been evaluated for space fractional diffusion equation.

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