scholarly journals Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Diffusion Equation ◽  
Elliptic Operator ◽  
Fractional Power ◽  
Finite Element Approximation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Evolutionary Equation ◽  
Element Approximation ◽  
First Order ◽  
Space Fractional Diffusion Equation

AbstractAn unsteady problem is considered for a space-fractional diffusion equation in abounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.

scholarly journals An inverse problem for a fractional diffusion equation with fractional power type nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Li Li
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Approximation Property ◽  
Fractional Power ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Well Posedness ◽  
Power Type ◽  
First Order ◽  
Dirichlet To Neumann Map

<p style='text-indent:20px;'>We study the well-posedness of a semi-linear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.</p>

scholarly journals Numerical Solution of Time-Dependent Problems with Fractional Power Elliptic Operator

2018 ◽  
Vol 18 (1) ◽  
pp. 111-128 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Elliptic Operator ◽  
Fractional Power ◽  
Finite Element Approximation ◽  
Evolutionary Equation ◽  
Quadrature Formulas ◽  
Element Approximation ◽  
Discrete Problem ◽  
Time Standard ◽  
Time Dependent Problems ◽  
Fractional Equation

AbstractAn unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Padé-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.

scholarly journals NUMERICAL SOLVING UNSTEADY SPACE-FRACTIONAL PROBLEMS WITH THE SQUARE ROOT OF AN ELLIPTIC OPERATOR

2016 ◽  
Vol 21 (2) ◽  
pp. 220-238 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Elliptic Operator ◽  
Finite Element Approximation ◽  
Second Order ◽  
Evolutionary Equation ◽  
Element Approximation ◽  
Square Root ◽  
Dimensional Problem ◽  
Order Accuracy ◽  
First Order ◽  
Fractional Equation

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two- level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.

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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