scholarly journals DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION

Fractals ◽  
2004 ◽  
Vol 12 (01) ◽  
pp. 23-32 ◽  
Author(s):  
MARK NABER
Keyword(s):  
Diffusion Equation ◽  
Time Dependence ◽  
Lower Bounds ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Decay Rates ◽  
Upper And Lower Bounds ◽  
Decay Properties ◽  
Distributed Order

A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper, sub-diffusive cases are considered. That is, the order of the time derivative ranges from zero to one. The equation is solved for Dirichlet, Neumann and Cauchy boundary conditions. The time dependence for each of the three cases is found to be a functional of the diffusion parameter. This functional is shown to have decay properties. Upper and lower bounds are computed for the functional. Examples are also worked out for comparative decay rates.

NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341001 ◽  
Author(s):  
LEEVAN LING ◽  
MASAHIRO YAMAMOTO
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Long Time ◽  
Liouville Fractional Derivative ◽  
Time Fractional Diffusion Equation

We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.

A High Order Finite Difference/Spectral Approximations to the Time Fractional Diffusion Equations

2014 ◽  
Vol 875-877 ◽  
pp. 781-785 ◽  
Author(s):  
Jun Ying Cao ◽  
Chuan Ju Xu ◽  
Zi Qiang Wang
Keyword(s):  
Diffusion Equation ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Polynomial Degree ◽  
Fractional Diffusion Equations ◽  
Spectral Approximations ◽  
High Order Finite Difference

In this paper, we consider the numerical solution of a time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order α, with 03-α+N-m) , where Δt,N and m are the time step size, the polynomial degree and the regularity of the exact solution, respectively.

scholarly journals Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time

2021 ◽  
Vol 6 (11) ◽  
pp. 12114-12132
Author(s):  
Shuang-Shuang Zhou ◽  
◽  
Saima Rashid ◽  
Asia Rauf ◽  
Khadija Tul Kubra ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Spectral Problem ◽  
Direct Problem ◽  
Fractional Derivatives ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value ◽  
The Given

<abstract><p>For a multi-term time-fractional diffusion equation comprising Hilfer fractional derivatives in time variables of different orders between $ 0 $ and $ 1 $, we have studied two problems (direct problem and inverse source problem). The spectral problem under consideration is self-adjoint. The solution to the given direct and inverse source problems is formulated utilizing the spectral problem. For the solution of the given direct problem, we proposed existence, uniqueness, and stability results. The existence, uniqueness, and consistency effects for the solution of the given inverse problem were addressed, as well as an inverse source for recovering space-dependent source term at certain $ T $. For the solution of the challenges, we proposed certain relevant cases.</p></abstract>

Inverse source problem for a distributed-order time fractional diffusion equation

2020 ◽  
Vol 28 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Xiaoliang Cheng ◽  
Lele Yuan ◽  
Kewei Liang
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Series Representation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order ◽  
Regularized Solution

AbstractThis paper studies an inverse source problem for a time fractional diffusion equation with the distributed order Caputo derivative. The space-dependent source term is recovered from a noisy final data. The uniqueness, ill-posedness and a conditional stability for this inverse source problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization method. Further, based on the series representation of the regularized solution, we give convergence rates of the regularized solution under an a-priori and an a-posteriori regularization parameter choice rule. With an adjoint technique for computing the gradient of the regularization functional, the conjugate gradient method is applied to reconstruct the space-dependent source term. Two numerical examples illustrate the effectiveness of the proposed method.

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