Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Diffusion Equation

2019 ◽  
Vol 2 (1) ◽  
pp. 57-72
Author(s):  
Yuxin Zhang ◽  
Hengfei Ding
Keyword(s):  
Diffusion Equation ◽  
Numerical Algorithm ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation

scholarly journals A Directly Numerical Algorithm for a Backward Time-Fractional Diffusion Equation Based on the Finite Element Method

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Zhousheng Ruan ◽  
Zewen Wang ◽  
Wen Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Algorithm ◽  
Optimization Problem ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
The Finite Element Method ◽  
Time Fractional Diffusion Equation ◽  
Regularized Optimization

We study a backward problem for a time-fractional diffusion equation, which is formulated into a regularized optimization problem. After solving a sequence of well-posed direct problems by the finite element method, a directly numerical algorithm is proposed for solving the regularized optimization problem. In order to obtain a reasonable regularization solution, we utilize the discrepancy principle with decreasing geometric sequence to choose regularization parameters. One- and two-dimensional examples are given to verify the efficiency and stability of the proposed method.

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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