Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate

2012 ◽  
Vol 28 (10) ◽  
pp. 105010 ◽  
Author(s):  
Masahiro Yamamoto ◽  
Ying Zhang
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Carleman Estimate ◽  
Conditional Stability ◽  
Zeroth Order ◽  
Order Coefficient

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.

Tikhonov Regularisation Method for Simultaneous Inversion of the Source Term and Initial Data in a Time-Fractional Diffusion Equation

2015 ◽  
Vol 5 (3) ◽  
pp. 273-300 ◽  
Author(s):  
Zhousheng Ruan ◽  
Jerry Zhijian Yang ◽  
Xiliang Lu
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Smoothness Assumption ◽  
Tikhonov Regularisation ◽  
Time Fractional Diffusion Equation ◽  
Regularisation Method

AbstractThe inverse problem of identifying the time-independent source term and initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated into an operator equation based on the Fourier method. Under a certain smoothness assumption, conditional stability is established. A standard Tikhonov regularisation method is proposed to solve the inverse problem. Furthermore, the convergence rate is given for an a priori and a posteriori regularisation parameter choice rule, respectively. Several numerical examples, including one-dimensional and two-dimensional cases, show the efficiency of our proposed method.

Carleman estimate for a fractional diffusion equation with half order and application

2011 ◽  
Vol 90 (9) ◽  
pp. 1355-1371 ◽  
Author(s):  
Xiang Xu ◽  
Jin Cheng ◽  
Masahiro Yamamoto
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Carleman Estimate

Solving a backward problem for a distributed-order time fractional diffusion equation by a new adjoint technique

2020 ◽  
Vol 28 (4) ◽  
pp. 471-488
Author(s):  
Lele Yuan ◽  
Xiaoliang Cheng ◽  
Kewei Liang
Keyword(s):  
Diffusion Equation ◽  
Convergence Rates ◽  
Series Representation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Initial Condition ◽  
Backward Problem ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order

AbstractThis paper studies a backward problem for a time fractional diffusion equation, with the distributed order Caputo derivative, of determining the initial condition from a noisy final datum. The uniqueness, ill-posedness and a conditional stability for this backward problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization. Based on the series representation of the regularized solution, we give convergence rates under an a-priori and an a-posteriori regularization parameter choice rule. With a new adjoint technique to compute the gradient of the functional, the conjugate gradient method is applied to reconstruct the initial condition. Numerical examples in one- and two-dimensional cases illustrate the effectiveness 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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