scholarly journals Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method

2018 ◽  
Vol 7 (4) ◽  
pp. 669-682 ◽  
Author(s):  
Zhousheng Ruan ◽  
◽  
Sen Zhang ◽  
Sican Xiong ◽  
◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Boundary Value ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Boundary Value Method ◽  
Time Fractional Diffusion Equation

scholarly journals Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
The Maximum Principle ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value

AbstractIn this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

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