scholarly journals Existence and Uniqueness of Time-Fractional Diffusion Equation on a Metric Star Graph

2021 ◽  
pp. 25-41
Author(s):  
Vaibhav Mehandiratta ◽  
Mani Mehra ◽  
Günter Leugering
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Star Graph ◽  
Time Fractional Diffusion Equation

Identification of the diffusion coefficient in a time fractional diffusion equation

2020 ◽  
Vol 28 (2) ◽  
pp. 299-306
Author(s):  
Amir Hossein Salehi Shayegan ◽  
Ali Zakeri ◽  
Soheila Bodaghi ◽  
M. Heshmati
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Stability Result ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Cost Functional ◽  
Time Fractional Diffusion Equation ◽  
The Cost ◽  
The Uniqueness Theorem ◽  
Admissible Functions

AbstractIn this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to {{}^{C}D_{t}^{\alpha}u-\nabla\cdot(k(x)\nabla u)=f}, where the function {k=k(x)} is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k. At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given.

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