Inverse Source Problems with Final Overdetermination

2021 ◽  
pp. 65-125
Author(s):  
Alemdar Hasanov Hasanoğlu ◽  
Vladimir G. Romanov
Keyword(s):  
Inverse Source Problems

A novel coupled complex boundary method for solving inverse source problems

2014 ◽  
Vol 30 (5) ◽  
pp. 055002 ◽  
Author(s):  
Xiaoliang Cheng ◽  
Rongfang Gong ◽  
Weimin Han ◽  
Xuan Zheng
Keyword(s):  
Boundary Method ◽  
Complex Boundary ◽  
Inverse Source Problems

scholarly journals Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Pengbin Feng ◽  
Erkinjon T. Karimov
Keyword(s):  
Wave Equation ◽  
Mixed Type ◽  
Orthonormal System ◽  
Type Equation ◽  
Hyperbolic Type ◽  
Initial Time ◽  
Inverse Source Problem ◽  
Ill Posed ◽  
Inverse Source Problems ◽  
Well Posed

AbstractIn the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is generally ill-posed. However, when the hyperbolic part is time-fractional, the problem is well-posed if the end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to the space variables. Finally, we prove uniqueness and stability of certain weak solutions for the problems under consideration.

scholarly journals Initial boundary value problems for a fractional differential equation with hyper-Bessel operator

2018 ◽  
Vol 21 (1) ◽  
pp. 200-219 ◽  
Author(s):  
Fatma Al-Musalhi ◽  
Nasser Al-Salti ◽  
Erkinjon Karimov
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Initial Boundary Value Problems ◽  
Bessel Differential Operator ◽  
Fractional Differential ◽  
Inverse Source Problems ◽  
Initial Boundary Value

AbstractDirect and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansions and results on existence and uniqueness are established. To solve the resultant equations, a solution to such kind of non-homogeneous fractional differential equation is also presented.

Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates

2018 ◽  
Vol 26 (5) ◽  
pp. 647-672 ◽  
Author(s):  
Atsushi Kawamoto
Keyword(s):  
Diffusion Equation ◽  
Additional Data ◽  
Fixed Time ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Carleman Estimates ◽  
Time Interval ◽  
Stability Estimates ◽  
Lipschitz Stability ◽  
Inverse Source Problems

Abstract In this article, we consider a fractional diffusion equation of half order in time. We study inverse problems of determining the space-dependent factor in the source term from additional data at a fixed time and interior or boundary data over an appropriate time interval. We establish the global Lipschitz stability estimates in the inverse source problems. Our methods are based on Carleman estimates. Here we prove and use the Carleman estimates for a fractional diffusion equation of half order in time.

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