scholarly journals An inverse source problem for the stochastic wave equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaoli Feng ◽  
Meixia Zhao ◽  
Peijun Li ◽  
Xu Wang
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Direct Problem ◽  
Inverse Source Problem ◽  
Time Data ◽  
Final Time ◽  
Stochastic Wave Equation ◽  
Unique Mild Solution ◽  
Well Posed ◽  
Class Of Functions

<p style='text-indent:20px;'>This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.</p>

scholarly journals An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

2021 ◽  
Vol 5 (3) ◽  
pp. 63
Author(s):  
Emilia Bazhlekova
Keyword(s):  
Boundary Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Estimates ◽  
Inverse Source Problem ◽  
Time Data ◽  
Final Time ◽  
Generalized Eigenfunction ◽  
The One ◽  
Initial Boundary Value

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

2021 ◽  
pp. 2140011
Author(s):  
Tomás Caraballo ◽  
Tran Bao Ngoc ◽  
Tran Ngoc Thach ◽  
Nguyen Huy Tuan
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Stochastic Integrals ◽  
Well Posedness ◽  
Time Data ◽  
Final Time ◽  
Fractional Brownian Motions ◽  
Nonclassical Diffusion Equation ◽  
Definition Of ◽  
Well Posed

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

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 On initial inverse problem for nonlinear couple heat with Kirchhoff type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Danh Hua Quoc Nam
Keyword(s):  
Inverse Problem ◽  
Fixed Point ◽  
Main Tool ◽  
Cauchy Data ◽  
Final Model ◽  
Kirchhoff Type ◽  
Biological Phenomena ◽  
Banach’S Fixed Point Theorem ◽  
Unique Mild Solution ◽  
Well Posed

AbstractThe main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach’s fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019].

An inverse source problem for a damped wave equation with memory

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Lukáš Šeliga ◽  
Marián Slodička
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Inverse Source Problem ◽  
Damped Wave Equation ◽  
With Memory

AbstractInverse problem of identifying the unknown spacewise dependent source

Direct and inverse source problems for a space fractional advection dispersion equation

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Abeer Aldoghaither ◽  
Taous-Meriem Laleg-Kirati ◽  
Da-Yan Liu
Keyword(s):  
Inverse Problem ◽  
Dispersion Equation ◽  
Direct Problem ◽  
Source Term ◽  
Analytic Solution ◽  
Finite Domain ◽  
Inverse Source Problem ◽  
Direct And Inverse Problems ◽  
Advection Dispersion ◽  
Inverse Source Problems

Abstract In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic solution to the direct problem which we use to prove the uniqueness and the unstability of the inverse source problem using final measurements. Finally, we illustrate the results with a numerical example.

scholarly journals Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ru-Yu Lai ◽  
Laurel Ohm
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Laplace Equation ◽  
Lower Order ◽  
Direct Problem ◽  
Nonlinear Perturbations ◽  
Well Posed ◽  
Lower Order Terms

<p style='text-indent:20px;'>We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings.</p>

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