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>

scholarly journals Analysis of direct and inverse problems for a fractional elastoplasticity model

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 699-708 ◽  
Author(s):  
Salih Tatar ◽  
Süleyman Ulusoy
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Unique Solution ◽  
Continuous Dependence ◽  
Direct Problem ◽  
Direct And Inverse Problems

This study is devoted to a nonlinear time fractional inverse coeficient problem. The unknown coeffecient depends on the gradient of the solution and belongs to a set of admissible coeffecients. First we prove that the direct problem has a unique solution. Afterwards we show the continuous dependence of the solution of the corresponding direct problem on the coeffecient, the existence of a quasi-solution of the inverse problem is obtained in the appropriate class of admissible coeffecients.

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 Inverse Problems for Degenerate Fractional Integro-Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 532
Author(s):  
Mohammed Al Horani ◽  
Mauro Fabrizio ◽  
Angelo Favini ◽  
Hiroki Tanabe
Keyword(s):  
Inverse Problem ◽  
Partial Differential Equations ◽  
Inverse Problems ◽  
Differential Equations ◽  
Banach Spaces ◽  
Unique Solution ◽  
Direct Problem ◽  
Regularity Of Solutions ◽  
Strict Solution ◽  
Fractional Equations

This paper deals with inverse problems related to degenerate fractional integro-differential equations in Banach spaces. We study existence, uniqueness and regularity of solutions to the problem, claiming to extend well known studies for the case of non-fractional equations. Our method is based on transforming the inverse problem to a direct problem and identifying the conditions under which this direct problem has a unique solution. The conditions under which the unique strict solution can be compared with the case of a mild solution, obtained in previous studies under quite restrictive requirements, are on the underlying functions. Applications from partial differential equations are given to illustrate our abstract results.

scholarly journals INVERSE HEAT TRANSPORT PROBLEMS FOR COEFFICIENTS IN TWO‐LAYER DOMAINS AND METHODS FOR THEIR SOLUTION

2002 ◽  
Vol 7 (2) ◽  
pp. 217-228 ◽  
Author(s):  
S. Guseinov ◽  
A. Buikis
Keyword(s):  
Inverse Problem ◽  
Thermal Diffusivity ◽  
Inverse Problems ◽  
Heat Transport ◽  
Transport Process ◽  
Direct Problem ◽  
Piecewise Constant Function ◽  
Algebraic Equations ◽  
Heat Transport Process

In various fields of science and technology it is often necessary to solve inverse problems, where from measurements of state of the system or process it is required to determine a certain typesetting of the causal characteristics. It is known that infringement of the natural causal relationships can entail incorrectness of the mathematical stating of inverse problems. Therefore the development of efficient methods for solving such problems allows one to considerably simplify experimental research and to increase the accuracy and reliability of the obtained results due to certain complication of algorithms for processing the experimental data. The problem of determination of thermal diffusivity coefficients considering other known characteristics of heat transport process is among incorrect inverse problems. These inverse problems for coefficients are quite difficult even in the case of homogeneous media. In this paper it is supposed that the heat transport equation is non‐homogeneous and an algorithm for determination of the thermal diffusivity coefficients for both the media is proposed. At the first step, the non‐homogeneous inverse problem with piecewise‐constant function of non‐homogeneity is solved. For this auxiliary inverse problem, the proposed method allows one to determine both the coefficients of thermal diffusivity and to restore the heat transport process without any additional information, i.e. the algorithm also solves the direct problem. Then the initial non‐homogeneous inverse problem with a piecewise‐continuous function of non‐homogeneity is solved. The proposed method reduces the non‐homogeneous inverse problem for coefficients to a set of two transcendent algebraic equations. Finally, the analytical solution to direct problem is obtained using Green's function.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

Variational-hemivariational inverse problem for electro-elastic unilateral frictional contact problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Othmane Baiz ◽  
Hicham Benaissa ◽  
Zakaria Faiz ◽  
Driss El Moutawakil
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Contact Problem ◽  
Existence And Uniqueness ◽  
Frictional Contact ◽  
Hemivariational Inequalities ◽  
Frictional Contact Problem ◽  
Uniqueness Of A Solution

AbstractIn the present paper, we study inverse problems for a class of nonlinear hemivariational inequalities. We prove the existence and uniqueness of a solution to inverse problems. Finally, we introduce an inverse problem for an electro-elastic frictional contact problem to illustrate our results.

scholarly journals Nonlinear evolution problems with singular coefficients in the lower order terms

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Lower Order ◽  
Time Horizon ◽  
Existence Result ◽  
Nonlinear Evolution ◽  
Order Term ◽  
Time Behavior ◽  
Infinite Time ◽  
Singular Coefficient ◽  
Singular Coefficients ◽  
Lower Order Terms

AbstractWe consider a Cauchy–Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite–time horizon.

scholarly journals Existence results for nonlinear degenerate elliptic equations with lower order terms

2020 ◽  
Vol 10 (1) ◽  
pp. 301-310
Author(s):  
Weilin Zou ◽  
Xinxin Li
Keyword(s):  
Lower Order ◽  
Elliptic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regularity Of Solutions ◽  
Degenerate Elliptic Equations ◽  
Degenerate Elliptic ◽  
Initial Boundary Value ◽  
Lower Order Terms ◽  
Nonlinear Degenerate

Abstract In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense.

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