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 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 Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Kiwoon Kwon
Keyword(s):  
Inverse Problem ◽  
Partial Differential Equations ◽  
Medical Imaging ◽  
Inverse Problems ◽  
Differential Equations ◽  
Optical Tomography ◽  
Elliptic Partial Differential Equations ◽  
Unique Determination ◽  
Partial Differential ◽  
Uniqueness Results

Unique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance tomography (EIT), diffuse optical tomography (DOT), and inverse scattering problem (ISP) which is an elliptic inverse problem closely related with DOT and EIT. If the coefficient inside the divergence is isotropic, many uniqueness results are known. However, it is known that inverse problem with anisotropic coefficients has many possible coefficients giving the same measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments. Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained.

scholarly journals VARIATIVE SOLUTION OF THE COEFFICIENT INVERSE PROBLEM FOR THE HEAT EQUATIONS

2020 ◽  
Vol 72 (4) ◽  
pp. 23-27
Author(s):  
L. Yermekkyzy ◽  
Keyword(s):  
Inverse Problem ◽  
Partial Differential Equations ◽  
Inverse Problems ◽  
Differential Equations ◽  
Generalized Solution ◽  
Heat Equations ◽  
Coefficient Inverse Problem ◽  
Partial Differential ◽  
Additional Information ◽  
Coefficient Inverse Problems

One of the main types of inverse problems for partial differential equations are problems in which the coefficients of the equations or the quantities included in them must be determined using some additional information. Such problems are called coefficient inverse problems for partial differential equations. Coefficient inverse problems (identification problems) have become the subject of close study, especially in recent years. Interest in them is caused primarily by their important applied values. They find applications in solving problems of planning the development of oil fields (determining the filtration parameters of fields), in creating new types of measuring equipment, in solving problems of environmental monitoring, etc. The standard formulation of the coefficient inverse problem contains a functional (discrepancy), physics. When formulating the statements of inverse problems, the statements of direct problems are assumed to be known. The solution to the problem is sought from the condition of its minimum. Inverse problems for partial differential equations can be posed in variational form, i.e., as optimal control problems for the corresponding systems. A variational statement of one coefficient inverse problem for a one-dimensional heat equation is considered. By the solution of the boundary value problem for each fixed control coefficient we mean a generalized solution from the Sobolev space. The questions of correctness of the considered coefficient inverse problem in the variational setting are investigated.

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.

NUMERICAL SOLUTION OF THE INVERSE PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS

2020 ◽  
pp. 106-110
Author(s):  
S.E. Kasenov ◽  
◽  
G.E. Kasenova ◽  
A.A. Sultangazin ◽  
B.D. Bakytbekova ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Numerical Solution ◽  
Direct Problem ◽  
Nonlinear Differential Equations ◽  
Direct And Inverse Problems ◽  
Additional Information ◽  
Several Variables ◽  
Gradient Of The Functional ◽  
Objective Functional

The article considers direct and inverse problems of a system of nonlinear differential equations. Such problems are often found in various fields of science, especially in medicine, chemistry and economics. One of the main methods for solving nonlinear differential equations is the numerical method. The initial direct problem is solved by the Rune-Kutta method with second accuracy and graphs of the numerical solution are shown. The inverse problem of finding the coefficients of a system of nonlinear differential equations with additional information on solving the direct problem is posed. The numerical solution of this inverse problem is reduced to minimizing the objective functional. One of the methods that is applicable to nonsmooth and noisy functionals, unconditional optimization of the functional of several variables, which does not use the gradient of the functional, is the Nelder-Mead method. The article presents the NellerMead algorithm. And also a numerical solution of the inverse problem is shown.

Inverse Problems for Partial Differential Equations

2012 ◽  
Vol 9 (1) ◽  
pp. 611-659
Author(s):  
Martin Hanke-Bourgeois ◽  
Andreas Kirsch ◽  
William Rundell ◽  
Matti Lassas
Keyword(s):  
Partial Differential Equations ◽  
Inverse Problems ◽  
Differential Equations ◽  
Partial Differential
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