This book studies the construction methods for solving one-dimensional and multidimensional inverse dynamical problems for hyperbolic equations with memory. The theorems of uniqueness, stability and existence of solutions of these inverse problems are obtained. This book discusses the processes, by using generalized solutions, the spread of elastic or electromagnetic waves arising from sources of the type of pulsed directional “impacts” or “explosions”. This book presents new results in the study of local and global solvability of kernel determination problems for a half-space. It describes the problems of reconstructing the coefficients of differential equations and the convolution kernel of hyperbolic integro-differential equations by the method of Dirichlet-to-Neumann. The book will be useful for researchers and students specializing in the field of inverse problems of mathematical physics.
Author(s): Durdimurod K. Durdiev, Zhanna D. Totieva
Series: Infosys Science Foundation Series
Publisher: Springer-ISF
Year: 2023
Language: English
Pages: 389
City: Bangalore
Preface
References
Contents
1 Local Solvability of One-Dimensional Kernel Determination Problems
1.1 The Inverse Problem for the Wave Equation in the Medium with Memory
1.2 The Inverse Problem for Hyperbolic Second-Order Integro-differential Equation with Variable Coefficients for Lower Derivatives
1.2.1 Formulation of the Problem. The Derivation of Integral Equations
1.2.2 The Existence and Uniqueness Theorem
1.2.3 The Stability Estimate
1.3 Inverse Problem for the System of Thermoelasticity Equations for a Vertically Inhomogeneous Cohesionless Medium with Memory
1.4 Inverse Problem for an Integro-differential Equation of Acoustics
1.5 Problem of Determining the Memory in a Two-Dimensional System of Maxwell Equations with Distributed Data
1.6 Conclusions
References
2 The Solvability of Multidimensional Inverse Problems of a Memory Kernel Determination
2.1 Definition of a Multidimensional Memory …
2.2 Definition of the Memory Kernel Standing at Some …
2.3 Definition of Memory Kernel in a Two-Dimensional System Maxwell Equations
2.4 Differential Properties of the Solution of the Memory …
2.5 Conclusions
References
3 Global Solvability of Memory Reconstruction Problems
3.1 The Problem of Determining the Memory Kernel from the Integro-Differential Equation of String Vibrations
3.2 Memory Identification Problem from Integro-Differential Wave Equation
3.3 Inverse Problem for an Integro-Differential Equation of Electrodynamics
3.4 Global Solvability of the Problem of Determining Two Unknowns in the Inverse Problem for the Integro-Differential Wave Equation
3.5 About Global Solvability of a Multidimensional Inverse Problem for an Equation with Memory
3.6 Conclusions
References
4 Stability in Inverse Problems for Determining of Two Unknowns
4.1 The Problem of Determining the Speed of Sound and Memory of Medium
4.1.1 Statement of the Problem and Preliminary Transformations
4.1.2 Properties of the Direct Problem Solution
4.1.3 Investigation of the Inverse Problem
4.2 The Problem of Determining the Memory of Medium and Regular Part of Impulsive Source
4.2.1 Statement of the Problem and Preliminary Constructions
4.2.2 Properties of the Direct Problem Solution. Statement and Proof of the Main Results
4.3 Conclusions
References
5 Two-Dimensional Special Kernel Determination Problems
5.1 A Problem of Identification of a Two-Dimensional Kernel
5.1.1 Setting of the Problem
5.1.2 Investigation of the Direct Problem
5.1.3 Inverse Problem—The Existence and Uniqueness Theorem
5.2 A Problem of Determining a Spatial Part of a Memory Kernel
5.3 Conclusions
References
6 Some Inverse Problems of Viscoelasticity System with the Known Kernel
6.1 The Multidimensional Linearized Problem of Determining the Density Function for a Viscoelasticity System
6.1.1 Statement of Problems and Main Results
6.1.2 Direct Problem of Isotropic Viscoelasticity and Properties of Its Solution
6.1.3 The Fundamental Solution of the Cauchy Problem for a Hyperbolic Operator with Memory
6.1.4 Direct and Inverse Linearized Problem
6.2 The Problem of Determining the Coefficient of Thermal Enlargement of Thermoviscoelasticity Equation
6.3 One-Dimensional Inverse Coefficient Problem of Anisotropic Viscoelasticity Equation
6.4 Conclusions
References
7 Kernel Identification Problems in a Viscoelasticity System
7.1 The Problem of Determining the One-Dimensional Kernel of Viscoelasticity Equation
7.2 The Problem of Determining the Multidimensional Kernel of Viscoelasticity Equation
7.3 The Problem of Determining the One-Dimensional Matrix Kernel of the System of Viscoelasticity
7.4 The Problem of Determining the One-Dimensional Kernel Thermoviscoelasticity Equation
7.5 Conclusions
References
8 Dirichlet-to-Neumann Maps Method in Kernel Determining Problems
8.1 The Dirichlet-to-Neumann Maps Method for a Half Space
8.2 Stability of Memory Reconstruction from the Dirichlet-to-Neumann Operator
8.2.1 The Problem Statement and Results
8.2.2 The Direct Problem
8.2.3 Auxiliary Assertions
8.2.4 Proof of the Main Result
8.3 Recovering a Time- and Space-Dependent Kernel in a Hyperbolic Integro-Differential Equation from a Restricted Dirichlet-to-Neymann Operator
8.3.1 Problem Formulation and Results
8.3.2 Auxiliary Results
8.3.3 Direct Problem
8.3.4 Uniqueness on the Boundary
8.4 Conclusions
References
9 Open Problems
Reference
