Multidimensional Inverse and Ill-Posed Problems for Differential Equations

✍ Scribed by Yu. E. Anikonov

• Publisher: De Gruyter
• Year: 2014
• Tongue: English
• Leaves: 140
• Series: Inverse and Ill-Posed Problems Series; 4
• Edition: Reprint 2014
• Category: Library

✦ Synopsis

Inverse problems are usually nonlinear and are separated into one-dimensional and multidimensional problems, depending on whether the sought function (or functions) is a function of one variable or of many. Multidimensionality of inverse problems has particular value at present, because practice shows that many investigating processes are described by an equation, of which the co-efficient essentially depends on many variables.

This monograph is devoted to statements of multidimensional inverse problems, in particular to methods of their investigation. Questions of the uniqueness of solution, solvability and stability are studied. Methods to construct a solution are given and, in certain cases, inversion formulas are given as well. Concrete applications of the theory developed here are also given. Where possible, the author has stopped to consider the method of investigation of the problems, thereby sometimes losing generality and quantity of the problems, which can be examined by such a method.

The book should be of interet to researchers in the field of applied mathematics, geophysics and mathematical biology.

✦ Table of Contents

1. Operator Equations and Inverse Problems
   1.1 Definition of Quasimonotonicity, the Uniqueness Theorem
   1.2 Inverse Problems for Hyperbolic Equations
   1.3 Multidimensional Inverse Kinematic Problem of Seismics
   1.4 On the Uniqueness of the Solution of the Fredholm and Volterra First Kind Integral Equations
   1.5 On the Uniqueness of the Solution of Integral Equations of the First Kind with Entire Kernel
   1.6 Existence and Uniqueness of a Solution to an Inverse Problem for a Parabolic Equation
   1.7 On Unique Solvability of an Inverse Problem for a Parabolic Equation
   1.8 Formulas in Multidimensional Inverse Problems for Evolution Equations
2. Inverse Problems for Kinetic Equations
   2.1 Kinetic Equations
   2.2 An Example of an Inverse Problem for Kinetic Equation
   2.3 One-dimensional Inverse Problems
   2.4 Multidimensional Inverse Problems
   2.5 An Uniqueness Theorem for the Solution of an Inverse Problem for a Kinetic Equation
   2.6 The General Uniqueness Theorem
   2.7 The Effect of the ‘Redundant’ Equation
   2.8 Problem of Separation
   2.9 Differential and Integro-differential Identities
   2.10 Solution Existence Problems
   2.11 An Inverse Problem of Mathematical Biology
3. Geometry of Convex Surfaces in the Large and Inverse Problems of Scattering Theory
   3.1 Geometrical Question of Scattering Theory
   3.2 Integral Equation of the First Kind
   3.3 Uniqueness
   3.4 Existence
   3.5 Stability
4. Integral Geometry
   4.1 Inversion Formulas
   4.2 The Uniqueness and Solvability
   4.3 Some Applications
   4.4 The Structure of Riemann Spaces and Problems of the Integral Geometry
   4.5 The Solvability of a Problem in Integral Geometry by Integration along Geodesics
   References