Kernel Determination Problems in Hyperbolic Integro-Differential Equations

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.

Focuses on the applied significance in the study of the properties of composites and polymers Studies direct and inverse problems for hyperbolic convolution-type integro-differential equations Discusses new research methods for multidimensional inverse problems of kernel recovery

Autorentext
Durdimurod K. Durdiev is the Head of the Institute of Mathematics (named after V.I. Romanovsky) in the Academy of Sciences of the Republic of Uzbekistan, Bukhara State University, Uzbekistan. He is also Professor at the Department of Differential Equations, Bukhara State University, Uzbekistan. Professor Durdiev is graduated from Novosibirsk State University, Russia, in 1990. In 1992, at this university, he defended his Ph.D. thesis in physical and mathematical sciences. He defended his doctoral dissertation at the Institute of Mathematics and Information Technologies in Tashkent, in 2010. He teaches mathematical analysis, partial differential equations, calculus of variations and optimal control, theory of inverse problems of mathematical physics and theory of integral equations.

His areas of scientific interest are in inverse problems for equations of mathematical physics, kernel determination inverse problems in integro-differential equations of hyperbolic and parabolic types, direct and inverse problems for equations with fractional derivatives. Author of 5 books, more than 120 research papers in national and international journals of repute, he has supervised 9 Ph.D. theses during his career at the Institute of Mathematics and Bukhara State University. The main scientific research of Professor Durdiev is devoted to inverse problems, namely, the problems of determining the kernels that describe the viscous properties of the medium in integro-differential equations of hyperbolic and parabolic types with a convolution-type integral. This is a young trend in the theory of inverse problems that has emerged and rapidly developed over the past three decades.

Zhanna D. Totieva is the leader scientist of the Southern Mathematical Institute of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences. She is also Assistant Professor at the Department of Mathematics and Computer Sciences at the North Ossetian State University, Russia, since 1993. Dr. Totieva is graduated from Novosibirsk State University, in 1993. In 1992, at Rostov-on Don State University, Russia, she defended her Ph.D. Her areas of scientific interest in mathematics are inverse problems for equations of mathematical physics, mathematical modelling and kernel determination inverse problems in integro-differential equations of hyperbolic types. She teaches mathematical analysis, mathematical modelling and linear algebra. Author of 2 books, more than 70 research papers in national and international journals of repute, her main scientific research is devoted to inverse problems, namely, the problems of determining the kernels that describe the viscous properties of the medium in integro-differential equations of hyperbolic types with a convolution-type integral.

Inhalt

Chapter 1: Local Solvability of One-dimensional Kernel Determination Problems.- Chapter 2: The Solvability of Multidimensional Inverse Problems of a Memory Kernel Determination.- Chapter 3: Global Solvability of Memory Reconstruction Problems.- Chapter 4: Stability in Inverse Problems for Determining Two Unknowns.- Chapter 5: Two-dimensional Special Kernel Determination Problems.- Chapter 6: Some Inverse Problems of Viscoelasticity System with the Known Kernel.- Chapter 7: Kernel Identification Problems in a Viscoelasticity System.- Chapter 1:Dirichlet-to-Neumann Maps Method in Kernel Determining Problems.