This book discusses various inverse problems for the time-fractional diffusion equation, such as inverse coefficient problems (nonlinear problems) and inverse problems for determining the right-hand sides of equations and initial functions (linear problems). The study of inverse problems requires a comprehensive investigation of direct problems (such as representation formulas, a priori estimates and differential properties of the solution). This is particularly evident in nonlinear problems, where obtaining solvability theorems necessitates careful tracking of the exact dependence of the differential properties of the solution to the direct problem on the smoothness of the coefficients and other problem data. Therefore, a significant portion of the book is devoted to direct problems, such as initial problems (Cauchy problems) and initial-boundary value problems with various boundary conditions.

Discusses the challenges of using observed data to estimate the values of unknown parameters to characterize a system Highlights the direct problems, such as initial problems (Cauchy problems) and initial-boundary value problems Presents new results on determining coefficients, convolution kernels and right-hand sides of time-fractional diffusion

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 NorthOssetian 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

Coeffcient Determination Problems with Local Boundary and Overdetermination Conditions.- Inverse Coeffcient Problems with Nonlocal Initial and Integral Overdetermination Conditions.- Coeffcient Determination Problems with Cauchy and Overdetermination Conditions.- Carleman Estimate Method in Inverse Problems for a Fractional Diffusion Equation.- Determination of Source and Initial Functions.- Convolution Kernel Determination Problems in Fractional Diffusion Equations.- Determining Two Unknown Functions in a Fractional Diffusion Equation.