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.

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Durdimurod K. Durdiev is Head of the Bukhara Branch of the Institute of Mathematics (named after V.I. Romanovskii) in the Academy of Sciences of the Republic of Uzbekistan, Uzbekistan. He is also Professor in the Department of Differential Equations, Bukhara State University, Uzbekistan. He 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. Further, he defended his doctoral dissertation at the Institute of Mathematics and Information Technologies in Tashkent, in 2010, Uzbekistan. 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.