This book explores a range of direct and inverse problems associated with time-fractional and time-space (fully fractional) wave equations. In the context of direct problems, it addresses Cauchy problems and initial-boundary value problems. For inverse problems, the book examines both nonlinear inverse coefficient problems and linear inverse problems focused on determining coefficients on the right-hand sides of the equations. In the fractional wave equations, the fractional derivatives are expressed using various definitions, including the Riemann–Liouville, Caputo, Caputo–Dzhrbashyan, generalized Riemann–Liouville (Hilfer), conformable fractional derivatives, and Riesz operators. The book is intended for a broad audience of mathematicians specializing in fractional differential equations and the theory of inverse problems associated with them.

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.