Boundary Value Problems and Applications for Mixed Type Loaded PDEs

This book is devoted to the analytical study of boundary value problems for third-order parabolic–hyperbolic equations containing nonlocal and fractional loaded terms.

This book is devoted to the analytical study of boundary value problems for third-order parabolic–hyperbolic equations containing nonlocal and fractional loaded terms. It develops a unified analytical framework for the investigation of local and nonlocal formulations, bringing together classical partial differential equation theory and fractional calculus. Particular attention is paid to the solvability theory of Tricomi-, Gellerstedt-, and Darboux-type problems, including the establishment of existence and uniqueness theorems and the derivation of explicit solution representations. The book extends traditional models through the incorporation of Riemann–Liouville fractional integro-differential operators and explores the connections between local and nonlocal formulations by means of integral transformation techniques and inverse problem methods. Special emphasis is placed on reduction procedures that transform third-order equations into lower-order systems, facilitating rigorous analytical treatment. The developed methodology is applied to a range of problems arising in mathematical physics and continuum mechanics, including transonic flow, diffusion in heterogeneous media, and environmental processes. Throughout the text, theoretical developments are complemented by constructive solution methods and detailed analyses of mixed-type loaded equations. Designed for researchers and specialists in partial differential equations, fractional calculus, mathematical physics, and applied mathematics, this book provides a comprehensive treatment of modern analytical approaches to nonlocal and mixed-type boundary value problems.