Fundamental Solution for a Nonlocal Second Order Problem of the Hyperbolic Type

This text discusses the importance and challenges of studying nonlocal boundary value problems, which arise frequently in applied fields such as soil moisture transfer, thermophysics, and diffusion processes. The paper emphasizes the complexity of these problems due to their nonlocal nature, which differs from traditional local boundary value problems. Despite the abundance of research in this area, particularly the works of A.M. Nakhushev and others, mathematical tools to handle nonlocal problems are insufficient, especially when addressing concepts like conjugacy or duality. These issues are particularly significant when coefficients are nonsmooth or measurable. The work introduces the concept of a conjugate problem and the fundamental solution, which generalizes existing functions like Green’s and Riemann’s functions. The study of these solutions for nonlocal boundary value problems is essential for advancing both theoretical understanding and practical applications.