Solving parabolic-hyperbolic type differential equations with spectral analysis method

The study investigates a parabolic-hyperbolic type differential equation with nonlocal boundary and initial conditions. The problem is approached using the spectral analysis method, allowing the solution to be expressed as a series expansion in terms of eigenfunctions of the associated spectral problem. The existence, uniqueness, and stability of the solution are rigorously established through analytical techniques, ensuring the well-posedness of the problem. Furthermore, the study carefully examines the issue of small denominators that arise in the series representation and derives sufficient conditions to guarantee their separation from zero. These results contribute to the broader mathematical theory of mixed-type differential equations, providing valuable insights into their structural properties. The findings have practical applications in various fields of physics and engineering, particularly in modeling wave propagation, heat conduction, and related dynamic processes. The theorems obtained ensure that under appropriate assumptions on the given data, the problem admits a unique and stable solution, reinforcing its theoretical and practical significance.