(untitled) (International journal of artificial intelligence, 2026)
This article presents a rigorous mathematical formulation of the direct (forward) problem for a class of conditionally parabolic partial differential equations subject to nonlocal boundary conditions of the Ionkin type. The Ionkin condition, which prescribes an integral-type constraint linking the solution values at opposite ends of the spatial domain, arises naturally in the modeling of heat conduction processes with nonlocal thermal interactions, diffusion phenomena in inhomogeneous media, and certain population dynamics models. The well-posedness of the direct problem is established in the sense of Hadamard by demonstrating the existence, uniqueness, and continuous dependence of the classical solution on the given data. The proof of existence employs the method of spectral decomposition with respect to a biorthogonal system of eigenfunctions and associated functions generated by a non-self-adjoint Sturm–Liouville operator corresponding to the Ionkin boundary conditions. Uniqueness is obtained via an energy-type inequality adapted to the nonlocal setting, and stability is verified by deriving a priori estimates in appropriate Sobolev-type norms. Illustrative examples are provided to demonstrate the applicability of the theoretical framework, and the relationship of the Ionkin problem to other nonlocal boundary value problems in the existing literature is discussed.