A Functional Parametrization Method for a Nonlocal Boundary‐Value Problem for the Heat Equation

This paper studies a boundary value problem for a linear parabolic equation with an involution perturbation and a nonlocal boundary condition of the Samarskii‐Ionkin type. A modification of Dzhumabaev's functional parameterization method is developed through the introduction of a time‐dependent functional parameter, extending the applicability of the method to nonlocal boundary value problems for partial differential equations. By introducing auxiliary functions and an appropriate shape function, the original problem is transformed into a Volterra‐type integral equation for the unknown functional parameter. An explicit integral representation of the solution is obtained in terms of Green's functions. Under suitable conditions on the resulting kernel, the problem is reduced to a generalized Abel integral equation. It is shown that the admissible class is not restricted to the standard linear function and may include nonlinear harmonic perturbations while preserving the solvability conditions. For these choices, the kernel admits an explicit convergent series representation and the required conditions can be verified directly. The proposed approach provides a constructive alternative to spectral methods and is particularly effective for nonlocal problems with involution, where the construction of complete systems of eigenfunctions is often difficult.