Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution

The Fourier method is used to obtain a classical solution of an initial-boundary value problem for a first-order partial differential equation with involution in the function and its derivative. The series $\Sigma$ produced by the Fourier method as a formal solution of the problem is represented as $\Sigma=S_0+(\Sigma-\Sigma_0)$, where $\Sigma_0$ is the formal solution of a special reference problem for which the sum $S_0$ can be explicitly calculated. Refined asymptotic formulas for the solution of the Dirac system are used to show that the series $\Sigma-\Sigma_0$ and the series obtained from it by termwise differentiation converge uniformly. Minimal smoothness assumptions are imposed on the initial data of the problem.