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 Σ produced by the Fourier method as a formal solution of the problem is represented as Σ = S 0 + (Σ − Σ0), where Σ0 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 Σ − Σ0 and the series obtained from it by termwise differentiation converge uniformly. Minimal smoothness assumptions are imposed on the initial data of the problem.

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