Behavior of the formal solution to a mixed problem for the wave equation

The behavior of the formal solution, obtained by the Fourier method, to a mixed problem for the wave equation with arbitrary two-point boundary conditions and the initial condition $\varphi(x)$ (for zero initial velocity) with weaker requirements than those for the classical solution is analyzed. An approach based on the Cauchy–Poincare technique, consisting in the contour integration of the resolvent of the operator generated by the corresponding spectral problem, is used. Conditions giving the solution to the mixed problem when the wave equation is satisfied only almost everywhere are found. When $\varphi(x)$ is an arbitrary function from $L_2[0, 1]$, the formal solution converges almost everywhere and is a generalized solution to the mixed problem.