Geodesic
Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 171-198

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The article gives a new short proof the V. A. Chernyatin theorem about the classical solution of the Fourier method of the mixed problem for the wave equation with fixed ends with minimum requirements on the initial data. Next, a similar problem for the simplest functional differential equation of the first order with involution in the case of the fixed end is considered, and also obtained definitive results. These results are due to a significant use of ideas A. N. Krylova to accelerate the convergence of series, like Fourier series. The results for other similar mixed problems given without proof. 
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     title = {Classical solution by the {Fourier} method of mixed problems with minimum requirements on the initial data},
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A. P. Khromov; M. Sh. Burlutskaya. Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 171-198. http://geodesic.mathdoc.fr/item/ISU_2014_14_2_a7/