Geodesic
Resolvent approach to Fourier method in a mixed problem for non-homogeneous wave equation
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 4, pp. 403-413

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Fourier method of obtaining classic solution is being justified in a mixed problem for non-homogeneous wave equation with a complex potential and fixed boundary conditions under minimal conditions on initial data. The proof is based on resolvent approach which does not need any information on eigen and associated functions of the corresponding spectral problem. 
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     title = {Resolvent approach to {Fourier} method in a mixed problem for non-homogeneous wave equation},
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V. V. Kornev; A. P. Khromov. Resolvent approach to Fourier method in a mixed problem for non-homogeneous wave equation. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 4, pp. 403-413. http://geodesic.mathdoc.fr/item/ISU_2016_16_4_a4/