Geodesic
Problems for the one-dimensional wave equation with conditionson characteristics and non-characteristic lines
Trudy Instituta matematiki, Tome 29 (2021) no. 1, pp. 106-112

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We consider problems for the one-dimensional wave equation with conditions given on both characteristic lines and non-characteristic lines in the paper. We construct the unique classical solutions of these problems in analytical form by the method of characteristics and indicate necessary and sufficient smoothness and matching conditions on given functions. 
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V. I. Korzyuk; O. A. Kovnatskaya; V. P. Serikov. Problems for the one-dimensional wave equation with conditionson characteristics and non-characteristic lines. Trudy Instituta matematiki, Tome 29 (2021) no. 1, pp. 106-112. http://geodesic.mathdoc.fr/item/TIMB_2021_29_1_a9/