Geodesic
The method of matching asymptotic expansions for the solution of a hyperbolic equation with a small parameter
Sbornik. Mathematics, Tome 48 (1984) no. 2, pp. 541-550 
T. N. Nesterova. The method of matching asymptotic expansions for the solution of a hyperbolic equation with a small parameter. Sbornik. Mathematics, Tome 48 (1984) no. 2, pp. 541-550. http://geodesic.mathdoc.fr/item/SM_1984_48_2_a16/
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     author = {T. N. Nesterova},
     title = {The method of matching asymptotic expansions for the solution of a~hyperbolic equation with a~small parameter},
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     volume = {48},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1984_48_2_a16/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The author considers an initial-boundary value problem for the hyperbolic equation $$ \varepsilon^2(u_{tt}-u_{xx})+a(x,t)u_t=f(x,t) $$ in a rectangle (here $\varepsilon$ is a small parameter and $a(x,t)\geqslant a_0>0$). It is assumed that the initial and boundary values of the function $u_\varepsilon(x,t)$ coincide at the lower corners of the rectangle. A complete asymptotic expansion of the solution in powers of $\varepsilon$ is constructed everywhere in the rectangle. Bibliography: 5 titles.