Geodesic
Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 12, pp. 2233-2246

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The Fourier method is used to obtain a classical solution of an initial-boundary value problem for a first-order partial differential equation with involution in the function and its derivative. The series $\Sigma$ produced by the Fourier method as a formal solution of the problem is represented as $\Sigma=S_0+(\Sigma-\Sigma_0)$, where $\Sigma_0$ is the formal solution of a special reference problem for which the sum $S_0$ can be explicitly calculated. Refined asymptotic formulas for the solution of the Dirac system are used to show that the series $\Sigma-\Sigma_0$ and the series obtained from it by termwise differentiation converge uniformly. Minimal smoothness assumptions are imposed on the initial data of the problem. 
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     author = {M. Sh. Burlutskaya and A. P. Khromov},
     title = {Fourier method in an initial-boundary value problem for a~first-order partial differential equation with involution},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
     volume = {51},
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     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_12_a7/}
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M. Sh. Burlutskaya; A. P. Khromov. Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 12, pp. 2233-2246. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_12_a7/