In article formulated and solved the problem of finding the temperature distribution in the active element solid-state laser in heating security mode considering radiation heat transfer component. Active element considered as a plate, whose length is much greater than the other sizes, so the problem is written in the form of one-dimensional boundary value problem of the heat equation [1, 2]. Initial boundary value problem, with introducing of dimensionless variables, reduces to the solution of a singularly perturbed boundary value problem of the heat equation with nonlinear boundary conditions on the moving boundaries. An approximate solution in the form of an asymptotic expansion of the solution in the sense of Poincare in powers of the small parameters, depending on the proximity to the borders of the point under consideration were obtained using "geometric-optical " asymptotic method [4-6].
активный элемент, твердотельный лазер, краевая задача, active element, solid-state laser, boundary value problem
1. Mezenov A.V. Termooptika tverdotel'nyh lazerov / A.V. Mezenov, L.N. Soms, A.I. Stepanov - L: Mashi-nostroenie, 1986. - 197 s.
2. Alpat'ev A.N. Osobennosti teplovyh i generacionnyh rezhimov opticheski plotnyh aktivnyh sred / A.N. Al-pat'ev, A.A. Danilov, N.Yu. Nikol'skiy, A.N. Pro-horov, V.B. Cvetkov, I.A. Scherbakov // Izd. AN SSSR. Trudy instituta obschey fiziki. - 1990. -T. 26. - S. 107-124.
3. Tihonov A.N. O funkcional'nyh uravneniyah Vol'terra i ih primeneniyah k nekotorym zadacham matematicheskoy fiziki / A.N. Tihonov // Byull. MGU. - 1938. - T.1(A). - № 8 - S. 1-25.
4. Nesenenko G.A. Asimptoticheskoe reshenie nestacionarnoy zadachi teploprovodnosti s nelineynym usloviem eksponencial'nogo tipa na podvizhnoy granice / G.A. Nesenenko, V.F. Kravchenko // Doklady RAN. - 1998. - T. 358, № 3. - S. 315-318.
5. Nesenenko G.A. Primenenie integral'nyh uravneniy k singulyarno vozmuschennoy nestacionarnoy kraevoy zadache teploprovodnosti s podvizhnymi granicami / G.A. Nesenenko, I.I. Latypov, V.F. Kravchenko // Differencial'nye uravneniya. - 1999. - T. 35, № 9. - S. 1171-1178.
6. Kravchenko V. F. Asimptotiki Puankare resheniy nelineynyh singulyarno vozmuschennyh zadach nestacionarnogo teplo- i massoperenosa / V.F. Krav-chenko, G.A. Nesenenko, V.I. Pustovoyt. - M.: Fizmatlit, 2006. - 325 s.
7. Nesenenko G.A. Priblizhennyy raschet temperaturnogo polya aktivnogo elementa lazera pri ohrannom nagreve / G.A. Nesenenko, I.I. Latypov, S.P. Nasel'skiy // Nelineynye kraevye zadachi mat. fiziki i ih prilozheniya - Kiev: Int. Matematiki AN Ukrainy, 1993.- S. 94-95.
8. Latypov I.I. Asimptotika resheniya nelineynoy singulyarno vozmuschennoy zadachi teplovoy zaschity poristym ohlazhdeniem / I.I. Latypov, G.A. Nesen-enko // Teplomassoobmen - MMF -Radiacionnyy i kombinirovannyy teploobmen. T. 2. - Minsk: ANK ITMO im. A. V. Lykova ANB, 1996. - S. 167-171.
9. Latypov I.I. Priblizhennyy raschet raspredeleniya temperaturnogo polya aktivnogo elementa tverdotel'nogo lazera. // Trudy kafedry eksperimental'noy i teoreticheskoy fiziki instituta fiziki molekul i kristallov UNC RAN. Vyp. 1. Ufa.: Gilem, 2001. - S. 82-92.
10. Latypov I.I. Modelirovanie ispareniya materiala korotkimi lazernymi impul'sami. Trudy chetvertoy Rossiyskoy nacional'noy konferencii po teploobmenu: V 8 tomah. T.5. Isparenie, kondensa- ciya. - M.: Izd. dom MEI, 2006. - S.138-142.
11. Serova V.N. Lazerno-aktivnye sredy na krasitelyah v sopolimetakrilatnyh matricah: osobennosti sinteza, stareniya i stabilizacii / V.N.Serova // Vestnik Kazanskogo tehnologicheskogo universiteta. - 2008. - № 5.- C. 50-65.
12. Serova V.N. Poluchenie polimernyh izdeliy s primeneniem lazernyh tehnologiy na primere Londonskogo universiteta Metropolitan / V.N.Sero-va // Vestnik Kazanskogo tehnologicheskogo univer-siteta. - 2012. - T. 15, № 6.- C. 76-78.
