The 14 references with contexts in paper I. Volkov K., P. Vlasov A., И. Волков К., П. Власов А. (2016) “Квазистационарное температурное поле двухслойного полупространства с подвижной границей // Quasi-Stationary Temperature Field of Two-Layer Half-Space with Moving Boundary” / spz:neicon:technomag:y:2015:i:5:p:126-136

1
Carslaw H.S., Jaeger J.C.Conduction of Heat in Solids. 2nd ed. Oxford University Press, 1959. (Russ. ed.: Karslou G., Eger D.Teploprovodnost' tverdykh tel. Moscow, Nauka Publ., 1964. 488 p.).
Total in-text references: 6
  1. In-text reference with the coordinate start=1950
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    Î ̨ ̃‚ ̊ ÒÎÓ‚‡:ÚÂÏÔÂ‡ÚÛÌÓ ÔÓÎÂ, ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚Ó, ÔÓ‰‚ËÊ̇ˇ „‡Ìˈ‡ ¬‚‰ÂÌË ¬ Ò‚ˇÁË Ò ËÌÚÂÌÒË‚Ì ̊Ï ‚̉ÂÌËÂÏ ‚ ËÌÊÂÌÂÌÛ ̨ Ô‡ÍÚËÍÛ ÏÂÚÓ‰Ó‚ χÚÂχÚË ̃ÂÒÍÓ„Ó ÏÓ‰ÂÎËÓ‚‡Ìˡ, ̇ˇ‰Û Ò ‚ ̊ ̃ËÒÎËÚÂÎ ̧Ì ̊ÏË ÏÂÚÓ‰‡ÏË ‚Ò ·ÓÎ ̧ ̄Û ̨ Á̇ ̃ËÏÓÒÚ ̧ ÔËÓ·ÂÚ‡ ̨Ú ‡Ì‡ÎËÚË ̃ÂÒÍË ÏÂÚÓ‰ ̊  ̄ÂÌˡ Á‡‰‡ ̃ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË
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    . ÕÂÒÏÓÚˇ ̇ ÂÒÚÂÒÚ‚ÂÌÌÛ ̨ Ó„‡ÌË ̃ÂÌÌÓÒÚ ̧ ӷ·ÒÚË ÔËÏÂÌËÏÓÒÚË ‡Ì‡ÎËÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚, ̇·Î ̨‰‡Âχˇ ÚẨÂÌˆËˇ Ó·ÛÒÎÓ‚ÎÂ̇ ÏÌÓ„ËÏË ÔË ̃Ë̇ÏË. ¬ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃, ÔÓÎÛ ̃ÂÌÌ ̊ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉÂ, ÔÓÁ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊ [4,5,6,7], ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒÎÂ

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    ÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9]. “Û‰ÌÓÒÚË, ÍÓÚÓ ̊ ‰ÓÎÊÂÌ ÔÂÓ‰ÓÎÂÚ ̧ ËÒÒΉӂ‡ÚÂÎ ̧ ÔË ÔÓÎÛ ̃ÂÌËË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ ÚÓÈ ËÎË ËÌÓÈ Á‡‰‡ ̃Ë Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, ıÓÓ ̄Ó ËÁ‚ÂÒÚÌ ̊
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    . “ÂÏ Ì ÏÂÌ ÓÌË Â ̆ ·ÓΠÛÒÛ„Û·Îˇ ̨ÚÒˇ, ÂÒÎË ‚ÓÁÌË͇ÂÚ ÌÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ‚ÎˡÌˡ ‡ÁÎË ̃Ì ̊ı ÙËÁËÍÓ-ıËÏË ̃ÂÒÍËı Ë ÏÂı‡ÌË ̃ÂÒÍËı ÔÓˆÂÒÒÓ‚, ÔË‚Ó‰ˇ ̆Ëı Í ËÁÏÂÌÂÌË ̨ ‚Ó ‚ÂÏÂÌË „‡Ìˈ ËÁÛ ̃‡ÂÏÓ„Ó Ú‚Â‰Ó„Ó Ú·. œË ̋ÚÓÏ ÔÓ·ÎÂχÚË ̃ÌÓÒÚ ̧ ̇ıÓʉÂÌˡ  ̄ÂÌËÈ Á‡‰‡ ̃ Ú‡ÍÓ„Ó Í·ÒÒ‡ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ ÒÓı‡ÌˇÂÚÒˇ ‰‡Ê ‚ ÚÂı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ [10,11].

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    ‡) ¬ ̇ ̃‡Î ̧Ì ̊È ÏÓÏÂÌÚ ‚ÂÏÂÌËt= 0ÚÂÏÔÂ‡ÚÛ‡ ‚ Î ̨·ÓÈ ÚÓ ̃Í ÒËÒÚÂÏ ̊ ÔÓÒÚÓˇÌ̇: T(r,z,0)≡T0= const; ·) Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁÓÚÓÔÌÓ„Ó ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚‡ ˇ‚ΡÂÚÒˇ ÎËÌÂÈÌ ̊Ï, Ú.Â.z= vt, „‰Âv= const; ‚) ËÁÓÚÓÔÌÓ ÔÓÍ ̊ÚË ËÏÂÂÚ ÔÓÒÚÓˇÌÌÛ ̨ ‚Ó ‚ÂÏÂÌË ÚÓÎ ̆ËÌÛh; „) ‚ ÒËÒÚÂÏ <ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚Ó | ÔÓÍ ̊ÚËÂ> ‡ÎËÁÛ ̨ÚÒˇ ÛÒÎӂˡ ˉ‡Π̧ÌÓ„Ó ÚÂÔÎÓ‚Ó„Ó ÍÓÌÚ‡ÍÚ‡
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    ; ‰) ÒÚ‡ˆËÓ̇Ì ̊È ÚÂÔÎÓ‚ÓÈ ÔÓÚÓÍ, ‚ÓÁ‰ÂÈÒÚ‚Û ̨ ̆ËÈ Ì‡ ‚Ì ̄Ì ̨ ̨ ÔÓ‚ÂıÌÓÒÚ ̧ ÔÓÍ ̊Úˡ ‚ ̇Ô‡‚ÎÂÌËË ÓÒËOzËÒÔÓÎ ̧ÁÛÂÏÓÈ ÒËÒÚÂÏ ̊ ÍÓÓ‰Ë̇Ú, ˇ‚ΡÂÚÒˇ ÓÒÂÒËÏÏÂÚË ̃Ì ̊Ï Ë ËÏÂÂÚ ËÌÚÂÌÒË‚ÌÓÒÚ ̧ „‡ÛÒÒÓ‚ÒÍÓ„Ó ÚËÔ‡ Ò ÓÔ‰ÂΡ ̨ ̆ËÏË Ô‡‡ÏÂÚ‡ÏËk2Ëq0; Â) ÔË Î ̨· ̊ı ÙËÍÒËÓ‚‡ÌÌ ̊ı Á̇ ̃ÂÌˡıt>0Ëz>vt−hÙÛÌ͈ËÓ̇ÎT(r,z,t)ËÌÚ„ËÛÂÏ Ò Í‚‡‰‡ÚÓÏ Ë ‚ÂÒÓÏṙ ÔÓÎÛ·ÂÒÍÓÌ ̃ÌÓÏ ËÌÚÂ‚‡ÎÂ[0; +∞), Ú.

  4. In-text reference with the coordinate start=7848
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    “ÂÏÔÂ‡ÚÛÌÓ ÔÓΠ”ÒÎӂˡ (5) ÔÓÁ‚ÓΡ ̨Ú ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓ ÔËÏÂÌËÚ ̧ Í ÒÏ ̄‡ÌÌÓÈ Á‡‰‡ ̃ (4) Ò̇ ̃‡Î‡ ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË √‡ÌÍÂΡ ÌÛÎÂ‚Ó„Ó ÔÓˇ‰Í‡ [12] ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÏÛ ÔÂÂÏÂÌÌÓÏÛρ H0[θ]≡ +∞∫ 0 θ(ρ)ρJ0(pρ)dρ,H−10[κ]≡ +∞∫ 0 κ(p)pJ0(pρ)dp,(6) ‡ Á‡ÚÂÏ | ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË À‡Ô·҇
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    ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛτ. œÓ·„‡ˇ A(p,X,τ) =H0[θ(ρ,X,τ)],(7) B(p,X,s) =L[A(p,X,τ)],(8) „‰ÂL| ÓÔÂ‡ÚÓ ÔˇÏÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇, Ë ËÒÔÓÎ ̧ÁÛˇ Ò‚ÓÈÒÚ‚‡ ËÌÚ„‡Î ̧Ì ̊ı ÔÂÓ·‡ÁÓ‚‡ÌËÈ √‡ÌÍÂΡ Ë À‡Ô·҇ [1,2,3,12] ‚ÍÛÔÂ Ò Ú‡·Îˈ‡ÏË <ËÁÓ·‡ÊÂÌËÂ-ÓË„Ë̇Î> [13], ÔËıÓ‰ËÏ Í Á‡‰‡ ̃ ‰Îˇ ÓÔ‰ÂÎÂÌˡ ËÁÓ·‡ÊÂÌˡB(p,X,s): d2B dX2 +V dB dX −(s+p2)B= 0,X>0;(9) d2B dX2 +α2V dB dX −(α2s+p2)B= 0,−ε<X<0;(

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    ÔÂÓ·‡ÁÓ‚‡ÌË √‡ÌÍÂΡ ÌÛÎÂ‚Ó„Ó ÔÓˇ‰Í‡ [12] ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÏÛ ÔÂÂÏÂÌÌÓÏÛρ H0[θ]≡ +∞∫ 0 θ(ρ)ρJ0(pρ)dρ,H−10[κ]≡ +∞∫ 0 κ(p)pJ0(pρ)dp,(6) ‡ Á‡ÚÂÏ | ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË À‡Ô·҇ [1,2] ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛτ. œÓ·„‡ˇ A(p,X,τ) =H0[θ(ρ,X,τ)],(7) B(p,X,s) =L[A(p,X,τ)],(8) „‰ÂL| ÓÔÂ‡ÚÓ ÔˇÏÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇, Ë ËÒÔÓÎ ̧ÁÛˇ Ò‚ÓÈÒÚ‚‡ ËÌÚ„‡Î ̧Ì ̊ı ÔÂÓ·‡ÁÓ‚‡ÌËÈ √‡ÌÍÂΡ Ë À‡Ô·҇
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    ‚ÍÛÔÂ Ò Ú‡·Îˈ‡ÏË <ËÁÓ·‡ÊÂÌËÂ-ÓË„Ë̇Î> [13], ÔËıÓ‰ËÏ Í Á‡‰‡ ̃ ‰Îˇ ÓÔ‰ÂÎÂÌˡ ËÁÓ·‡ÊÂÌˡB(p,X,s): d2B dX2 +V dB dX −(s+p2)B= 0,X>0;(9) d2B dX2 +α2V dB dX −(α2s+p2)B= 0,−ε<X<0;(10) dB dX ∣ ∣ ∣ ∣ ∣ X=−ε =− Q0 2K2s exp ( − p2 4K2 ) ;(11) B(p,0−0,s) =B(p,0 + 0,s);(12) Λ dB dX ∣ ∣ ∣ ∣∣ X=0−0 = dB dX ∣ ∣ ∣ ∣∣ X=0+0 ,(13) „‰Â ‚ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò Ô‰ÔÓÎÓÊÂÌˡÏË (5) ‰ÓÎÊÌÓ ‚ ̊ÔÓÎÌˇÚ ̧Òˇ ÛÒÎÓ‚Ë B(p,·,s

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    ̧ ̨ ÓÔ‰ÂÎÂÌÓ ‡‚ÂÌÒÚ‚‡ÏË (15){(17), (21), (20), (22), (23), (24). œË Ó·ÒÛʉÂÌËË ÒÔˆËÙËÍË Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË (4) · ̊Î Ò‰ÂÎ‡Ì ‚ ̊‚Ó‰ Ó ÒÛ ̆ÂÒÚ‚Ó‚‡ÌËË Í‚‡ÁËÒÚ‡ˆËÓ̇ÌÓ„Ó ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ Ó·Î‡ÒÚË, ÔÓ‰ ÍÓÚÓ ̊Ï ÔÓÌËχÂÚÒˇ  ÒÚ‡ˆËÓ̇ÌÓ ÚÂÏÔÂ‡ÚÛÌÓ ÔÓΠ‚ ÔÓ‰‚ËÊÌÓÈ ÒËÒÚÂÏ ÍÓÓ‰ËÌ‡Ú (3). — Û ̃ÂÚÓÏ ̋ÚÓ„Ó Ù‡ÍÚ‡ Ë Ô‰ÂÎ ̧ÌÓÈ ÚÂÓÂÏ ̊ ËÌÚ„‡Î ̧ÌÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇
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    Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ Ô‰ÒÚ‡‚ÎÂÌˡ (8) ËÏÂÂÏ: A(p,X,+∞) = lim s→0 sB(p,X,s)≡B(p,X,0),(25) „‰Â ÙÛÌÍˆËˇB(p,X,0)ÓÔ‰ÂÎÂ̇ ‡‚ÂÌÒÚ‚‡ÏË (15){(17), (21), (20), C0(p,s)≡φ(p,s)(26) Ë (23), (24) ÔËs= 0. “‡ÍËÏ Ó·‡ÁÓÏ, Òӄ·ÒÌÓ (7) Ë (6), ‚ ÔÓ‰‚ËÊÌÓÈ ÒËÒÚÂÏ ÍÓÓ‰ËÌ‡Ú (3) ËÒÍÓÏÓ ÚÂÏÔÂ‡ÚÛÌÓ ÔÓΠÏÓÊÂÚ · ̊Ú ̧ Ô‰ÒÚ‡‚ÎÂÌÓ ‚ ‚ˉ θ(ρ,X,+∞) = +∞∫ 0 A(p,X,+∞)pJ0(pρ)dp,(27) „‰Â ËÁÓ·‡ÊÂÌËÂA(p,X,+∞)ÓÔ‰ÂÎÂÌÓ

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Lykov A.V.Teoriya teploprovodnosti[Theory of thermal conductivity]. Moscow, Vysshaya shkola Publ., 1967. 600 p. (in Russian).
Total in-text references: 6
  1. In-text reference with the coordinate start=1950
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    Î ̨ ̃‚ ̊ ÒÎÓ‚‡:ÚÂÏÔÂ‡ÚÛÌÓ ÔÓÎÂ, ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚Ó, ÔÓ‰‚ËÊ̇ˇ „‡Ìˈ‡ ¬‚‰ÂÌË ¬ Ò‚ˇÁË Ò ËÌÚÂÌÒË‚Ì ̊Ï ‚̉ÂÌËÂÏ ‚ ËÌÊÂÌÂÌÛ ̨ Ô‡ÍÚËÍÛ ÏÂÚÓ‰Ó‚ χÚÂχÚË ̃ÂÒÍÓ„Ó ÏÓ‰ÂÎËÓ‚‡Ìˡ, ̇ˇ‰Û Ò ‚ ̊ ̃ËÒÎËÚÂÎ ̧Ì ̊ÏË ÏÂÚÓ‰‡ÏË ‚Ò ·ÓÎ ̧ ̄Û ̨ Á̇ ̃ËÏÓÒÚ ̧ ÔËÓ·ÂÚ‡ ̨Ú ‡Ì‡ÎËÚË ̃ÂÒÍË ÏÂÚÓ‰ ̊  ̄ÂÌˡ Á‡‰‡ ̃ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË
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    . ÕÂÒÏÓÚˇ ̇ ÂÒÚÂÒÚ‚ÂÌÌÛ ̨ Ó„‡ÌË ̃ÂÌÌÓÒÚ ̧ ӷ·ÒÚË ÔËÏÂÌËÏÓÒÚË ‡Ì‡ÎËÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚, ̇·Î ̨‰‡Âχˇ ÚẨÂÌˆËˇ Ó·ÛÒÎÓ‚ÎÂ̇ ÏÌÓ„ËÏË ÔË ̃Ë̇ÏË. ¬ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃, ÔÓÎÛ ̃ÂÌÌ ̊ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉÂ, ÔÓÁ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊ [4,5,6,7], ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒÎÂ

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    ÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9]. “Û‰ÌÓÒÚË, ÍÓÚÓ ̊ ‰ÓÎÊÂÌ ÔÂÓ‰ÓÎÂÚ ̧ ËÒÒΉӂ‡ÚÂÎ ̧ ÔË ÔÓÎÛ ̃ÂÌËË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ ÚÓÈ ËÎË ËÌÓÈ Á‡‰‡ ̃Ë Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, ıÓÓ ̄Ó ËÁ‚ÂÒÚÌ ̊
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    . “ÂÏ Ì ÏÂÌ ÓÌË Â ̆ ·ÓΠÛÒÛ„Û·Îˇ ̨ÚÒˇ, ÂÒÎË ‚ÓÁÌË͇ÂÚ ÌÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ‚ÎˡÌˡ ‡ÁÎË ̃Ì ̊ı ÙËÁËÍÓ-ıËÏË ̃ÂÒÍËı Ë ÏÂı‡ÌË ̃ÂÒÍËı ÔÓˆÂÒÒÓ‚, ÔË‚Ó‰ˇ ̆Ëı Í ËÁÏÂÌÂÌË ̨ ‚Ó ‚ÂÏÂÌË „‡Ìˈ ËÁÛ ̃‡ÂÏÓ„Ó Ú‚Â‰Ó„Ó Ú·. œË ̋ÚÓÏ ÔÓ·ÎÂχÚË ̃ÌÓÒÚ ̧ ̇ıÓʉÂÌˡ  ̄ÂÌËÈ Á‡‰‡ ̃ Ú‡ÍÓ„Ó Í·ÒÒ‡ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ ÒÓı‡ÌˇÂÚÒˇ ‰‡Ê ‚ ÚÂı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ [10,11].

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    ‡) ¬ ̇ ̃‡Î ̧Ì ̊È ÏÓÏÂÌÚ ‚ÂÏÂÌËt= 0ÚÂÏÔÂ‡ÚÛ‡ ‚ Î ̨·ÓÈ ÚÓ ̃Í ÒËÒÚÂÏ ̊ ÔÓÒÚÓˇÌ̇: T(r,z,0)≡T0= const; ·) Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁÓÚÓÔÌÓ„Ó ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚‡ ˇ‚ΡÂÚÒˇ ÎËÌÂÈÌ ̊Ï, Ú.Â.z= vt, „‰Âv= const; ‚) ËÁÓÚÓÔÌÓ ÔÓÍ ̊ÚË ËÏÂÂÚ ÔÓÒÚÓˇÌÌÛ ̨ ‚Ó ‚ÂÏÂÌË ÚÓÎ ̆ËÌÛh; „) ‚ ÒËÒÚÂÏ <ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚Ó | ÔÓÍ ̊ÚËÂ> ‡ÎËÁÛ ̨ÚÒˇ ÛÒÎӂˡ ˉ‡Π̧ÌÓ„Ó ÚÂÔÎÓ‚Ó„Ó ÍÓÌÚ‡ÍÚ‡
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    ; ‰) ÒÚ‡ˆËÓ̇Ì ̊È ÚÂÔÎÓ‚ÓÈ ÔÓÚÓÍ, ‚ÓÁ‰ÂÈÒÚ‚Û ̨ ̆ËÈ Ì‡ ‚Ì ̄Ì ̨ ̨ ÔÓ‚ÂıÌÓÒÚ ̧ ÔÓÍ ̊Úˡ ‚ ̇Ô‡‚ÎÂÌËË ÓÒËOzËÒÔÓÎ ̧ÁÛÂÏÓÈ ÒËÒÚÂÏ ̊ ÍÓÓ‰Ë̇Ú, ˇ‚ΡÂÚÒˇ ÓÒÂÒËÏÏÂÚË ̃Ì ̊Ï Ë ËÏÂÂÚ ËÌÚÂÌÒË‚ÌÓÒÚ ̧ „‡ÛÒÒÓ‚ÒÍÓ„Ó ÚËÔ‡ Ò ÓÔ‰ÂΡ ̨ ̆ËÏË Ô‡‡ÏÂÚ‡ÏËk2Ëq0; Â) ÔË Î ̨· ̊ı ÙËÍÒËÓ‚‡ÌÌ ̊ı Á̇ ̃ÂÌˡıt>0Ëz>vt−hÙÛÌ͈ËÓ̇ÎT(r,z,t)ËÌÚ„ËÛÂÏ Ò Í‚‡‰‡ÚÓÏ Ë ‚ÂÒÓÏṙ ÔÓÎÛ·ÂÒÍÓÌ ̃ÌÓÏ ËÌÚÂ‚‡ÎÂ[0; +∞), Ú.

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    ÔË̇‰ÎÂÊËÚ ÔÓÒÚ‡ÌÒÚ‚ÛL2r[0; +∞)[12]; Ê) ÔË Î ̨· ̊ı ÙËÍÒËÓ‚‡ÌÌ ̊ı Á̇ ̃ÂÌˡıt>0Ër>0ÙÛÌ͈ËÓ̇ÎT(r,z,t)ËÌÚ„ËÛÂÏ Ò Í‚‡‰‡ÚÓÏ Ì‡ ÔÓÎÛ·ÂÒÍÓÌ ̃ÌÓÏ ËÌÚÂ‚‡ÎÂ[vt−h; +∞), Ú.Â. ÔË̇‰ÎÂÊËÚ ÔÓÒÚ‡ÌÒÚ‚Û L2[vt−h; +∞)[12]; Á) ÔË Î ̨· ̊ı ÙËÍÒËÓ‚‡ÌÌ ̊ı Á̇ ̃ÂÌˡır>0Ëz>vt−hÙÛÌ͈ËÓ̇ÎT(r,z,t)ˇ‚ΡÂÚÒˇ ÓË„Ë̇ÎÓÏ ËÌÚ„‡Î ̧ÌÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇
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    ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛt, Ú.Â. ÔË̇‰ÎÂÊËÚ ÔÓÒÚ‡ÌÒÚ‚ÛL[0,+∞). œËÌˇÚË ‰ÓÔÛ ̆ÂÌËÈ ‡){‰) ÔË‚Ó‰ËÚ Í ÒÎÂ‰Û ̨ ̆ÂÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË ËÁÛ ̃‡ÂÏÓ„Ó ÔÓˆÂÒÒ‡, Ô‰ÒÚ‡‚Ρ ̨ ̆ÂÈ ÒÓ·ÓÈ ÒÏ ̄‡ÌÌÛ ̨ Á‡‰‡ ̃Û ‰Îˇ ÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ ‚ ̃‡ÒÚÌ ̊ı ÔÓËÁ‚Ó‰Ì ̊ı Ô‡‡·ÓÎË ̃ÂÒÍÓ„Ó ÚËÔ‡:                                  ∂θ ∂Fo = 1 ρ ∂ ∂ρ ρ ∂θ ∂ρ +

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    “ÂÏÔÂ‡ÚÛÌÓ ÔÓΠ”ÒÎӂˡ (5) ÔÓÁ‚ÓΡ ̨Ú ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓ ÔËÏÂÌËÚ ̧ Í ÒÏ ̄‡ÌÌÓÈ Á‡‰‡ ̃ (4) Ò̇ ̃‡Î‡ ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË √‡ÌÍÂΡ ÌÛÎÂ‚Ó„Ó ÔÓˇ‰Í‡ [12] ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÏÛ ÔÂÂÏÂÌÌÓÏÛρ H0[θ]≡ +∞∫ 0 θ(ρ)ρJ0(pρ)dρ,H−10[κ]≡ +∞∫ 0 κ(p)pJ0(pρ)dp,(6) ‡ Á‡ÚÂÏ | ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË À‡Ô·҇
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    ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛτ. œÓ·„‡ˇ A(p,X,τ) =H0[θ(ρ,X,τ)],(7) B(p,X,s) =L[A(p,X,τ)],(8) „‰ÂL| ÓÔÂ‡ÚÓ ÔˇÏÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇, Ë ËÒÔÓÎ ̧ÁÛˇ Ò‚ÓÈÒÚ‚‡ ËÌÚ„‡Î ̧Ì ̊ı ÔÂÓ·‡ÁÓ‚‡ÌËÈ √‡ÌÍÂΡ Ë À‡Ô·҇ [1,2,3,12] ‚ÍÛÔÂ Ò Ú‡·Îˈ‡ÏË <ËÁÓ·‡ÊÂÌËÂ-ÓË„Ë̇Î> [13], ÔËıÓ‰ËÏ Í Á‡‰‡ ̃ ‰Îˇ ÓÔ‰ÂÎÂÌˡ ËÁÓ·‡ÊÂÌˡB(p,X,s): d2B dX2 +V dB dX −(s+p2)B= 0,X>0;(9) d2B dX2 +α2V dB dX −(α2s+p2)B= 0,−ε<X<0;(

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    ÔÂÓ·‡ÁÓ‚‡ÌË √‡ÌÍÂΡ ÌÛÎÂ‚Ó„Ó ÔÓˇ‰Í‡ [12] ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÏÛ ÔÂÂÏÂÌÌÓÏÛρ H0[θ]≡ +∞∫ 0 θ(ρ)ρJ0(pρ)dρ,H−10[κ]≡ +∞∫ 0 κ(p)pJ0(pρ)dp,(6) ‡ Á‡ÚÂÏ | ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË À‡Ô·҇ [1,2] ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛτ. œÓ·„‡ˇ A(p,X,τ) =H0[θ(ρ,X,τ)],(7) B(p,X,s) =L[A(p,X,τ)],(8) „‰ÂL| ÓÔÂ‡ÚÓ ÔˇÏÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇, Ë ËÒÔÓÎ ̧ÁÛˇ Ò‚ÓÈÒÚ‚‡ ËÌÚ„‡Î ̧Ì ̊ı ÔÂÓ·‡ÁÓ‚‡ÌËÈ √‡ÌÍÂΡ Ë À‡Ô·҇
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    ‚ÍÛÔÂ Ò Ú‡·Îˈ‡ÏË <ËÁÓ·‡ÊÂÌËÂ-ÓË„Ë̇Î> [13], ÔËıÓ‰ËÏ Í Á‡‰‡ ̃ ‰Îˇ ÓÔ‰ÂÎÂÌˡ ËÁÓ·‡ÊÂÌˡB(p,X,s): d2B dX2 +V dB dX −(s+p2)B= 0,X>0;(9) d2B dX2 +α2V dB dX −(α2s+p2)B= 0,−ε<X<0;(10) dB dX ∣ ∣ ∣ ∣ ∣ X=−ε =− Q0 2K2s exp ( − p2 4K2 ) ;(11) B(p,0−0,s) =B(p,0 + 0,s);(12) Λ dB dX ∣ ∣ ∣ ∣∣ X=0−0 = dB dX ∣ ∣ ∣ ∣∣ X=0+0 ,(13) „‰Â ‚ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò Ô‰ÔÓÎÓÊÂÌˡÏË (5) ‰ÓÎÊÌÓ ‚ ̊ÔÓÎÌˇÚ ̧Òˇ ÛÒÎÓ‚Ë B(p,·,s

3
Kartashov E.M.Analiticheskie metody v teorii teploprovodnosti tverdykh tel[Analytic methods in the theory of thermal conductivity of solids]. Moscow, Vysshaya shkola Publ., 1985. 553 p. (in Russian).
Total in-text references: 4
  1. In-text reference with the coordinate start=1950
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    Î ̨ ̃‚ ̊ ÒÎÓ‚‡:ÚÂÏÔÂ‡ÚÛÌÓ ÔÓÎÂ, ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚Ó, ÔÓ‰‚ËÊ̇ˇ „‡Ìˈ‡ ¬‚‰ÂÌË ¬ Ò‚ˇÁË Ò ËÌÚÂÌÒË‚Ì ̊Ï ‚̉ÂÌËÂÏ ‚ ËÌÊÂÌÂÌÛ ̨ Ô‡ÍÚËÍÛ ÏÂÚÓ‰Ó‚ χÚÂχÚË ̃ÂÒÍÓ„Ó ÏÓ‰ÂÎËÓ‚‡Ìˡ, ̇ˇ‰Û Ò ‚ ̊ ̃ËÒÎËÚÂÎ ̧Ì ̊ÏË ÏÂÚÓ‰‡ÏË ‚Ò ·ÓÎ ̧ ̄Û ̨ Á̇ ̃ËÏÓÒÚ ̧ ÔËÓ·ÂÚ‡ ̨Ú ‡Ì‡ÎËÚË ̃ÂÒÍË ÏÂÚÓ‰ ̊  ̄ÂÌˡ Á‡‰‡ ̃ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË
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    . ÕÂÒÏÓÚˇ ̇ ÂÒÚÂÒÚ‚ÂÌÌÛ ̨ Ó„‡ÌË ̃ÂÌÌÓÒÚ ̧ ӷ·ÒÚË ÔËÏÂÌËÏÓÒÚË ‡Ì‡ÎËÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚, ̇·Î ̨‰‡Âχˇ ÚẨÂÌˆËˇ Ó·ÛÒÎÓ‚ÎÂ̇ ÏÌÓ„ËÏË ÔË ̃Ë̇ÏË. ¬ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃, ÔÓÎÛ ̃ÂÌÌ ̊ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉÂ, ÔÓÁ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊ [4,5,6,7], ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒÎÂ

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    ÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9]. “Û‰ÌÓÒÚË, ÍÓÚÓ ̊ ‰ÓÎÊÂÌ ÔÂÓ‰ÓÎÂÚ ̧ ËÒÒΉӂ‡ÚÂÎ ̧ ÔË ÔÓÎÛ ̃ÂÌËË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ ÚÓÈ ËÎË ËÌÓÈ Á‡‰‡ ̃Ë Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, ıÓÓ ̄Ó ËÁ‚ÂÒÚÌ ̊
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    . “ÂÏ Ì ÏÂÌ ÓÌË Â ̆ ·ÓΠÛÒÛ„Û·Îˇ ̨ÚÒˇ, ÂÒÎË ‚ÓÁÌË͇ÂÚ ÌÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ‚ÎˡÌˡ ‡ÁÎË ̃Ì ̊ı ÙËÁËÍÓ-ıËÏË ̃ÂÒÍËı Ë ÏÂı‡ÌË ̃ÂÒÍËı ÔÓˆÂÒÒÓ‚, ÔË‚Ó‰ˇ ̆Ëı Í ËÁÏÂÌÂÌË ̨ ‚Ó ‚ÂÏÂÌË „‡Ìˈ ËÁÛ ̃‡ÂÏÓ„Ó Ú‚Â‰Ó„Ó Ú·. œË ̋ÚÓÏ ÔÓ·ÎÂχÚË ̃ÌÓÒÚ ̧ ̇ıÓʉÂÌˡ  ̄ÂÌËÈ Á‡‰‡ ̃ Ú‡ÍÓ„Ó Í·ÒÒ‡ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ ÒÓı‡ÌˇÂÚÒˇ ‰‡Ê ‚ ÚÂı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ [10,11].

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    ‡) ¬ ̇ ̃‡Î ̧Ì ̊È ÏÓÏÂÌÚ ‚ÂÏÂÌËt= 0ÚÂÏÔÂ‡ÚÛ‡ ‚ Î ̨·ÓÈ ÚÓ ̃Í ÒËÒÚÂÏ ̊ ÔÓÒÚÓˇÌ̇: T(r,z,0)≡T0= const; ·) Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁÓÚÓÔÌÓ„Ó ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚‡ ˇ‚ΡÂÚÒˇ ÎËÌÂÈÌ ̊Ï, Ú.Â.z= vt, „‰Âv= const; ‚) ËÁÓÚÓÔÌÓ ÔÓÍ ̊ÚË ËÏÂÂÚ ÔÓÒÚÓˇÌÌÛ ̨ ‚Ó ‚ÂÏÂÌË ÚÓÎ ̆ËÌÛh; „) ‚ ÒËÒÚÂÏ <ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚Ó | ÔÓÍ ̊ÚËÂ> ‡ÎËÁÛ ̨ÚÒˇ ÛÒÎӂˡ ˉ‡Π̧ÌÓ„Ó ÚÂÔÎÓ‚Ó„Ó ÍÓÌÚ‡ÍÚ‡
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    ; ‰) ÒÚ‡ˆËÓ̇Ì ̊È ÚÂÔÎÓ‚ÓÈ ÔÓÚÓÍ, ‚ÓÁ‰ÂÈÒÚ‚Û ̨ ̆ËÈ Ì‡ ‚Ì ̄Ì ̨ ̨ ÔÓ‚ÂıÌÓÒÚ ̧ ÔÓÍ ̊Úˡ ‚ ̇Ô‡‚ÎÂÌËË ÓÒËOzËÒÔÓÎ ̧ÁÛÂÏÓÈ ÒËÒÚÂÏ ̊ ÍÓÓ‰Ë̇Ú, ˇ‚ΡÂÚÒˇ ÓÒÂÒËÏÏÂÚË ̃Ì ̊Ï Ë ËÏÂÂÚ ËÌÚÂÌÒË‚ÌÓÒÚ ̧ „‡ÛÒÒÓ‚ÒÍÓ„Ó ÚËÔ‡ Ò ÓÔ‰ÂΡ ̨ ̆ËÏË Ô‡‡ÏÂÚ‡ÏËk2Ëq0; Â) ÔË Î ̨· ̊ı ÙËÍÒËÓ‚‡ÌÌ ̊ı Á̇ ̃ÂÌˡıt>0Ëz>vt−hÙÛÌ͈ËÓ̇ÎT(r,z,t)ËÌÚ„ËÛÂÏ Ò Í‚‡‰‡ÚÓÏ Ë ‚ÂÒÓÏṙ ÔÓÎÛ·ÂÒÍÓÌ ̃ÌÓÏ ËÌÚÂ‚‡ÎÂ[0; +∞), Ú.

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    ÔÂÓ·‡ÁÓ‚‡ÌË √‡ÌÍÂΡ ÌÛÎÂ‚Ó„Ó ÔÓˇ‰Í‡ [12] ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÏÛ ÔÂÂÏÂÌÌÓÏÛρ H0[θ]≡ +∞∫ 0 θ(ρ)ρJ0(pρ)dρ,H−10[κ]≡ +∞∫ 0 κ(p)pJ0(pρ)dp,(6) ‡ Á‡ÚÂÏ | ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË À‡Ô·҇ [1,2] ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛτ. œÓ·„‡ˇ A(p,X,τ) =H0[θ(ρ,X,τ)],(7) B(p,X,s) =L[A(p,X,τ)],(8) „‰ÂL| ÓÔÂ‡ÚÓ ÔˇÏÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇, Ë ËÒÔÓÎ ̧ÁÛˇ Ò‚ÓÈÒÚ‚‡ ËÌÚ„‡Î ̧Ì ̊ı ÔÂÓ·‡ÁÓ‚‡ÌËÈ √‡ÌÍÂΡ Ë À‡Ô·҇
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    ‚ÍÛÔÂ Ò Ú‡·Îˈ‡ÏË <ËÁÓ·‡ÊÂÌËÂ-ÓË„Ë̇Î> [13], ÔËıÓ‰ËÏ Í Á‡‰‡ ̃ ‰Îˇ ÓÔ‰ÂÎÂÌˡ ËÁÓ·‡ÊÂÌˡB(p,X,s): d2B dX2 +V dB dX −(s+p2)B= 0,X>0;(9) d2B dX2 +α2V dB dX −(α2s+p2)B= 0,−ε<X<0;(10) dB dX ∣ ∣ ∣ ∣ ∣ X=−ε =− Q0 2K2s exp ( − p2 4K2 ) ;(11) B(p,0−0,s) =B(p,0 + 0,s);(12) Λ dB dX ∣ ∣ ∣ ∣∣ X=0−0 = dB dX ∣ ∣ ∣ ∣∣ X=0+0 ,(13) „‰Â ‚ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò Ô‰ÔÓÎÓÊÂÌˡÏË (5) ‰ÓÎÊÌÓ ‚ ̊ÔÓÎÌˇÚ ̧Òˇ ÛÒÎÓ‚Ë B(p,·,s

4
Formaljov V.F., Kuznetsova E.L.Teplomassoperenos v anizotropnykh telakh pri aerogazodinamicheskom nagreve[Heat and mass transfer in anisotropic solids under aerogasdynamic heating]. Moscow, MAI Publ., 2010. 308 p. (in Russian).
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  1. In-text reference with the coordinate start=1950
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    Î ̨ ̃‚ ̊ ÒÎÓ‚‡:ÚÂÏÔÂ‡ÚÛÌÓ ÔÓÎÂ, ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚Ó, ÔÓ‰‚ËÊ̇ˇ „‡Ìˈ‡ ¬‚‰ÂÌË ¬ Ò‚ˇÁË Ò ËÌÚÂÌÒË‚Ì ̊Ï ‚̉ÂÌËÂÏ ‚ ËÌÊÂÌÂÌÛ ̨ Ô‡ÍÚËÍÛ ÏÂÚÓ‰Ó‚ χÚÂχÚË ̃ÂÒÍÓ„Ó ÏÓ‰ÂÎËÓ‚‡Ìˡ, ̇ˇ‰Û Ò ‚ ̊ ̃ËÒÎËÚÂÎ ̧Ì ̊ÏË ÏÂÚÓ‰‡ÏË ‚Ò ·ÓÎ ̧ ̄Û ̨ Á̇ ̃ËÏÓÒÚ ̧ ÔËÓ·ÂÚ‡ ̨Ú ‡Ì‡ÎËÚË ̃ÂÒÍË ÏÂÚÓ‰ ̊  ̄ÂÌˡ Á‡‰‡ ̃ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË
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    . ÕÂÒÏÓÚˇ ̇ ÂÒÚÂÒÚ‚ÂÌÌÛ ̨ Ó„‡ÌË ̃ÂÌÌÓÒÚ ̧ ӷ·ÒÚË ÔËÏÂÌËÏÓÒÚË ‡Ì‡ÎËÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚, ̇·Î ̨‰‡Âχˇ ÚẨÂÌˆËˇ Ó·ÛÒÎÓ‚ÎÂ̇ ÏÌÓ„ËÏË ÔË ̃Ë̇ÏË. ¬ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃, ÔÓÎÛ ̃ÂÌÌ ̊ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉÂ, ÔÓÁ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊ [4,5,6,7], ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒÎÂ

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    ÕÂÒÏÓÚˇ ̇ ÂÒÚÂÒÚ‚ÂÌÌÛ ̨ Ó„‡ÌË ̃ÂÌÌÓÒÚ ̧ ӷ·ÒÚË ÔËÏÂÌËÏÓÒÚË ‡Ì‡ÎËÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚, ̇·Î ̨‰‡Âχˇ ÚẨÂÌˆËˇ Ó·ÛÒÎÓ‚ÎÂ̇ ÏÌÓ„ËÏË ÔË ̃Ë̇ÏË. ¬ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃, ÔÓÎÛ ̃ÂÌÌ ̊ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉÂ, ÔÓÁ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊
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    , ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒΉӂ‡Ú ̧ ÒÔˆËÙË ̃ÂÒÍË ÓÒÓ·ÂÌÌÓÒÚË ÔÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9].

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    Prefix
    ÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9]. “Û‰ÌÓÒÚË, ÍÓÚÓ ̊ ‰ÓÎÊÂÌ ÔÂÓ‰ÓÎÂÚ ̧ ËÒÒΉӂ‡ÚÂÎ ̧ ÔË ÔÓÎÛ ̃ÂÌËË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ ÚÓÈ ËÎË ËÌÓÈ Á‡‰‡ ̃Ë Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, ıÓÓ ̄Ó ËÁ‚ÂÒÚÌ ̊
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    . “ÂÏ Ì ÏÂÌ ÓÌË Â ̆ ·ÓΠÛÒÛ„Û·Îˇ ̨ÚÒˇ, ÂÒÎË ‚ÓÁÌË͇ÂÚ ÌÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ‚ÎˡÌˡ ‡ÁÎË ̃Ì ̊ı ÙËÁËÍÓ-ıËÏË ̃ÂÒÍËı Ë ÏÂı‡ÌË ̃ÂÒÍËı ÔÓˆÂÒÒÓ‚, ÔË‚Ó‰ˇ ̆Ëı Í ËÁÏÂÌÂÌË ̨ ‚Ó ‚ÂÏÂÌË „‡Ìˈ ËÁÛ ̃‡ÂÏÓ„Ó Ú‚Â‰Ó„Ó Ú·. œË ̋ÚÓÏ ÔÓ·ÎÂχÚË ̃ÌÓÒÚ ̧ ̇ıÓʉÂÌˡ  ̄ÂÌËÈ Á‡‰‡ ̃ Ú‡ÍÓ„Ó Í·ÒÒ‡ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ ÒÓı‡ÌˇÂÚÒˇ ‰‡Ê ‚ ÚÂı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ [10,11].

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Nechepurenko Y.M., Ovchinnikov G.V. An estimation of voltage settling time for RC circuits. Russian Journal of Numerical Analysis and Mathematical Modelling, 2010, vol. 25, no. 3, pp. 253{259.
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    ÕÂÒÏÓÚˇ ̇ ÂÒÚÂÒÚ‚ÂÌÌÛ ̨ Ó„‡ÌË ̃ÂÌÌÓÒÚ ̧ ӷ·ÒÚË ÔËÏÂÌËÏÓÒÚË ‡Ì‡ÎËÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚, ̇·Î ̨‰‡Âχˇ ÚẨÂÌˆËˇ Ó·ÛÒÎÓ‚ÎÂ̇ ÏÌÓ„ËÏË ÔË ̃Ë̇ÏË. ¬ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃, ÔÓÎÛ ̃ÂÌÌ ̊ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉÂ, ÔÓÁ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊
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    , ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒΉӂ‡Ú ̧ ÒÔˆËÙË ̃ÂÒÍË ÓÒÓ·ÂÌÌÓÒÚË ÔÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9].

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Olshevsky V., Tyrtyshnikov E., Zlobich P. Tellegen's principle, non-minimal realizations of systems and inversion of polynomial Vandermonde matricies.Russian Journal of Numerical Analysis and Mathematical Modelling, 2012, vol. 27, no. 2, pp. 131{154.
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    ÕÂÒÏÓÚˇ ̇ ÂÒÚÂÒÚ‚ÂÌÌÛ ̨ Ó„‡ÌË ̃ÂÌÌÓÒÚ ̧ ӷ·ÒÚË ÔËÏÂÌËÏÓÒÚË ‡Ì‡ÎËÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚, ̇·Î ̨‰‡Âχˇ ÚẨÂÌˆËˇ Ó·ÛÒÎÓ‚ÎÂ̇ ÏÌÓ„ËÏË ÔË ̃Ë̇ÏË. ¬ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃, ÔÓÎÛ ̃ÂÌÌ ̊ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉÂ, ÔÓÁ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊
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    , ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒΉӂ‡Ú ̧ ÒÔˆËÙË ̃ÂÒÍË ÓÒÓ·ÂÌÌÓÒÚË ÔÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9].

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Bella T., Olshevsky V., Zlobich P. A quasiseparable approach to five-diagonal CMV and Fiedler matricies.Linear Algebra and its Applications, 2011, vol. 434, no. 4, pp. 957{976.
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    ÕÂÒÏÓÚˇ ̇ ÂÒÚÂÒÚ‚ÂÌÌÛ ̨ Ó„‡ÌË ̃ÂÌÌÓÒÚ ̧ ӷ·ÒÚË ÔËÏÂÌËÏÓÒÚË ‡Ì‡ÎËÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚, ̇·Î ̨‰‡Âχˇ ÚẨÂÌˆËˇ Ó·ÛÒÎÓ‚ÎÂ̇ ÏÌÓ„ËÏË ÔË ̃Ë̇ÏË. ¬ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌˡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı Á‡‰‡ ̃, ÔÓÎÛ ̃ÂÌÌ ̊ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉÂ, ÔÓÁ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊
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    , ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒΉӂ‡Ú ̧ ÒÔˆËÙË ̃ÂÒÍË ÓÒÓ·ÂÌÌÓÒÚË ÔÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9].

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Pudovkin M.A., Volkov I.K.Kraevye zadachi matematicheskoi teorii teploprovodnosti v prilozhenii k raschetam temperaturnykh polei v neftyanykh plastakh pri zavodnenii[Boundaryvalue problems of mathematical theory of heat conduction in the annex to the calculations of temperature fields in oil reservoirs during flooding]. Kazan, Kazan University Publ., 1978. 188 p. (in Russian).
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  1. In-text reference with the coordinate start=2644
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    ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊ [4,5,6,7], ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒΉӂ‡Ú ̧ ÒÔˆËÙË ̃ÂÒÍË ÓÒÓ·ÂÌÌÓÒÚË ÔÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ
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    [8,9]
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    . “Û‰ÌÓÒÚË, ÍÓÚÓ ̊ ‰ÓÎÊÂÌ ÔÂÓ‰ÓÎÂÚ ̧ ËÒÒΉӂ‡ÚÂÎ ̧ ÔË ÔÓÎÛ ̃ÂÌËË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ ÚÓÈ ËÎË ËÌÓÈ Á‡‰‡ ̃Ë Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, ıÓÓ ̄Ó ËÁ‚ÂÒÚÌ ̊ [1,2,3,4,8].

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    ÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ [8,9]. “Û‰ÌÓÒÚË, ÍÓÚÓ ̊ ‰ÓÎÊÂÌ ÔÂÓ‰ÓÎÂÚ ̧ ËÒÒΉӂ‡ÚÂÎ ̧ ÔË ÔÓÎÛ ̃ÂÌËË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ ÚÓÈ ËÎË ËÌÓÈ Á‡‰‡ ̃Ë Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, ıÓÓ ̄Ó ËÁ‚ÂÒÚÌ ̊
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    . “ÂÏ Ì ÏÂÌ ÓÌË Â ̆ ·ÓΠÛÒÛ„Û·Îˇ ̨ÚÒˇ, ÂÒÎË ‚ÓÁÌË͇ÂÚ ÌÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ‚ÎˡÌˡ ‡ÁÎË ̃Ì ̊ı ÙËÁËÍÓ-ıËÏË ̃ÂÒÍËı Ë ÏÂı‡ÌË ̃ÂÒÍËı ÔÓˆÂÒÒÓ‚, ÔË‚Ó‰ˇ ̆Ëı Í ËÁÏÂÌÂÌË ̨ ‚Ó ‚ÂÏÂÌË „‡Ìˈ ËÁÛ ̃‡ÂÏÓ„Ó Ú‚Â‰Ó„Ó Ú·. œË ̋ÚÓÏ ÔÓ·ÎÂχÚË ̃ÌÓÒÚ ̧ ̇ıÓʉÂÌˡ  ̄ÂÌËÈ Á‡‰‡ ̃ Ú‡ÍÓ„Ó Í·ÒÒ‡ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ ÒÓı‡ÌˇÂÚÒˇ ‰‡Ê ‚ ÚÂı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ [10,11].

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    =0−0 = ∂θ ∂X ∣ ∣ ∣ ∣ ∣ X=0+0 . (4) œË ̋ÚÓÏ ÛÒÎӂˡ (2) ÔËÏÛÚ ‚ˉ      θ(·,X,τ)∈L2ρ[0,+∞),X>−ε,τ>0, θ(ρ,·,τ)∈L2[−ε,+∞),ρ>0,τ>0, θ(ρ,X,·)∈L[0,+∞),ρ>0,X>−ε. (5) —ΉÛÂÚ Á‡ÏÂÚËÚ ̧, ̃ÚÓ Òӄ·ÒÌÓ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË (4) ‚ ÔÓ‰‚ËÊÌÓÈ ÒËÒÚÂÏ ÍÓÓ‰ËÌ‡Ú (3) ÔÓˆÂÒÒ ÙÓÏËÓ‚‡Ìˡ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ Ó·Î‡ÒÚË Ô‰ÒÚ‡‚ΡÂÚ ÒÓ·ÓÈ ÍÓÏÔÓÁËˆË ̨ ÍÓ̉ÛÍÚË‚ÌÓ„Ó Ë ÍÓÌ‚ÂÍÚË‚ÌÓ„Ó ÚÂÔÎÓÔÂÂÌÓÒ‡
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    . œÂ‚ ̊È ËÁ ÌËı ‡ÎËÁÛÂÚ ‰ËÙÙÛÁËÓÌÌ ̊È <ÔÂÂÌÓÒ ÚÂÔ·>, ÔÓÒÚÛÔ‡ ̨ ̆Â„Ó ‚ ÒËÒÚÂÏÛ ÔË ‚ÓÁ‰ÂÈÒÚ‚ËË ‚Ì ̄ÌÂ„Ó ÚÂÔÎÓ‚Ó„Ó ÔÓÚÓ͇, ÓÚ ‚Ì ̄ÌÂÈ „‡Ìˈ ̊ ÔÓÍ ̊Úˡ Í ÔÂËÙÂËË ËÁÓÚÓÔÌÓ„Ó ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚‡, ‡ ‚ÚÓÓÈ | ÍÓÌ‚ÂÍÚË‚Ì ̊È <ÔÂÂÌÓÒ ıÓÎÓ‰‡> ÓÚ ÔÂËÙÂËË ËÁÓÚÓÔÌÓ„Ó ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚‡ Í ‚Ì ̄ÌÂÈ „‡ÌËˆÂ Â„Ó ÔÓÍ ̊Úˡ ‚ ̇Ô‡‚ÎÂÌËË ‚ÂÍÚÓ‡ ‚Ì ̄ÌÂÈ ÌÓχÎË Í ÌÂÈ. œÓÒÍÓÎ ̧ÍÛ ÒÍÓÓÒÚ ̧V ̋ÚÓ„Ó ÔÂÂÌÓÒ‡

9
Attetkov A.V., Volkov I.K., Tverskaya E.S. The Hierarchy of Mathematical Models of the Process of Temperature Field Formation in the System \Plane Isotropic Wall | Thermal Active Layer | Anisotropic Coating".Teplovye protsessy v tekhnike=Thermal Processes in Engineering, 2013, no. 5, pp. 224{228. (in Russian).
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    ‚ÓΡ ̨Ú ÚÂÒÚËÓ‚‡Ú ̧ ÌÓ‚ ̊ ‚ ̊ÒÓÍÓÔÓËÁ‚Ó‰ËÚÂÎ ̧Ì ̊ ‡Î„ÓËÚÏ ̊ [4,5,6,7], ÔÓ‚Ó‰ËÚ ̧ Ô‡‡ÏÂÚË ̃ÂÒÍËÈ ‡Ì‡ÎËÁ ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ËÁÛ ̃‡ÂÏÓÈ ÒËÒÚÂÏ ̊ Ë ËÒÒΉӂ‡Ú ̧ ÒÔˆËÙË ̃ÂÒÍË ÓÒÓ·ÂÌÌÓÒÚË ÔÓˆÂÒÒ‡  ÙÓÏËÓ‚‡Ìˡ, ÙÓÏÛÎËÓ‚‡Ú ̧ Ë  ̄‡Ú ̧ Á‡‰‡ ̃Ë ÓÔÚËÏËÁ‡ˆËË, ‡ Ú‡ÍÊ ËÒÒΉӂ‡Ú ̧ ‚ÓÁÏÓÊÌÓÒÚË ÛÔÓ ̆ÂÌˡ ËÒÔÓÎ ̧ÁÛÂÏÓÈ Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÏÓ‰ÂÎË Ò ÒÓı‡ÌÂÌËÂÏ Â ‡‰ÂÍ‚‡ÚÌÓÒÚË ËÁÛ ̃‡ÂÏÓÏÛ ÔÓˆÂÒÒÛ
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    . “Û‰ÌÓÒÚË, ÍÓÚÓ ̊ ‰ÓÎÊÂÌ ÔÂÓ‰ÓÎÂÚ ̧ ËÒÒΉӂ‡ÚÂÎ ̧ ÔË ÔÓÎÛ ̃ÂÌËË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ ÚÓÈ ËÎË ËÌÓÈ Á‡‰‡ ̃Ë Ï‡ÚÂχÚË ̃ÂÒÍÓÈ ÚÂÓËË ÚÂÔÎÓÔÓ‚Ó‰ÌÓÒÚË, ıÓÓ ̄Ó ËÁ‚ÂÒÚÌ ̊ [1,2,3,4,8].

10
Kartashov E.M. Analytical Methods of Solution of Boundary Value Problems of Nonstationary Heat Conduction in Regions with Moving Boundaries.Proceedings of the RAS. Power engineering, 1999, no. 5, pp. 3{34. (in Russian).
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    “ÂÏ Ì ÏÂÌ ÓÌË Â ̆ ·ÓΠÛÒÛ„Û·Îˇ ̨ÚÒˇ, ÂÒÎË ‚ÓÁÌË͇ÂÚ ÌÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ‚ÎˡÌˡ ‡ÁÎË ̃Ì ̊ı ÙËÁËÍÓ-ıËÏË ̃ÂÒÍËı Ë ÏÂı‡ÌË ̃ÂÒÍËı ÔÓˆÂÒÒÓ‚, ÔË‚Ó‰ˇ ̆Ëı Í ËÁÏÂÌÂÌË ̨ ‚Ó ‚ÂÏÂÌË „‡Ìˈ ËÁÛ ̃‡ÂÏÓ„Ó Ú‚Â‰Ó„Ó Ú·. œË ̋ÚÓÏ ÔÓ·ÎÂχÚË ̃ÌÓÒÚ ̧ ̇ıÓʉÂÌˡ  ̄ÂÌËÈ Á‡‰‡ ̃ Ú‡ÍÓ„Ó Í·ÒÒ‡ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ ÒÓı‡ÌˇÂÚÒˇ ‰‡Ê ‚ ÚÂı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ
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    . ŒÒÌӂ̇ˇ ˆÂÎ ̧ Ôӂ‰ÂÌÌ ̊ı ËÒÒΉӂ‡ÌËÈ | Ô‰ÒÚ‡‚ÎÂÌË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ Á‡‰‡ ̃Ë Ì‡ıÓʉÂÌˡ Í‚‡ÁËÒÚ‡ˆËÓ̇ÌÓ„Ó ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ÒËÒÚÂÏ ̊, ËÏËÚËÛÂÏÓÈ ËÁÓÚÓÔÌ ̊Ï ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚ÓÏ, ‚Ì ̄Ìˇˇ „‡Ìˈ‡ ÍÓÚÓÓ„Ó, ӷ·‰‡ ̨ ̆‡ˇ ËÁÓÚÓÔÌ ̊Ï ÔÓÍ ̊ÚËÂÏ ÔÓÒÚÓˇÌÌÓÈ ÚÓÎ ̆ËÌ ̊ Ë Ì‡ıÓ‰ˇ ̆‡ˇÒˇ ÔÓ‰ ‚ÓÁ‰ÂÈÒÚ‚ËÂÏ ÒÚ‡ˆËÓ̇ÌÓ„Ó ÚÂÔÎÓ‚Ó„Ó ÔÓÚÓ͇ „‡ÛÒÒÓ‚ÒÍÓ„Ó ÚËÔ‡, ÔÂÂÏ ̆‡ÂÚÒˇ Ò ÔÓÒÚÓˇÌÌ

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Kartashov E.M., Lyubov B.Ya. Analitic Methods in Solving Boundary Value Problems of Heat Conduction in a Region with Moving Boundaries.Heat Transfer | Soviet Research, 1976, vol. 8, no. 2, pp. 1{39.
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    “ÂÏ Ì ÏÂÌ ÓÌË Â ̆ ·ÓΠÛÒÛ„Û·Îˇ ̨ÚÒˇ, ÂÒÎË ‚ÓÁÌË͇ÂÚ ÌÂÓ·ıÓ‰ËÏÓÒÚ ̧ Û ̃ÂÚ‡ ‚ÎˡÌˡ ‡ÁÎË ̃Ì ̊ı ÙËÁËÍÓ-ıËÏË ̃ÂÒÍËı Ë ÏÂı‡ÌË ̃ÂÒÍËı ÔÓˆÂÒÒÓ‚, ÔË‚Ó‰ˇ ̆Ëı Í ËÁÏÂÌÂÌË ̨ ‚Ó ‚ÂÏÂÌË „‡Ìˈ ËÁÛ ̃‡ÂÏÓ„Ó Ú‚Â‰Ó„Ó Ú·. œË ̋ÚÓÏ ÔÓ·ÎÂχÚË ̃ÌÓÒÚ ̧ ̇ıÓʉÂÌˡ  ̄ÂÌËÈ Á‡‰‡ ̃ Ú‡ÍÓ„Ó Í·ÒÒ‡ ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ ÒÓı‡ÌˇÂÚÒˇ ‰‡Ê ‚ ÚÂı ÒÎÛ ̃‡ˇı, ÍÓ„‰‡ Á‡ÍÓÌ ‰‚ËÊÂÌˡ „‡Ìˈ ̊ ËÁ‚ÂÒÚÂÌ
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    . ŒÒÌӂ̇ˇ ˆÂÎ ̧ Ôӂ‰ÂÌÌ ̊ı ËÒÒΉӂ‡ÌËÈ | Ô‰ÒÚ‡‚ÎÂÌË ‚ ‡Ì‡ÎËÚË ̃ÂÒÍË Á‡ÏÍÌÛÚÓÏ ‚ˉ  ̄ÂÌˡ Á‡‰‡ ̃Ë Ì‡ıÓʉÂÌˡ Í‚‡ÁËÒÚ‡ˆËÓ̇ÌÓ„Ó ÚÂÏÔÂ‡ÚÛÌÓ„Ó ÔÓΡ ÒËÒÚÂÏ ̊, ËÏËÚËÛÂÏÓÈ ËÁÓÚÓÔÌ ̊Ï ÔÓÎÛÔÓÒÚ‡ÌÒÚ‚ÓÏ, ‚Ì ̄Ìˇˇ „‡Ìˈ‡ ÍÓÚÓÓ„Ó, ӷ·‰‡ ̨ ̆‡ˇ ËÁÓÚÓÔÌ ̊Ï ÔÓÍ ̊ÚËÂÏ ÔÓÒÚÓˇÌÌÓÈ ÚÓÎ ̆ËÌ ̊ Ë Ì‡ıÓ‰ˇ ̆‡ˇÒˇ ÔÓ‰ ‚ÓÁ‰ÂÈÒÚ‚ËÂÏ ÒÚ‡ˆËÓ̇ÌÓ„Ó ÚÂÔÎÓ‚Ó„Ó ÔÓÚÓ͇ „‡ÛÒÒÓ‚ÒÍÓ„Ó ÚËÔ‡, ÔÂÂÏ ̆‡ÂÚÒˇ Ò ÔÓÒÚÓˇÌÌ

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Koshliakov N.S., Gleaner E.B., Smirnov M.M.Uravneniya v chastnykh proizvodnykh matematicheskoi fiziki[Partial differential equations of mathematical physics]. Moscow, Vysshaya shkola Publ., 1970. 712 p. (in Russian).
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    Í ÌÂÈ. œÓÒÍÓÎ ̧ÍÛ ÒÍÓÓÒÚ ̧V ̋ÚÓ„Ó ÔÂÂÌÓÒ‡ ˇ‚ΡÂÚÒˇ ÔÓÒÚÓˇÌÌÓÈ, ‡ ÏÓ ̆ÌÓÒÚ ̧ ‚Ì ̄ÌÂ„Ó ÚÂÔÎÓ‚Ó„Ó ÔÓÚÓ͇ Ú‡ÍÊ Ì Á‡‚ËÒËÚ ÓÚ ‚ÂÏÂÌË, ÚÓ ËÁ ÙËÁË ̃ÂÒÍËı ÒÓÓ·‡ÊÂÌËÈ ÒÚ‡ÌÓ‚ËÚÒˇ Ó ̃Â‚Ë‰Ì ̊Ï ÒÛ ̆ÂÒÚ‚Ó‚‡ÌË ËÒÍÓÏÓ„Ó Í‚‡ÁËÒÚ‡ˆËÓ̇ÌÓ„Ó ÔÓΡ. 2. “ÂÏÔÂ‡ÚÛÌÓ ÔÓΠ”ÒÎӂˡ (5) ÔÓÁ‚ÓΡ ̨Ú ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓ ÔËÏÂÌËÚ ̧ Í ÒÏ ̄‡ÌÌÓÈ Á‡‰‡ ̃ (4) Ò̇ ̃‡Î‡ ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË √‡ÌÍÂΡ ÌÛÎÂ‚Ó„Ó ÔÓˇ‰Í‡
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    ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÏÛ ÔÂÂÏÂÌÌÓÏÛρ H0[θ]≡ +∞∫ 0 θ(ρ)ρJ0(pρ)dρ,H−10[κ]≡ +∞∫ 0 κ(p)pJ0(pρ)dp,(6) ‡ Á‡ÚÂÏ | ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË À‡Ô·҇ [1,2] ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛτ. œÓ·„‡ˇ A(p,X,τ) =H0[θ(ρ,X,τ)],(7) B(p,X,s) =L[A(p,X,τ)],(8) „‰ÂL| ÓÔÂ‡ÚÓ ÔˇÏÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇, Ë ËÒÔÓÎ ̧ÁÛˇ Ò‚ÓÈÒÚ‚‡ ËÌÚ„‡Î ̧Ì ̊ı ÔÂÓ·‡ÁÓ‚‡ÌËÈ √‡ÌÍÂΡ Ë À‡Ô·҇ [1,2,3,12] ‚ÍÛÔÂ Ò Ú‡·Îˈ‡ÏË <ËÁÓ·‡ÊÂÌËÂ-ÓË

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    ÔÂÓ·‡ÁÓ‚‡ÌË √‡ÌÍÂΡ ÌÛÎÂ‚Ó„Ó ÔÓˇ‰Í‡ [12] ÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÏÛ ÔÂÂÏÂÌÌÓÏÛρ H0[θ]≡ +∞∫ 0 θ(ρ)ρJ0(pρ)dρ,H−10[κ]≡ +∞∫ 0 κ(p)pJ0(pρ)dp,(6) ‡ Á‡ÚÂÏ | ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË À‡Ô·҇ [1,2] ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛτ. œÓ·„‡ˇ A(p,X,τ) =H0[θ(ρ,X,τ)],(7) B(p,X,s) =L[A(p,X,τ)],(8) „‰ÂL| ÓÔÂ‡ÚÓ ÔˇÏÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇, Ë ËÒÔÓÎ ̧ÁÛˇ Ò‚ÓÈÒÚ‚‡ ËÌÚ„‡Î ̧Ì ̊ı ÔÂÓ·‡ÁÓ‚‡ÌËÈ √‡ÌÍÂΡ Ë À‡Ô·҇
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    ‚ÍÛÔÂ Ò Ú‡·Îˈ‡ÏË <ËÁÓ·‡ÊÂÌËÂ-ÓË„Ë̇Î> [13], ÔËıÓ‰ËÏ Í Á‡‰‡ ̃ ‰Îˇ ÓÔ‰ÂÎÂÌˡ ËÁÓ·‡ÊÂÌˡB(p,X,s): d2B dX2 +V dB dX −(s+p2)B= 0,X>0;(9) d2B dX2 +α2V dB dX −(α2s+p2)B= 0,−ε<X<0;(10) dB dX ∣ ∣ ∣ ∣ ∣ X=−ε =− Q0 2K2s exp ( − p2 4K2 ) ;(11) B(p,0−0,s) =B(p,0 + 0,s);(12) Λ dB dX ∣ ∣ ∣ ∣∣ X=0−0 = dB dX ∣ ∣ ∣ ∣∣ X=0+0 ,(13) „‰Â ‚ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò Ô‰ÔÓÎÓÊÂÌˡÏË (5) ‰ÓÎÊÌÓ ‚ ̊ÔÓÎÌˇÚ ̧Òˇ ÛÒÎÓ‚Ë B(p,·,s

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Erdelyi A., ed.Tables of integral transforms. Vol. 2. Based, in part, on notes left by Harry Bateman. McGraw-Hill, 1954. (Bateman Manuscript Project). (Russ. ed.: Bateman H., Erdelyi A.Tablitsy integral'nykh preobrazovaniy. T. 2. Preobrazovaniya Besselya. Integraly ot spetsial'nykh funktsiy. Moscow, Nauka Publ., 1970. 328 p.).
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  1. In-text reference with the coordinate start=8152
    Prefix
    ‡ÌÒÚ‚ÂÌÌÓÏÛ ÔÂÂÏÂÌÌÓÏÛρ H0[θ]≡ +∞∫ 0 θ(ρ)ρJ0(pρ)dρ,H−10[κ]≡ +∞∫ 0 κ(p)pJ0(pρ)dp,(6) ‡ Á‡ÚÂÏ | ËÌÚ„‡Î ̧ÌÓ ÔÂÓ·‡ÁÓ‚‡ÌË À‡Ô·҇ [1,2] ÔÓ ‚ÂÏÂÌÌ-ÓÏÛ ÔÂÂÏÂÌÌÓÏÛτ. œÓ·„‡ˇ A(p,X,τ) =H0[θ(ρ,X,τ)],(7) B(p,X,s) =L[A(p,X,τ)],(8) „‰ÂL| ÓÔÂ‡ÚÓ ÔˇÏÓ„Ó ÔÂÓ·‡ÁÓ‚‡Ìˡ À‡Ô·҇, Ë ËÒÔÓÎ ̧ÁÛˇ Ò‚ÓÈÒÚ‚‡ ËÌÚ„‡Î ̧Ì ̊ı ÔÂÓ·‡ÁÓ‚‡ÌËÈ √‡ÌÍÂΡ Ë À‡Ô·҇ [1,2,3,12] ‚ÍÛÔÂ Ò Ú‡·Îˈ‡ÏË <ËÁÓ·‡ÊÂÌËÂ-ÓË„Ë̇Î>
    Exact
    [13]
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    , ÔËıÓ‰ËÏ Í Á‡‰‡ ̃ ‰Îˇ ÓÔ‰ÂÎÂÌˡ ËÁÓ·‡ÊÂÌˡB(p,X,s): d2B dX2 +V dB dX −(s+p2)B= 0,X>0;(9) d2B dX2 +α2V dB dX −(α2s+p2)B= 0,−ε<X<0;(10) dB dX ∣ ∣ ∣ ∣ ∣ X=−ε =− Q0 2K2s exp ( − p2 4K2 ) ;(11) B(p,0−0,s) =B(p,0 + 0,s);(12) Λ dB dX ∣ ∣ ∣ ∣∣ X=0−0 = dB dX ∣ ∣ ∣ ∣∣ X=0+0 ,(13) „‰Â ‚ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò Ô‰ÔÓÎÓÊÂÌˡÏË (5) ‰ÓÎÊÌÓ ‚ ̊ÔÓÎÌˇÚ ̧Òˇ ÛÒÎÓ‚Ë B(p,·,s)∈L2[−ε,+∞),p∈R+,s∈C,(14) ÍÓÚÓÓ ڇÍÊ ˄‡ÂÚ

14
Elsgolts L.E.Differentsial'nye uravneniya i variatsionnoe ischislenie[Differential equations and calculus of variations]. Moscow, Nauka Publ., 1969. 424 p. (in Russian).
Total in-text references: 1
  1. In-text reference with the coordinate start=8709
    Prefix
    B(p,0 + 0,s);(12) Λ dB dX ∣ ∣ ∣ ∣∣ X=0−0 = dB dX ∣ ∣ ∣ ∣∣ X=0+0 ,(13) „‰Â ‚ ÒÓÓÚ‚ÂÚÒÚ‚ËË Ò Ô‰ÔÓÎÓÊÂÌˡÏË (5) ‰ÓÎÊÌÓ ‚ ̊ÔÓÎÌˇÚ ̧Òˇ ÛÒÎÓ‚Ë B(p,·,s)∈L2[−ε,+∞),p∈R+,s∈C,(14) ÍÓÚÓÓ ڇÍÊ ˄‡ÂÚ ÓÎ ̧ „‡ÌË ̃ÌÓ„Ó ÛÒÎӂˡ ‚ ·ÂÒÍÓÌ ̃ÌÓÒÚË. — Û ̃ÂÚÓÏ Ó„‡ÌË ̃ÂÌˡ (14)  ̄ÂÌË ÒËÒÚÂÏ ̊ Ó· ̊ÍÌÓ‚ÂÌÌ ̊ı ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ı Û‡‚ÌÂÌËÈ ‚ÚÓÓ„Ó ÔÓˇ‰Í‡ (9){(10) ÏÓÊÂÚ · ̊Ú ̧ ̇ȉÂÌÓ Òڇ̉‡ÚÌ ̊ÏË ÏÂÚÓ‰‡ÏË
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    : B(p,X,s) =C0(p,s) exp { − [ V 2 + √ ∆(p,s) ] X } ,X>0;(15) B(p,X,s) =C1(p,s) exp { − [ α2 V 2 + √ ∆(p,s) ] X } + +C2(p,s) exp { − [ α2 V 2 − √ ∆(p,s) ] X } ,−ε<X<0,(16) „‰Â ÙÛÌ͈ËÓ̇Π̊δ(p,s)Ë∆(p,s)ÓÔ‰ÂÎÂÌ ̊ ‡‚ÂÌÒÚ‚‡ÏË δ(p,s) =s+p2+ V2 4 ,∆(p,s) =α2s+p2+ α4V2 4 ,(17) ‡ ÙÛÌ͈ËÓ̇Π̊Cj(p,s),j=0,2, Û‰Ó‚ÎÂÚ‚Óˇ ̨Ú ÒËÒÚÂÏ ÎËÌÂÈÌ ̊ı ‡Î„·‡Ë ̃ÂÒÍËı Û‡‚ÌÂÌËÈ, ÓÚ‚Â ̃‡ ̨ ̆Ëı „‡ÌË ̃ÌÓÏÛ ÛÒÎÓ‚Ë ̨ (11