The 14 references with contexts in paper O. Miakinnik O., V. Feoktistov V., В. Феоктистов В., О. Мякинник О. (2016) “Малые возмущения идеального газа: волны, инварианты и задача Коши // Ideal Gas Small Perturbations: Waves, Invariants and Cauchy Problem” / spz:neicon:technomag:y:2014:i:2:p:387-406

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GodunovS.K.Uravneniyamatematicheskoyfiziki[Equationsof MathematicalPhysics]. Moscow,NaukaPubl.,1979.392 p. (in Russian).
Total in-text references: 9
  1. In-text reference with the coordinate start=2087
    Prefix
    ÕÂÒÚ‡ˆËÓ̇Ì ̊ÂÛ‡‚ÌÂÌˡ„‡ÁÓ‚ÓȉË̇ÏËÍË,‚ ̊‡Ê‡ ̨ ̆ËÂÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ,ˇ‚Ρ ̨Ú Ò ˇÛ‡‚ÌÂÌˡÏË„ËÔÂ·ÓÎË ̃ÂÒÍÓ„ÓÚËÔ‡.—‚ÓÈÒÚ‚Ó„ËÔÂ·ÓÎË ̃ÌÓÒÚËÚÂÒÌÓÒ‚ˇÁ‡ÌÓÒ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌËÂÏ ̄ÂÌËÈÒÔˆˇΠ̧Ìӄӂˉ‡,ÍÓÚÓ ̊Âӷ·‰‡ ̨ÚÒ‚ÓÈÒÚ‚ÓÏËÌ‚‡ˇÌÚÌÓÒÚË. “‡ÍËÂ ̄ÂÌËˇÌ‡Á ̊‚‡ ̨Ú·Â„Û ̆ËÏË‚ÓÎ̇ÏË
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    [1, 2, 3, 4]
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    . Õ‡ÔÓÏÌËÏ, ̃ÚÓ „ËÔÂ·ÓÎË ̃ÂÒ͇ˇÒËÒÚÂχn,n≥1, Ó‰ÌÓÓ‰Ì ̊ı‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡ÓÚÌÓÒËÚÂÎ ̧ÌÓÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı(t, x), x=xs, Ë Ó‰ÌÓÓ‰ÌÓÂÛ‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ıÔÓˇ‰Í‡m,m≥1, ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÔÂÂÏÂÌÌ ̊ı(t, x),x= (x1, . . . , xs),s≥1, ËÏ ̨Ú ̄ÂÌˡ,ÍÓÚÓ ̊ÂÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌˡ ̇ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËıÔˇÏ ̊ıË „ËÔÂÔÎÓÒÍÓÒÚˇıÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ[2, 3].

  2. In-text reference with the coordinate start=3136
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    ̊ıÔÂÂÏÂÌÌ ̊ı ‚ ‚ ̊‡ÊÂÌËË(ω0t+ω·x),ω·x=ω1x1+. . .+ωsxs,  ̄Ë‚‰Îˇ ̋ÚӄӇ΄·‡Ë ̃ÂÒÍÓÂÛ‡‚ÌÂÌËÂ, ÔÓÒÚÓÂÌÌÓÂÔÓ ÓÔ‰ÂÎÂÌÌÓÏÛÔ‡‚ËÎÛËÁ ̃ËÒÎÓ‚ ̊ıÍÓ ̋ÙÙˈËÂÌÚÓ‚‰ËÙÙÂÂ̈ˇΠ̧ÌÓ„Ó Û‡‚ÌÂÌˡËÎË ÒËÒÚÂÏ ̊Û‡‚ÌÂÌËÈ. ’ÓÓ ̄ÓËÁ‚ÂÒÚÌ ̊ÏËÔËÏÂ‡Ïˡ‚Ρ ̨ÚÒˇ:ÒËÒÚÂχÛ‡‚ÌÂÌËȇÍÛÒÚËÍË(n= 2) Ë Â  ̄ÂÌË‚ ‚ˉ ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ(xs−at)Ë(xs+at), ‡ÒÔÓÒÚ‡Ìˇ ̨ ̆ËıÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨ Á‚Û͇a‚‰ÓÎ ̧ÍÓÓ‰Ë̇ÚÌÓÈÓÒËOXs
    Exact
    [1, 2, 6]
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    ; ‚ÓÎÌÓ‚ÓÂÛ‡‚ÌÂÌËÂ(m= 2) ‚ ÔÓÒÚ‡ÌÒÚ‚Âs ÔÂÂÏÂÌÌ ̊ı,ÍÓÚÓÓÂÔÓÓʉ‡ÂÚ‚ÓÎÌÛf(ω0t+ω·x)(Á‰ÂÒ ̧f| ÔÓËÁ‚ÓÎ ̧̇ˇ„·‰Í‡ˇ ÙÛÌÍˆËˇ),‡ÒÔÓÒÚ‡Ìˇ ̨ ̆‡ˇÒˇ‚ ̇Ô‡‚ÎÂÌËË‚ÂÍÚÓ‡(ω0, ω)[1, 3]. ¬ ‡·ÓÚ‡ı[7, 8, 9] ‰Îˇ ÒËÒÚÂÏ ̊ E ∂ ~B ∂t = ∑s k=1 Ak ∂ ~B ∂xk ,~B= (B1(t, x), .

  3. In-text reference with the coordinate start=3317
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    ’ÓÓ ̄ÓËÁ‚ÂÒÚÌ ̊ÏËÔËÏÂ‡Ïˡ‚Ρ ̨ÚÒˇ:ÒËÒÚÂχÛ‡‚ÌÂÌËȇÍÛÒÚËÍË(n= 2) Ë Â  ̄ÂÌË‚ ‚ˉ ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ(xs−at)Ë(xs+at), ‡ÒÔÓÒÚ‡Ìˇ ̨ ̆ËıÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨ Á‚Û͇a‚‰ÓÎ ̧ÍÓÓ‰Ë̇ÚÌÓÈÓÒËOXs[1, 2, 6]; ‚ÓÎÌÓ‚ÓÂÛ‡‚ÌÂÌËÂ(m= 2) ‚ ÔÓÒÚ‡ÌÒÚ‚Âs ÔÂÂÏÂÌÌ ̊ı,ÍÓÚÓÓÂÔÓÓʉ‡ÂÚ‚ÓÎÌÛf(ω0t+ω·x)(Á‰ÂÒ ̧f| ÔÓËÁ‚ÓÎ ̧̇ˇ„·‰Í‡ˇ ÙÛÌÍˆËˇ),‡ÒÔÓÒÚ‡Ìˇ ̨ ̆‡ˇÒˇ‚ ̇Ô‡‚ÎÂÌËË‚ÂÍÚÓ‡(ω0, ω)
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    [1, 3]
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    . ¬ ‡·ÓÚ‡ı[7, 8, 9] ‰Îˇ ÒËÒÚÂÏ ̊ E ∂ ~B ∂t = ∑s k=1 Ak ∂ ~B ∂xk ,~B= (B1(t, x), . . . Bn(t, x)) Ú ,x= (x1, . . . , xs),(1) „ ‰ ÂAk,k=1, s, | ̃ËÒÎÓ‚ ̊ÂÍ‚‡‰‡ÚÌ ̊ÂχÚˈ ̊ÔÓˇ‰Í‡n,E| ‰ËÌË ̃̇ˇÏ‡Úˈ‡, ·ÂÁÓÚÌÓÒËÚÂÎ ̧ÌÓÍ ÓÔ‰ÂΡÂÏÓÏÛχÚˈ‡ÏË-ÍÓ ̋ÙÙˈËÂÌÚ‡ÏËÚËÔÛ‡ÒÒÏÓÚÂÌ ̊ ̄ÂÌˡ (Exk+Akt),k=1, s, Ë ÙÛÌ͈ËËÓÚ ÌËı.œÓ ‡Ì‡ÎÓ„ËËÒ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏËÚ‡ÍËÂ ̄ÂÌˡ ̇Á‚‡Ì ̊·Â„Û ̆ËÏË‚ÓÎ̇ÏËÒ Ï‡ÚË

  4. In-text reference with the coordinate start=5391
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    Á‡‰‡ ̨ ̆ËÂÒËÒÚÂÏÛ.ƒÎˇ ÔÓÎÛ ̃ÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄ˉÓÒÚ‡ÚÓ ̃ÌÓ‚ Ó· ̆ÂÏ ̄ÂÌËËÁ‡ÏÂÌËÚ ̧ ÌÂÓÔ‰ÂÎÂÌÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊̇ ‚ÂÍÚÓÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊‡ÁÎÓÊÂÌˡ‡Ì‡ÎËÚË ̃ÂÒÍÓÈ̇ ̃‡Î ̧ÌÓÈÙÛÌ͈ËË‚ ˇ‰ “ÂÈÎÓ‡[8]. ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚ [7, 8, 9] ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊
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    [1, 5, 12]
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    .¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡[2, 5, 12], ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.¬ ͇ ̃ÂÒÚ‚ÂÁ‡Ï ͇̊ ̨ ̆ËıÛ‡‚ÌÂÌËÈ[12] ‰Îˇ Û‡ÌÂÌËÈÒÓı‡ÌÂÌˡÒÍÓÓÒÚËË ËÏÔÛÎ ̧Ò‡‡ÒÒÏÓÚËÏÛ‡‚ÌÂÌËÂÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË, ‚ ̊‡Á˂ ̃ÂÂÁ ̋ÌڇΠ̧ÔË ̨,Ë Û‡‚ÌÂÌËÂÒÓÒÚÓˇÌˡ,Ò‚ˇÁ ̊‚‡ ̨ ̆ ̋ÌڇΠ̧ÔË ̨Ò ÔÎÓÚÌÓÒÚ ̧ ̨Ë ‰‡‚ÎÂÌËÂÏ„‡Á‡.

  5. In-text reference with the coordinate start=11978
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    ∂~V ∂x .(10) —ËÒÚÂχ(10) ÒÓÒÚÓËÚËÁ ÌÂÁ‡‚ËÒËÏ ̊ıÛ‡‚ÌÂÌËÈ ∂Vi ∂t =λi ∂Vi ∂x ,i=1,3,(11) ͇ʉÓÂËÁ ÍÓÚÓ ̊ıÒÓ‰ÂÊËÚÓ‰ÌÛÌÂËÁ‚ÂÒÚÌÛ ̨ÙÛÌÍˆË ̨Ë ËÏÂÂÚ ̄ÂÌË Vi=fi(x+λit),i=1,3,(12) „ ‰ Âfi| ÔÓËÁ‚ÓÎ ̧Ì ̊ÂÌÂÔÂ ̊‚ÌÓ-‰ËÙÙÂÂ̈ËÛÂÏ ̊ÂÙÛÌ͈ËË.œˇÏ ̊ x+λit=ci,ci= const,i=1,3,(13) ˇ‚Ρ ̨ÚÒˇı‡‡ÍÚÂËÒÚË͇ÏËÒËÒÚÂÏ ̊Ë ÔË ÔÂÂÒ ̃ÂÌËËÓ·‡ÁÛ ̨Ú‚ ÔÎÓÒÍÓÒÚË(t, x)Ú ۄÓÎ ̧ÌËÍ,̇Á ̊‚‡ÂÏ ̊Èı‡‡ÍÚÂËÒÚË ̃ÂÒÍËÏ
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    . ŒÒÌÓ‚‡ÌËÂÚÂÛ„ÓÎ ̧ÌË͇,ÓÚ‚Â ̃‡ ̨ ̆ÂÂÔˇÏÓÈ(13)ÔËi= 1, Ì Á‡‚ËÒËÚÓÚ Ô‡‡ÏÂÚÓ‚„‡Á‡.ƒ‡ÌÌ ̊ÈÙ‡ÍÚÒӄ·ÒÛÂÚÒˇÒ Á‡ÍÓÌÓÏ ‡ÒÔÓÒÚ‡ÌÂÌËˇÏ‡Î ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ÔÓÍÓˇ ̆ÂÏÒˇ„‡Á [1] Ë ÔË ÓÚÌÓÒËÚÂÎ ̧ÌÓω‚ËÊÂÌËË „‡Á‡,ÍÓÚÓÓÂ‡ÒÒχÚË‚‡ÂÚÒˇ‚ ‡Á‰ÂÎÂ4.

  6. In-text reference with the coordinate start=12151
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    12) „ ‰ Âfi| ÔÓËÁ‚ÓÎ ̧Ì ̊ÂÌÂÔÂ ̊‚ÌÓ-‰ËÙÙÂÂ̈ËÛÂÏ ̊ÂÙÛÌ͈ËË.œˇÏ ̊ x+λit=ci,ci= const,i=1,3,(13) ˇ‚Ρ ̨ÚÒˇı‡‡ÍÚÂËÒÚË͇ÏËÒËÒÚÂÏ ̊Ë ÔË ÔÂÂÒ ̃ÂÌËËÓ·‡ÁÛ ̨Ú‚ ÔÎÓÒÍÓÒÚË(t, x)Ú ۄÓÎ ̧ÌËÍ,̇Á ̊‚‡ÂÏ ̊Èı‡‡ÍÚÂËÒÚË ̃ÂÒÍËÏ[1]. ŒÒÌÓ‚‡ÌËÂÚÂÛ„ÓÎ ̧ÌË͇,ÓÚ‚Â ̃‡ ̨ ̆ÂÂÔˇÏÓÈ(13)ÔËi= 1, Ì Á‡‚ËÒËÚÓÚ Ô‡‡ÏÂÚÓ‚„‡Á‡.ƒ‡ÌÌ ̊ÈÙ‡ÍÚÒӄ·ÒÛÂÚÒˇÒ Á‡ÍÓÌÓÏ ‡ÒÔÓÒÚ‡ÌÂÌËˇÏ‡Î ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ÔÓÍÓˇ ̆ÂÏÒˇ„‡ÁÂ
    Exact
    [1]
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    Ë ÔË ÓÚÌÓÒËÚÂÎ ̧ÌÓω‚ËÊÂÌËË „‡Á‡,ÍÓÚÓÓÂ‡ÒÒχÚË‚‡ÂÚÒˇ‚ ‡Á‰ÂÎÂ4. Õ‡ÔÓÏÌËÏ, ̃ÚÓ ‰Îˇ Û‡‚ÌÂÌËȇÍÛÒÚËÍË‚ ͇ ̃ÂÒÚ‚Â ÓÒÌÓ‚‡Ìˡı‡‡ÍÚÂËÒÚË ̃ÂÒÍÓ„ÓÚÂÛ„ÓÎ ̧ÌË͇‚ ̊·Ë‡ÂÚÒˇËÌÚÂ‚‡Î̇ ÓÒËOX[1]. ‘ÛÌ͈ËËfi,i=1,3, ÒÓı‡Ìˇ ̨ÚÁ̇ ̃ÂÌËˇÌ‡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËıÔ‡‡ÎÎÂÎ ̧Ì ̊ıÔˇÏ ̊ı (13) Ë ‚ÏÂÒÚÂÒÓ Ò‚ÓËÏˇ„ÛÏÂÌÚ‡ÏË̇Á ̊‚‡ ̨ÚÒˇ·Â„Û ̆ËÏË‚ÓÎ̇ÏË[1, 3].

  7. In-text reference with the coordinate start=12339
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    ŒÒÌÓ‚‡ÌËÂÚÂÛ„ÓÎ ̧ÌË͇,ÓÚ‚Â ̃‡ ̨ ̆ÂÂÔˇÏÓÈ(13)ÔËi= 1, Ì Á‡‚ËÒËÚÓÚ Ô‡‡ÏÂÚÓ‚„‡Á‡.ƒ‡ÌÌ ̊ÈÙ‡ÍÚÒӄ·ÒÛÂÚÒˇÒ Á‡ÍÓÌÓÏ ‡ÒÔÓÒÚ‡ÌÂÌËˇÏ‡Î ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ÔÓÍÓˇ ̆ÂÏÒˇ„‡Á [1] Ë ÔË ÓÚÌÓÒËÚÂÎ ̧ÌÓω‚ËÊÂÌËË „‡Á‡,ÍÓÚÓÓÂ‡ÒÒχÚË‚‡ÂÚÒˇ‚ ‡Á‰ÂÎÂ4. Õ‡ÔÓÏÌËÏ, ̃ÚÓ ‰Îˇ Û‡‚ÌÂÌËȇÍÛÒÚËÍË‚ ͇ ̃ÂÒÚ‚Â ÓÒÌÓ‚‡Ìˡı‡‡ÍÚÂËÒÚË ̃ÂÒÍÓ„ÓÚÂÛ„ÓÎ ̧ÌË͇‚ ̊·Ë‡ÂÚÒˇËÌÚÂ‚‡Î̇ ÓÒËOX
    Exact
    [1]
    Suffix
    . ‘ÛÌ͈ËËfi,i=1,3, ÒÓı‡Ìˇ ̨ÚÁ̇ ̃ÂÌËˇÌ‡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËıÔ‡‡ÎÎÂÎ ̧Ì ̊ıÔˇÏ ̊ı (13) Ë ‚ÏÂÒÚÂÒÓ Ò‚ÓËÏˇ„ÛÏÂÌÚ‡ÏË̇Á ̊‚‡ ̨ÚÒˇ·Â„Û ̆ËÏË‚ÓÎ̇ÏË[1, 3]. ”‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ò ‰‚ÛÏˇÌÂÁ‡‚ËÒËÏ ̊ÏËÔÂÂÏÂÌÌ ̊ÏË(10) ÏÓ‰ÂÎËÛÂÚÓ ‰ ÌÓÏÂÌ ̊È‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ(‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ‚ Ó‰ÌÓÏÂÌÓÈÒ‰Â)Ë ˇ‚ΡÂÚÒˇÒ‡Ï ̊ÏÔÓÒÚ ̊ÏËÁ Û‡‚ÌÂÌËÈ,Í ÍÓÚÓ ̊ÏÔËÏÂÌˇÂÚÒˇÓÔ‰ÂÎÂÌËÂ<‚ÓÎÌÓ‚ÓÂ>.œÓ ̋ÚÓÏÛ‡„ÛÏ

  8. In-text reference with the coordinate start=12483
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    Õ‡ÔÓÏÌËÏ, ̃ÚÓ ‰Îˇ Û‡‚ÌÂÌËȇÍÛÒÚËÍË‚ ͇ ̃ÂÒÚ‚Â ÓÒÌÓ‚‡Ìˡı‡‡ÍÚÂËÒÚË ̃ÂÒÍÓ„ÓÚÂÛ„ÓÎ ̧ÌË͇‚ ̊·Ë‡ÂÚÒˇËÌÚÂ‚‡Î̇ ÓÒËOX[1]. ‘ÛÌ͈ËËfi,i=1,3, ÒÓı‡Ìˇ ̨ÚÁ̇ ̃ÂÌËˇÌ‡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËıÔ‡‡ÎÎÂÎ ̧Ì ̊ıÔˇÏ ̊ı (13) Ë ‚ÏÂÒÚÂÒÓ Ò‚ÓËÏˇ„ÛÏÂÌÚ‡ÏË̇Á ̊‚‡ ̨ÚÒˇ·Â„Û ̆ËÏË‚ÓÎ̇ÏË
    Exact
    [1, 3]
    Suffix
    . ”‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ò ‰‚ÛÏˇÌÂÁ‡‚ËÒËÏ ̊ÏËÔÂÂÏÂÌÌ ̊ÏË(10) ÏÓ‰ÂÎËÛÂÚÓ ‰ ÌÓÏÂÌ ̊È‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ(‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ‚ Ó‰ÌÓÏÂÌÓÈÒ‰Â)Ë ˇ‚ΡÂÚÒˇÒ‡Ï ̊ÏÔÓÒÚ ̊ÏËÁ Û‡‚ÌÂÌËÈ,Í ÍÓÚÓ ̊ÏÔËÏÂÌˇÂÚÒˇÓÔ‰ÂÎÂÌËÂ<‚ÓÎÌÓ‚ÓÂ>.œÓ ̋ÚÓÏÛ‡„ÛÏÂÌÚ ̊ ÙÛÌ͈ËÈfi̇Á ̊‚‡ ̨ÚÚ‡ÍÊ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏË[4] (‚ [5] | ÔÓÒÚ ̊ÏË‚ÓÎ̇ÏË).¬ [2] ‰Îˇ ‚ÓÎÌ(12)ÔËÏÂÌÂÌÚÂÏËÌ"·Â„Û ̆Ë‚ÓÎÌ ̊χΠ̊ı‚ÓÁÏÛ ̆ÂÌË

  9. In-text reference with the coordinate start=13521
    Prefix
    ‡Î ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‡ÒÔÓÒÚ‡Ìˇ ̨ÚÒˇÒ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËÏË ÒÍÓÓÒÚˇÏË. ¬ ̊ÔÓÎÌÂÌËÂÔÂÓ·‡ÁÓ‚‡ÌˡÔӉӷˡ(9) ÔÓ ÓÚÌÓ ̄ÂÌË ̨Í „ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏÂ(8) ÔË‚ÂÎÓÍ ÌÂÁ‡‚ËÒËÏ ̊ÏÛ‡‚ÌÂÌˡÏË ÔÓˇ‚ÎÂÌË ̨ ̋ÎÂÏÂÌÚ‡Ì ̊ı·Â„Û ̆Ëı‚ÓÎÌ‚ ͇ ̃ÂÒÚ‚Â  ̄ÂÌˡ. ՇȉÂÌÌ ̊‚ ̊‡ÊÂÌˡ Vi≡ ( S−1~B ) i ,i=1,3, Á‡ÔËÒ‡ÌÌ ̊ ̃ÂÂÁÙËÁË ̃ÂÒÍËÁ̇ ̃ËÏ ̊ÂÙÛÌ͈ËË,ÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌËˇÌ‡ ı‡‡ÍÚÂËÒÚËÍ ‡ ı Ë, Òӄ·ÒÌÓÓÔ‰ÂÎÂÌË ̨
    Exact
    [1, 2]
    Suffix
    , Ô‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈËÌ‚‡ˇÌÚ ̊–Ëχ̇ÒËÒÚÂÏ ̊(8): (γ−1) γ ρ(t, x)− 1 γ I(t, x) =f1(x−t),(14) 1 2 u(t, x)− 1 2γM ρ(t, x)− 1 2γM I(t, x) =f2 ( x+ (1−M) M t ) ,(15) 1 2 u(t, x) + 1 2γM ρ(t, x) + 1 2γM I(t, x) =f3 ( x− (1 +M) M t ) .(16) ‡Ê‰ ̊ÈËÁ ÚÂıËÌ‚‡ˇÌÚÓ‚(14){(16)‡ÒÔÓÒÚ‡ÌˇÂÚÒˇ·Â Á ËÒ͇ÊÂÌˡÙÓÏ ̊Ò ı‡‡ÍÚÂËÒÚË ̃ÂÒÍÓÈÒÍÓÓÒÚ ̧ ̨λi,i=1,3.

2
KulikovskiyA.G.,SveshnikovaE.I.,ChugaynovaA.P.LektsionnyekursyNOTs.Vyp.16. Matematicheskiemetodyizucheniyarazryvnykhresheniynelineynykhgiperbolicheskikhsistem uravneniy[Lecturecoursesof Scientificand EducationalCenter.Is. 16. Mathematicalmethods for studyingdiscontinuoussolutionsof nonlinearhyperbolicsystemsof equations].Moscow, SteklovMathematicalInstituteof RASPubl.,2010.122 p. (in Russian).
Total in-text references: 15
  1. In-text reference with the coordinate start=2087
    Prefix
    ÕÂÒÚ‡ˆËÓ̇Ì ̊ÂÛ‡‚ÌÂÌˡ„‡ÁÓ‚ÓȉË̇ÏËÍË,‚ ̊‡Ê‡ ̨ ̆ËÂÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ,ˇ‚Ρ ̨Ú Ò ˇÛ‡‚ÌÂÌˡÏË„ËÔÂ·ÓÎË ̃ÂÒÍÓ„ÓÚËÔ‡.—‚ÓÈÒÚ‚Ó„ËÔÂ·ÓÎË ̃ÌÓÒÚËÚÂÒÌÓÒ‚ˇÁ‡ÌÓÒ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌËÂÏ ̄ÂÌËÈÒÔˆˇΠ̧Ìӄӂˉ‡,ÍÓÚÓ ̊Âӷ·‰‡ ̨ÚÒ‚ÓÈÒÚ‚ÓÏËÌ‚‡ˇÌÚÌÓÒÚË. “‡ÍËÂ ̄ÂÌËˇÌ‡Á ̊‚‡ ̨Ú·Â„Û ̆ËÏË‚ÓÎ̇ÏË
    Exact
    [1, 2, 3, 4]
    Suffix
    . Õ‡ÔÓÏÌËÏ, ̃ÚÓ „ËÔÂ·ÓÎË ̃ÂÒ͇ˇÒËÒÚÂχn,n≥1, Ó‰ÌÓÓ‰Ì ̊ı‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡ÓÚÌÓÒËÚÂÎ ̧ÌÓÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı(t, x), x=xs, Ë Ó‰ÌÓÓ‰ÌÓÂÛ‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ıÔÓˇ‰Í‡m,m≥1, ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÔÂÂÏÂÌÌ ̊ı(t, x),x= (x1, . . . , xs),s≥1, ËÏ ̨Ú ̄ÂÌˡ,ÍÓÚÓ ̊ÂÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌˡ ̇ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËıÔˇÏ ̊ıË „ËÔÂÔÎÓÒÍÓÒÚˇıÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ[2, 3].

  2. In-text reference with the coordinate start=2532
    Prefix
    Õ‡ÔÓÏÌËÏ, ̃ÚÓ „ËÔÂ·ÓÎË ̃ÂÒ͇ˇÒËÒÚÂχn,n≥1, Ó‰ÌÓÓ‰Ì ̊ı‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡ÓÚÌÓÒËÚÂÎ ̧ÌÓÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı(t, x), x=xs, Ë Ó‰ÌÓÓ‰ÌÓÂÛ‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ıÔÓˇ‰Í‡m,m≥1, ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÔÂÂÏÂÌÌ ̊ı(t, x),x= (x1, . . . , xs),s≥1, ËÏ ̨Ú ̄ÂÌˡ,ÍÓÚÓ ̊ÂÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌˡ ̇ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËıÔˇÏ ̊ıË „ËÔÂÔÎÓÒÍÓÒÚˇıÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ
    Exact
    [2, 3]
    Suffix
    . —ÎÛ ̃‡Èn= 1, s= 1Ëm= 1,s= 1ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÔÓÒÚÂÈ ̄ÂÏÛÛ‡‚ÌÂÌË ̨‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı,  ̄ÂÌËÂÏÍÓÚÓÓ„Óˇ‚ΡÂÚÒˇ ̋ÎÂÏÂÌÚ‡̇ˇ‚ÓÎ̇[4, 5]. ƒÎˇ ÔÓÒÚÓÂÌˡ‚ÓÎÌ ̊ÒΉÛÂÚÓÔ‰ÂÎËÚ ̧ÍÓ ̋ÙÙˈËÂÌÚ ̊ÔË ÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı ‚ ‚ ̊‡ÊÂÌËË(ω0t+ω·x),ω·x=ω1x1+. . .+ωsxs,  ̄Ë‚‰Îˇ ̋ÚӄӇ΄·‡Ë ̃ÂÒÍÓÂÛ‡‚ÌÂÌËÂ, ÔÓÒÚÓÂÌÌÓÂÔÓ ÓÔ‰ÂÎÂÌÌÓÏÛÔ‡‚ËÎÛËÁ ̃ËÒÎÓ‚ ̊ıÍÓ ̋ÙÙˈËÂÌÚÓ‚‰ËÙÙÂÂ̈ˇΠ̧ÌÓ„Ó Û‡‚ÌÂÌˡËÎË

  3. In-text reference with the coordinate start=3136
    Prefix
    ̊ıÔÂÂÏÂÌÌ ̊ı ‚ ‚ ̊‡ÊÂÌËË(ω0t+ω·x),ω·x=ω1x1+. . .+ωsxs,  ̄Ë‚‰Îˇ ̋ÚӄӇ΄·‡Ë ̃ÂÒÍÓÂÛ‡‚ÌÂÌËÂ, ÔÓÒÚÓÂÌÌÓÂÔÓ ÓÔ‰ÂÎÂÌÌÓÏÛÔ‡‚ËÎÛËÁ ̃ËÒÎÓ‚ ̊ıÍÓ ̋ÙÙˈËÂÌÚÓ‚‰ËÙÙÂÂ̈ˇΠ̧ÌÓ„Ó Û‡‚ÌÂÌˡËÎË ÒËÒÚÂÏ ̊Û‡‚ÌÂÌËÈ. ’ÓÓ ̄ÓËÁ‚ÂÒÚÌ ̊ÏËÔËÏÂ‡Ïˡ‚Ρ ̨ÚÒˇ:ÒËÒÚÂχÛ‡‚ÌÂÌËȇÍÛÒÚËÍË(n= 2) Ë Â  ̄ÂÌË‚ ‚ˉ ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ(xs−at)Ë(xs+at), ‡ÒÔÓÒÚ‡Ìˇ ̨ ̆ËıÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨ Á‚Û͇a‚‰ÓÎ ̧ÍÓÓ‰Ë̇ÚÌÓÈÓÒËOXs
    Exact
    [1, 2, 6]
    Suffix
    ; ‚ÓÎÌÓ‚ÓÂÛ‡‚ÌÂÌËÂ(m= 2) ‚ ÔÓÒÚ‡ÌÒÚ‚Âs ÔÂÂÏÂÌÌ ̊ı,ÍÓÚÓÓÂÔÓÓʉ‡ÂÚ‚ÓÎÌÛf(ω0t+ω·x)(Á‰ÂÒ ̧f| ÔÓËÁ‚ÓÎ ̧̇ˇ„·‰Í‡ˇ ÙÛÌÍˆËˇ),‡ÒÔÓÒÚ‡Ìˇ ̨ ̆‡ˇÒˇ‚ ̇Ô‡‚ÎÂÌËË‚ÂÍÚÓ‡(ω0, ω)[1, 3]. ¬ ‡·ÓÚ‡ı[7, 8, 9] ‰Îˇ ÒËÒÚÂÏ ̊ E ∂ ~B ∂t = ∑s k=1 Ak ∂ ~B ∂xk ,~B= (B1(t, x), .

  4. In-text reference with the coordinate start=5519
    Prefix
    ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚ [7, 8, 9] ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊[1, 5, 12].¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡
    Exact
    [2, 5, 12]
    Suffix
    , ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.¬ ͇ ̃ÂÒÚ‚ÂÁ‡Ï ͇̊ ̨ ̆ËıÛ‡‚ÌÂÌËÈ[12] ‰Îˇ Û‡ÌÂÌËÈÒÓı‡ÌÂÌˡÒÍÓÓÒÚËË ËÏÔÛÎ ̧Ò‡‡ÒÒÏÓÚËÏÛ‡‚ÌÂÌËÂÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË, ‚ ̊‡Á˂ ̃ÂÂÁ ̋ÌڇΠ̧ÔË ̨,Ë Û‡‚ÌÂÌËÂÒÓÒÚÓˇÌˡ,Ò‚ˇÁ ̊‚‡ ̨ ̆ ̋ÌڇΠ̧ÔË ̨Ò ÔÎÓÚÌÓÒÚ ̧ ̨Ë ‰‡‚ÎÂÌËÂÏ„‡Á‡.

  5. In-text reference with the coordinate start=6382
    Prefix
    t, x), Â„Ó ÔÎÓÚÌÓÒÚËρ(t, x)Ë ̋ÌڇΠ̧ÔËËI(t, x), „ ‰ Âx= (x1, . . . , xs),s≤3. œÓÒÍÓÎ ̧ÍÛ ̋ÌڇΠ̧ÔˡÔÓÒ‰ÒÚ‚ÓÏÍÓ ̋ÙÙˈËÂÌÚ‡ ÚÂÔÎÓÂÏÍÓÒÚËÒ‚ˇÁ‡Ì‡Ò ÚÂÏÔÂ‡ÚÛÓÈ„‡Á‡,͇ʉ‡ˇËÁ ÍÓÏÔÓÌÂÌÚËÒÍÓÏÓ„Ó‚ÂÍÚÓ‡~B(t, x) ˇ‚ΡÂÚÒˇ‚ÂÎË ̃ËÌÓÈ,ÍÓÚÓ‡ˇÏÓÊÂÚ· ̊Ú ̧ËÁÏÂÂ̇‚ ı Ó ‰  ̋ÍÒÔÂËÏÂÌÚ‡. ◊ÚÓ· ̊ËÁ Í‚‡ÁËÎËÌÂÈÌÓ„ÓÔÓˆÂÒÒ‡‚ ̊‰ÂÎËÚ ̧ÎËÌÂÈÌ ̊‚ÓÎÌ ̊,ÔÓÓʉ‡ÂÏ ̊ÂÒËÒÚÂÏÓÈ(1), ÔËÏÂÌËÏÏÂÚӉχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ(̇ÔËÏÂ,
    Exact
    [2]
    Suffix
    ).¬ÓÁÌË͇ ̨ ̆Û ̨ÔË ̋ÚÓÏ Á‡‰‡ ̃ÛÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ,ÍÓÚÓ‡ˇÔ‰ÒÚ‡‚ΡÂÚÒ‡ÏÓÒÚÓˇÚÂÎ ̧Ì ̊ÈËÌÚÂÂÒ[2, 5, 13], ‡ÒÒÏÓÚËÏÍ ‡ Í ÚÂÒÚÓ‚Û ̨‰Îˇ ÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒÚ‚ËˇÌ‡ ÔËÏÂ ‰‚ËÊÂÌˡˉ‡Π̧ÌÓ„Ó„‡Á‡ ÔË Á‡‰‡ÌÌ ̊ınËs. œÓ ÒÚÛÍÚÛÂÔÓÒÚÓÂÌÌÓÈÏÓ‰ÂÎËÓ· ̆ÂÂ ̄ÂÌËÂÒËÒÚÂÏ ̊ E ∂ ~B ∂t =Ak ∂ ~B ∂xk ,~B= (B1(t, xk), .

  6. In-text reference with the coordinate start=6495
    Prefix
    ◊ÚÓ· ̊ËÁ Í‚‡ÁËÎËÌÂÈÌÓ„ÓÔÓˆÂÒÒ‡‚ ̊‰ÂÎËÚ ̧ÎËÌÂÈÌ ̊‚ÓÎÌ ̊,ÔÓÓʉ‡ÂÏ ̊ÂÒËÒÚÂÏÓÈ(1), ÔËÏÂÌËÏÏÂÚӉχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ(̇ÔËÏÂ,[2]).¬ÓÁÌË͇ ̨ ̆Û ̨ÔË ̋ÚÓÏ Á‡‰‡ ̃ÛÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ,ÍÓÚÓ‡ˇÔ‰ÒÚ‡‚ΡÂÚÒ‡ÏÓÒÚÓˇÚÂÎ ̧Ì ̊ÈËÌÚÂÂÒ
    Exact
    [2, 5, 13]
    Suffix
    , ‡ÒÒÏÓÚËÏÍ ‡ Í ÚÂÒÚÓ‚Û ̨‰Îˇ ÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒÚ‚ËˇÌ‡ ÔËÏÂ ‰‚ËÊÂÌˡˉ‡Π̧ÌÓ„Ó„‡Á‡ ÔË Á‡‰‡ÌÌ ̊ınËs. œÓ ÒÚÛÍÚÛÂÔÓÒÚÓÂÌÌÓÈÏÓ‰ÂÎËÓ· ̆ÂÂ ̄ÂÌËÂÒËÒÚÂÏ ̊ E ∂ ~B ∂t =Ak ∂ ~B ∂xk ,~B= (B1(t, xk), .

  7. In-text reference with the coordinate start=7124
    Prefix
    s,(2) Ô‰ÒÚ‡‚ÎÂÌÌÓÂÍ ‡ Í ‡ÁÎÓÊÂÌËÂÔÓ ‚ÓÎÌÂ(Exk+Akt), ˇ‚ΡÂÚÒˇ ̃‡ÒÚÌ ̊Ï ̄ÂÌËÂÏÒËÒÚÂÏ ̊(1) Ë ÒÓÒÚ‡‚ÌÓÈ ̃‡ÒÚ ̧ ̨ ӷ ̆„Ó ̄ÂÌˡ,Ô‰ÒÚ‡‚ÎÂÌÌÓ„Ó‚ ‚ˉÂ‡ÁÎÓÊÂÌˡÔÓs ‚Á‡ËÏÓ‰ÂÈÒÚ‚Û ̨ ̆ËÏ‚ÓÎ̇Ï[7, 9]. œÛÒÚ ̧(2) ÓÚ‚Â ̃‡ÂÚ‡ÒÔÓÒÚ‡ÌÂÌË ̨χΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË Ó‰ÌÓÏÂÌÓÏ Ú ̃ÂÌËË.ՇȉÂÏ ̄ÂÌË ̋ÚÓÈÒËÒÚÂÏ ̊Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏÚ‡‰ËˆËÓÌÌÓ„ÓÔÓ‰ıÓ‰‡Í Ó‰ÌÓÏÂÌÓÈ„ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏÂ
    Exact
    [2, 6]
    Suffix
    Ë Ò‰ÒÚ‚‡ÏËÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ ·Â Á ËÒÔÓÎ ̧ÁÓ‚‡ÌˡÛÒÎӂˡ„ËÔÂ·ÓÎË ̃ÌÓÒÚË[7]. œÓ͇ÊÂÏÚÓʉÂÒÚ‚ÂÌÌÓÒÚ ̧ÔÓÎÛ ̃ÂÌÌ ̊ı ̄ÂÌËÈË ÔÂËÏÛ ̆ÂÒÚ‚‡‚ÓÎÌÓ‚Ó„ÓÔÓ‰ıÓ‰‡ÔË ËÒÒΉӂ‡ÌËËÒËÒÚÂÏ ̊(2). ÷ÂÎ ̧ ̨‡·ÓÚ ̊ˇ‚ΡÂÚÒˇ‡ÒÒÏÓÚÂÌËÂÔ‰·„‡ÂÏÓ„Ó‡‚ÚÓ‡ÏËÌӂӄӇ̇ÎËÚË ̃ÂÒÍÓ„Ó ÏÂÚÓ‰‡ ̄ÂÌˡÌÓχΠ̧ÌÓÈÒËÒÚÂÏ ̊ÎËÌÂÈÌ ̊ıÓ‰ÌÓÓ‰Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ì‡ ÔËÏÂÂÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ

  8. In-text reference with the coordinate start=7929
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    Ó ÔÓˇ‰Í‡Ì‡ ÔËÏÂÂÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ „‡ÁÂÔË Ó‰ÌÓÏÂÌÓÏÚ ̃ÂÌËËÒ ÔÓÒÎÂ‰Û ̨ ̆ËÏËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏÔÓÎÛ ̃ÂÌÌÓ„Ó ̄ÂÌˡ‰Îˇ ÔÓÒÚÓÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃ËÔË Ú ̃ÂÌËË„‡Á‡ ‚ ÔÓÒÚ‡ÌÒÚ‚Â. 2. ”‡‚ÌÂÌˡ‡ÒÔÓÒÚ‡ÌÂÌËˇÏ‡Î ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ‚‡ÁËÎËÌÂÈÌ ̊ÂÛ‡‚ÌÂÌˡӉÌÓÏÂÌÓ„Ó‰‚ËÊÂÌˡˉ‡Π̧ÌÓ„Ó„‡Á‡ ÔË ÓÚÒÛÚÒÚ‚ËËÏ‡Ò ÒÓ‚ ̊ıÒËÎ ‚ ‰Â͇ÚÓ‚ÓÈÒËÒÚÂÏÂÍÓÓ‰Ë̇ÚËÏ ̨ڂˉ
    Exact
    [2, 5, 13]
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    :          ∂u ̃ ∂ ̃t + ̃u ∂u ̃ ∂ ̃x =− 1 ρ ̃ ∂ ̃P ∂ ̃x , ∂ ̃ρ ∂ ̃t + ∂( ̃ρ ̃u) ∂ ̃x = 0, ∂I ̃ ∂ ̃t + ̃u ∂I ̃ ∂ ̃x = 1 ρ ̃ ( ∂ ̃P ∂ ̃t + ̃u ∂ ̃P ∂ ̃x ) , x ̃∈R1,(3) „ ‰  ̃u( ̃t, ̃x), ̃ρ( ̃t, ̃x), ̃P( ̃t, ̃x), ̃I( ̃t ̃x)| ÒÍÓÓÒÚ ̧‰‚ËÊÂÌˡ„‡Á‡,Â„Ó ÔÎÓÚÌÓÒÚ ̧,‰‡‚ÎÂÌËÂË ̋ÌڇΠ̧ÔˡÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ.

  9. In-text reference with the coordinate start=9710
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    , St ∂ρ ∂t =−ρ ∂u ∂x −u ∂ρ ∂x , St ∂I ∂t =−(γ−1)I ∂u ∂x −u ∂I ∂x . (5) ¬ (5)St| ̃ËÒÎÓ—ÚÛı‡Î‡;St=L0/U0T0;M| ̃ËÒÎÓÇı‡;M=U0/a,a| ÒÍÓÓÒÚ ̧Á‚Û͇ ‚ „‡ÁÓ‚ÓÈÒ‰Â,ÔË ̋ÚÓÏa2= (γ−1)I0. œÓÒÍÓÎ ̧ÍÛÒ‰Ë‡ÁÏÂÌ ̊ı‚ÂÎË ̃ËÌ,Á‡‰‡ ̨ ̆Ëı ̃ËÒÎÓ—ÚÛı‡Î‡,‰‚ ˇ‚Ρ ̨ÚÒˇÌÂÁ‡‚ËÒËÏ ̊ÏË,ÏÓÊÌÓ‚ ̊·‡Ú ̧St= 1. –‡ÒÒÏÓÚËÏÁ‡‰‡ ̃ÛÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡ÁÂÔË Ó‰ÌÓÏÂÌÓÏÔ‡‡ÎÎÂÎ ̧ÌÓÏÓÒËOX‰‚ËÊÂÌËË
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    [2, 5]
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    . œÛÒÚ ̧‚ ‰‚ËÊÛ ̆ÂÏÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨U0 ˉ‡Π̧ÌÓÏ„‡ÁÂ,ÍÓÚÓ ̊ÈËÏÂÂÚÔÎÓÚÌÓÒÚ ̧ρ0Ë ̋ÌڇΠ̧ÔË ̨I0, ÒÓÁ‰‡ ̨ÚÒˇÏ‡Î ̊‚ÓÁÏÛ ̆ÂÌˡ ÒÍÓÓÒÚËu∗(t, x), ÔÎÓÚÌÓÒÚËρ∗(t, x)Ë ̋ÌڇΠ̧ÔËËI∗(t, x). “Ó„‰‡‰‚ËÊÂÌË ̨‚ÓÁÏÛ ̆ÂÌÌÓ„Ó „‡Á‡ ÓÚ‚Â ̃‡ ̨ÚÙÛÌ͈ËË u(t, x) = 1 +u∗(t, x),ρ(t, x) = 1 +ρ∗(t, x),I(t, x) = 1 +I∗(t, x).(6) ◊ÚÓ· ̊ÌÂÎËÌÂÈ̇ˇÒËÒÚÂχ(5) ËÏ· ̄ÂÌË‚ˉ‡(6), ÙÛÌ͈ËË,Á‡‰‡ ̨ ̆ËÂχΠ̊‚ÓÁÏÛ ̆ÂÌˡ,‰ÓÎÊÌ ̊

  10. In-text reference with the coordinate start=10152
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    “Ó„‰‡‰‚ËÊÂÌË ̨‚ÓÁÏÛ ̆ÂÌÌÓ„Ó „‡Á‡ ÓÚ‚Â ̃‡ ̨ÚÙÛÌ͈ËË u(t, x) = 1 +u∗(t, x),ρ(t, x) = 1 +ρ∗(t, x),I(t, x) = 1 +I∗(t, x).(6) ◊ÚÓ· ̊ÌÂÎËÌÂÈ̇ˇÒËÒÚÂχ(5) ËÏ· ̄ÂÌË‚ˉ‡(6), ÙÛÌ͈ËË,Á‡‰‡ ̨ ̆ËÂχΠ̊‚ÓÁÏÛ ̆ÂÌˡ,‰ÓÎÊÌ ̊Û‰Ó‚ÎÂÚ‚ÓˇÚ ̧ÒËÒÚÂÏÂÎËÌÂÈÌ ̊ıÛ‡‚ÌÂÌËÈ
    Exact
    [2]
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    :        ∂u ∂t =− ∂u ∂x − 1 γM2 ∂ρ ∂x − 1 γM2 ∂I ∂x , ∂ρ ∂t =− ∂u ∂x − ∂ρ ∂x , ∂I ∂t =−(γ−1) ∂u ∂x − ∂I ∂x (7) (‰ÎˇÛÔÓ ̆ÂÌˡӷÓÁ̇ ̃ÂÌËÈÁ‚ÂÁ‰Ó ̃ÍËÔË ËÒÍÓÏ ̊ıÙÛÌÍˆËˇıÓÔÛ ̆ÂÌ ̊).¬ ̊ÔÓÎÌÂÌ̇ˇ Ôӈ‰Û‡Ì‡Á ̊‚‡ÂÚÒˇÎË̇ËÁ‡ˆËÂÈÒËÒÚÂÏ ̊(5) ÓÍÓÎÓÒÓÒÚÓˇÌˡ(U0, ρ0, I0)[2]. 3. œ˂‰ÂÌËÂÍ ÌÂÁ‡‚ËÒËÏ ̊Ï‚ÓÎÌÓ‚ ̊ÏÛ‡‚ÌÂÌˡÏË Ì‡ıÓʉÂÌËÂËÌ‚‡ˇÌÚÓ‚ «‡ÔË ̄ÂÏÔÓÎÛ ̃ÂÌÌÛ ̨ÎËÌÂÈÌÛ ̨ÒËÒÚÂÏÛ(7) ‚ χÚ

  11. In-text reference with the coordinate start=10390
    Prefix
    ‡(5) ËÏ· ̄ÂÌË‚ˉ‡(6), ÙÛÌ͈ËË,Á‡‰‡ ̨ ̆ËÂχΠ̊‚ÓÁÏÛ ̆ÂÌˡ,‰ÓÎÊÌ ̊Û‰Ó‚ÎÂÚ‚ÓˇÚ ̧ÒËÒÚÂÏÂÎËÌÂÈÌ ̊ıÛ‡‚ÌÂÌËÈ[2]:        ∂u ∂t =− ∂u ∂x − 1 γM2 ∂ρ ∂x − 1 γM2 ∂I ∂x , ∂ρ ∂t =− ∂u ∂x − ∂ρ ∂x , ∂I ∂t =−(γ−1) ∂u ∂x − ∂I ∂x (7) (‰ÎˇÛÔÓ ̆ÂÌˡӷÓÁ̇ ̃ÂÌËÈÁ‚ÂÁ‰Ó ̃ÍËÔË ËÒÍÓÏ ̊ıÙÛÌÍˆËˇıÓÔÛ ̆ÂÌ ̊).¬ ̊ÔÓÎÌÂÌ̇ˇ Ôӈ‰Û‡Ì‡Á ̊‚‡ÂÚÒˇÎË̇ËÁ‡ˆËÂÈÒËÒÚÂÏ ̊(5) ÓÍÓÎÓÒÓÒÚÓˇÌˡ(U0, ρ0, I0)
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    [2]
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    . 3. œ˂‰ÂÌËÂÍ ÌÂÁ‡‚ËÒËÏ ̊Ï‚ÓÎÌÓ‚ ̊ÏÛ‡‚ÌÂÌˡÏË Ì‡ıÓʉÂÌËÂËÌ‚‡ˇÌÚÓ‚ «‡ÔË ̄ÂÏÔÓÎÛ ̃ÂÌÌÛ ̨ÎËÌÂÈÌÛ ̨ÒËÒÚÂÏÛ(7) ‚ χÚË ̃ÌÓςˉÂÓÚÌÓÒËÚÂÎ ̧ÌÓËÒÍÓÏÓÈ ‚ÂÍÚÓ-ÙÛÌ͈ËË~B(t, x) = (u(t, x), ρ(t, x), I(t, x)) Ú ,x∈R1: E ∂ ~B ∂t =A ∂ ~B ∂x ,A=     −1− 1 γM2 − 1 γM2 −1−10 −(γ−1)0−1     ,(8) „ ‰ ÂE| ‰ËÌË ̃̇ˇÏ‡Úˈ‡ÔÓˇ‰Í‡3.

  12. In-text reference with the coordinate start=11398
    Prefix
    Ë ‡ÁÎË ̃Ì ̊‰ÂÈÒÚ‚ËÚÂÎ ̧Ì ̊ÂÒÓ·ÒÚ‚ÂÌÌ ̊ Á̇ ̃ÂÌˡ λ1=−1,λ2= 1−M M ,λ3=− 1 +M M Ë ÔӉӷ̇‰Ë‡„Ó̇Π̧ÌÓÈÊÓ‰‡ÌÓ‚ÓÈχÚˈÂJc χÚˈÂÈÔӉӷˡS(J=S−1AS), „ ‰  S=     011 1−MM −1−(γ−1)M(γ−1)M     ,S−1=     0(γ−1)/γ−1/γ 1/2−1/2γM−1/2γM 1/21/2γM1/2γM     . ÇÚˈ‡SÒÓÒÚ‡‚ÎÂ̇ËÁ ÒÓ·ÒÚ‚ÂÌÌ ̊ı‚ÂÍÚÓӂχÚˈ ̊A, χÚˈ‡S−1̇ȉÂÌ‡Í ‡ Í Ó·‡Ú̇ˇ ÍS. —ӄ·ÒÌÓÓÔ‰ÂÎÂÌË ̨(̇ÔËÏÂ,
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    ) ÒËÒÚÂχ(8) (ÒËÒÚÂχ(7)) ˇ‚ΡÂÚÒˇ„ËÔÂ·ÓÎË ̃ÂÒÍÓÈ(‚ ÛÁÍÓÏÒÏ ̊ÒÎÂ). ¬ ̊ÔÓÎÌËÏÁ‡ÏÂÌÛÔÂÂÏÂÌÌ ̊ı B~=S~V ,~V(t, x) = (V1(t, x), V2(t, x), V3(t, x))Ú,(9) Ë ‡ÒÒÏÓÚËÏÒËÒÚÂÏÛÛ‡‚ÌÂÌËÈ E ∂~V ∂t =S−1AS ∂~V ∂x .(10) —ËÒÚÂχ(10) ÒÓÒÚÓËÚËÁ ÌÂÁ‡‚ËÒËÏ ̊ıÛ‡‚ÌÂÌËÈ ∂Vi ∂t =λi ∂Vi ∂x ,i=1,3,(11) ͇ʉÓÂËÁ ÍÓÚÓ ̊ıÒÓ‰ÂÊËÚÓ‰ÌÛÌÂËÁ‚ÂÒÚÌÛ ̨ÙÛÌÍˆË ̨Ë ËÏÂÂÚ ̄ÂÌË Vi=fi(x+λit),i=1,3,(12) „ ‰ Âfi| ÔÓËÁ‚ÓÎ ̧Ì ̊ÂÌ

  13. In-text reference with the coordinate start=12829
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    ”‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ò ‰‚ÛÏˇÌÂÁ‡‚ËÒËÏ ̊ÏËÔÂÂÏÂÌÌ ̊ÏË(10) ÏÓ‰ÂÎËÛÂÚÓ ‰ ÌÓÏÂÌ ̊È‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ(‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ‚ Ó‰ÌÓÏÂÌÓÈÒ‰Â)Ë ˇ‚ΡÂÚÒˇÒ‡Ï ̊ÏÔÓÒÚ ̊ÏËÁ Û‡‚ÌÂÌËÈ,Í ÍÓÚÓ ̊ÏÔËÏÂÌˇÂÚÒˇÓÔ‰ÂÎÂÌËÂ<‚ÓÎÌÓ‚ÓÂ>.œÓ ̋ÚÓÏÛ‡„ÛÏÂÌÚ ̊ ÙÛÌ͈ËÈfi̇Á ̊‚‡ ̨ÚÚ‡ÍÊ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏË[4] (‚ [5] | ÔÓÒÚ ̊ÏË‚ÓÎ̇ÏË).¬
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    ‰Îˇ ‚ÓÎÌ(12)ÔËÏÂÌÂÌÚÂÏËÌ"·Â„Û ̆Ë‚ÓÎÌ ̊χΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ",ÍÓÚÓ ̊È,ÓÚ‡Ê‡ˇ Ò‚ˇÁ ̧ ̋ÚËı‚ÓÎÌÒ ËÒıÓ‰ÌÓÈÍ‚‡ÁËÎËÌÂÈÌÓÈÒËÒÚÂÏÓÈË ÏÂÚÓ‰ÓÏ ÎË̇ËÁ‡ˆËË,Û͇Á ̊‚‡ÂÚ̇ ‚‡ÊÌÓÒÚ ̧Á‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‰Îˇ ËÒÒΉӂ‡Ìˡ‚ÓÎÌÓ‚ ̊ı ÔÓˆÂÒÒÓ‚.

  14. In-text reference with the coordinate start=13521
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    ‡Î ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‡ÒÔÓÒÚ‡Ìˇ ̨ÚÒˇÒ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËÏË ÒÍÓÓÒÚˇÏË. ¬ ̊ÔÓÎÌÂÌËÂÔÂÓ·‡ÁÓ‚‡ÌˡÔӉӷˡ(9) ÔÓ ÓÚÌÓ ̄ÂÌË ̨Í „ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏÂ(8) ÔË‚ÂÎÓÍ ÌÂÁ‡‚ËÒËÏ ̊ÏÛ‡‚ÌÂÌˡÏË ÔÓˇ‚ÎÂÌË ̨ ̋ÎÂÏÂÌÚ‡Ì ̊ı·Â„Û ̆Ëı‚ÓÎÌ‚ ͇ ̃ÂÒÚ‚Â  ̄ÂÌˡ. ՇȉÂÌÌ ̊‚ ̊‡ÊÂÌˡ Vi≡ ( S−1~B ) i ,i=1,3, Á‡ÔËÒ‡ÌÌ ̊ ̃ÂÂÁÙËÁË ̃ÂÒÍËÁ̇ ̃ËÏ ̊ÂÙÛÌ͈ËË,ÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌËˇÌ‡ ı‡‡ÍÚÂËÒÚËÍ ‡ ı Ë, Òӄ·ÒÌÓÓÔ‰ÂÎÂÌË ̨
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    , Ô‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈËÌ‚‡ˇÌÚ ̊–Ëχ̇ÒËÒÚÂÏ ̊(8): (γ−1) γ ρ(t, x)− 1 γ I(t, x) =f1(x−t),(14) 1 2 u(t, x)− 1 2γM ρ(t, x)− 1 2γM I(t, x) =f2 ( x+ (1−M) M t ) ,(15) 1 2 u(t, x) + 1 2γM ρ(t, x) + 1 2γM I(t, x) =f3 ( x− (1 +M) M t ) .(16) ‡Ê‰ ̊ÈËÁ ÚÂıËÌ‚‡ˇÌÚÓ‚(14){(16)‡ÒÔÓÒÚ‡ÌˇÂÚÒˇ·Â Á ËÒ͇ÊÂÌˡÙÓÏ ̊Ò ı‡‡ÍÚÂËÒÚË ̃ÂÒÍÓÈÒÍÓÓÒÚ ̧ ̨λi,i=1,3.

  15. In-text reference with the coordinate start=23795
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    “Ó„‰‡,ÔÓÒÍÓÎ ̧ÍÛÛ‡‚ÌÂÌË ∂B(t, x) ∂t =a ∂B(t, x) ∂x ,a11=a,B1(t, x) =B(t, x),(36) ËÏÂÂÚ ̄ÂÌËÂB=f(x+at), „ ‰ Âf∈C1, ‰Îˇ  ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄Ë‚ ͇ ̃ÂÒÚ‚ÂÙÛÌ͈ËËf ÒΉÛÂÚ‚ ̊·‡Ú ̧ÙÛÌÍˆË ̨,ÍÓÚÓ‡ˇÁ‡‰‡ÂÚ̇ ̃‡Î ̧ÌÓÂÛÒÎÓ‚ËÂ[4]: B(t, x) =φ(x+at),φ(x) =φ1(x).(37) Œ ̃‚ˉÌÓ(̇ÔËÏÂ,
    Exact
    [2, 6]
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    ), ̃ÚÓ ‰Îˇ „ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏ ̊(32) ‚ˉ ÙÛÌ͈ËÈfi,i=1, n, ÍÓÚÓ ̊ÂÓÔ‰ÂΡ ̨ÚÒÓı‡Ìˇ ̨ ̆Û ̨ÒˇÙÓÏÛi-È ·Â„Û ̆ÂÈ‚ÓÎÌ ̊,ÛÒڇ̇‚ÎË‚‡ÂÚÒˇÔË ÔÓÏÓ ̆ËχÚˈ ̊ÔӉӷˡ: fi(0, x) = (S−1~φ(x))i,S−1AS=J ,J=diag(λ1, . . . , λn).(38) “Ó„‰‡ÔËÁ‡ÏÂÌ‚i-È ÍÓÏÔÓÌÂÌÚ‚ÂÍÚÓ‡~φ(x)‡„ÛÏÂÌÚ‡ẋ ‚ÓÎÌÛ(x+λit)ËÁ (9) ÔÓÎÛ ̃ËÏÒÎÂ‰Û ̨ ̆ÂÂ ̄ÂÌËÂ ̄ÂÌËÂÁ‡‰‡ ̃Ë Ó ̄ˉΡ ÒËÒÚÂÏ ̊(8), ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ÂÈ ‡ÒÔÓÒÚ‡ÌÂÌË ̨Ï

3
CourantR.Methodsof mathematicalphysics.Vol. 2. PartialDifferentialEquations. New York-London:IntersciencePubl.,1962.(Russ.ed.:CourantR.Uravneniyas chastnymi proizvodnymi. Moscow,Mir Publ.,1964.830 p.).
Total in-text references: 6
  1. In-text reference with the coordinate start=2087
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    ÕÂÒÚ‡ˆËÓ̇Ì ̊ÂÛ‡‚ÌÂÌˡ„‡ÁÓ‚ÓȉË̇ÏËÍË,‚ ̊‡Ê‡ ̨ ̆ËÂÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ,ˇ‚Ρ ̨Ú Ò ˇÛ‡‚ÌÂÌˡÏË„ËÔÂ·ÓÎË ̃ÂÒÍÓ„ÓÚËÔ‡.—‚ÓÈÒÚ‚Ó„ËÔÂ·ÓÎË ̃ÌÓÒÚËÚÂÒÌÓÒ‚ˇÁ‡ÌÓÒ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌËÂÏ ̄ÂÌËÈÒÔˆˇΠ̧Ìӄӂˉ‡,ÍÓÚÓ ̊Âӷ·‰‡ ̨ÚÒ‚ÓÈÒÚ‚ÓÏËÌ‚‡ˇÌÚÌÓÒÚË. “‡ÍËÂ ̄ÂÌËˇÌ‡Á ̊‚‡ ̨Ú·Â„Û ̆ËÏË‚ÓÎ̇ÏË
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    [1, 2, 3, 4]
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    . Õ‡ÔÓÏÌËÏ, ̃ÚÓ „ËÔÂ·ÓÎË ̃ÂÒ͇ˇÒËÒÚÂχn,n≥1, Ó‰ÌÓÓ‰Ì ̊ı‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡ÓÚÌÓÒËÚÂÎ ̧ÌÓÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı(t, x), x=xs, Ë Ó‰ÌÓÓ‰ÌÓÂÛ‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ıÔÓˇ‰Í‡m,m≥1, ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÔÂÂÏÂÌÌ ̊ı(t, x),x= (x1, . . . , xs),s≥1, ËÏ ̨Ú ̄ÂÌˡ,ÍÓÚÓ ̊ÂÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌˡ ̇ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËıÔˇÏ ̊ıË „ËÔÂÔÎÓÒÍÓÒÚˇıÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ[2, 3].

  2. In-text reference with the coordinate start=2532
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    Õ‡ÔÓÏÌËÏ, ̃ÚÓ „ËÔÂ·ÓÎË ̃ÂÒ͇ˇÒËÒÚÂχn,n≥1, Ó‰ÌÓÓ‰Ì ̊ı‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡ÓÚÌÓÒËÚÂÎ ̧ÌÓÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı(t, x), x=xs, Ë Ó‰ÌÓÓ‰ÌÓÂÛ‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ıÔÓˇ‰Í‡m,m≥1, ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÔÂÂÏÂÌÌ ̊ı(t, x),x= (x1, . . . , xs),s≥1, ËÏ ̨Ú ̄ÂÌˡ,ÍÓÚÓ ̊ÂÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌˡ ̇ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËıÔˇÏ ̊ıË „ËÔÂÔÎÓÒÍÓÒÚˇıÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ
    Exact
    [2, 3]
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    . —ÎÛ ̃‡Èn= 1, s= 1Ëm= 1,s= 1ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÔÓÒÚÂÈ ̄ÂÏÛÛ‡‚ÌÂÌË ̨‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı,  ̄ÂÌËÂÏÍÓÚÓÓ„Óˇ‚ΡÂÚÒˇ ̋ÎÂÏÂÌÚ‡̇ˇ‚ÓÎ̇[4, 5]. ƒÎˇ ÔÓÒÚÓÂÌˡ‚ÓÎÌ ̊ÒΉÛÂÚÓÔ‰ÂÎËÚ ̧ÍÓ ̋ÙÙˈËÂÌÚ ̊ÔË ÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı ‚ ‚ ̊‡ÊÂÌËË(ω0t+ω·x),ω·x=ω1x1+. . .+ωsxs,  ̄Ë‚‰Îˇ ̋ÚӄӇ΄·‡Ë ̃ÂÒÍÓÂÛ‡‚ÌÂÌËÂ, ÔÓÒÚÓÂÌÌÓÂÔÓ ÓÔ‰ÂÎÂÌÌÓÏÛÔ‡‚ËÎÛËÁ ̃ËÒÎÓ‚ ̊ıÍÓ ̋ÙÙˈËÂÌÚÓ‚‰ËÙÙÂÂ̈ˇΠ̧ÌÓ„Ó Û‡‚ÌÂÌˡËÎË

  3. In-text reference with the coordinate start=3317
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    ’ÓÓ ̄ÓËÁ‚ÂÒÚÌ ̊ÏËÔËÏÂ‡Ïˡ‚Ρ ̨ÚÒˇ:ÒËÒÚÂχÛ‡‚ÌÂÌËȇÍÛÒÚËÍË(n= 2) Ë Â  ̄ÂÌË‚ ‚ˉ ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ(xs−at)Ë(xs+at), ‡ÒÔÓÒÚ‡Ìˇ ̨ ̆ËıÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨ Á‚Û͇a‚‰ÓÎ ̧ÍÓÓ‰Ë̇ÚÌÓÈÓÒËOXs[1, 2, 6]; ‚ÓÎÌÓ‚ÓÂÛ‡‚ÌÂÌËÂ(m= 2) ‚ ÔÓÒÚ‡ÌÒÚ‚Âs ÔÂÂÏÂÌÌ ̊ı,ÍÓÚÓÓÂÔÓÓʉ‡ÂÚ‚ÓÎÌÛf(ω0t+ω·x)(Á‰ÂÒ ̧f| ÔÓËÁ‚ÓÎ ̧̇ˇ„·‰Í‡ˇ ÙÛÌÍˆËˇ),‡ÒÔÓÒÚ‡Ìˇ ̨ ̆‡ˇÒˇ‚ ̇Ô‡‚ÎÂÌËË‚ÂÍÚÓ‡(ω0, ω)
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    . ¬ ‡·ÓÚ‡ı[7, 8, 9] ‰Îˇ ÒËÒÚÂÏ ̊ E ∂ ~B ∂t = ∑s k=1 Ak ∂ ~B ∂xk ,~B= (B1(t, x), . . . Bn(t, x)) Ú ,x= (x1, . . . , xs),(1) „ ‰ ÂAk,k=1, s, | ̃ËÒÎÓ‚ ̊ÂÍ‚‡‰‡ÚÌ ̊ÂχÚˈ ̊ÔÓˇ‰Í‡n,E| ‰ËÌË ̃̇ˇÏ‡Úˈ‡, ·ÂÁÓÚÌÓÒËÚÂÎ ̧ÌÓÍ ÓÔ‰ÂΡÂÏÓÏÛχÚˈ‡ÏË-ÍÓ ̋ÙÙˈËÂÌÚ‡ÏËÚËÔÛ‡ÒÒÏÓÚÂÌ ̊ ̄ÂÌˡ (Exk+Akt),k=1, s, Ë ÙÛÌ͈ËËÓÚ ÌËı.œÓ ‡Ì‡ÎÓ„ËËÒ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏËÚ‡ÍËÂ ̄ÂÌˡ ̇Á‚‡Ì ̊·Â„Û ̆ËÏË‚ÓÎ̇ÏËÒ Ï‡ÚË

  4. In-text reference with the coordinate start=12483
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    Õ‡ÔÓÏÌËÏ, ̃ÚÓ ‰Îˇ Û‡‚ÌÂÌËȇÍÛÒÚËÍË‚ ͇ ̃ÂÒÚ‚Â ÓÒÌÓ‚‡Ìˡı‡‡ÍÚÂËÒÚË ̃ÂÒÍÓ„ÓÚÂÛ„ÓÎ ̧ÌË͇‚ ̊·Ë‡ÂÚÒˇËÌÚÂ‚‡Î̇ ÓÒËOX[1]. ‘ÛÌ͈ËËfi,i=1,3, ÒÓı‡Ìˇ ̨ÚÁ̇ ̃ÂÌËˇÌ‡ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËıÔ‡‡ÎÎÂÎ ̧Ì ̊ıÔˇÏ ̊ı (13) Ë ‚ÏÂÒÚÂÒÓ Ò‚ÓËÏˇ„ÛÏÂÌÚ‡ÏË̇Á ̊‚‡ ̨ÚÒˇ·Â„Û ̆ËÏË‚ÓÎ̇ÏË
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    . ”‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ò ‰‚ÛÏˇÌÂÁ‡‚ËÒËÏ ̊ÏËÔÂÂÏÂÌÌ ̊ÏË(10) ÏÓ‰ÂÎËÛÂÚÓ ‰ ÌÓÏÂÌ ̊È‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ(‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ‚ Ó‰ÌÓÏÂÌÓÈÒ‰Â)Ë ˇ‚ΡÂÚÒˇÒ‡Ï ̊ÏÔÓÒÚ ̊ÏËÁ Û‡‚ÌÂÌËÈ,Í ÍÓÚÓ ̊ÏÔËÏÂÌˇÂÚÒˇÓÔ‰ÂÎÂÌËÂ<‚ÓÎÌÓ‚ÓÂ>.œÓ ̋ÚÓÏÛ‡„ÛÏÂÌÚ ̊ ÙÛÌ͈ËÈfi̇Á ̊‚‡ ̨ÚÚ‡ÍÊ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏË[4] (‚ [5] | ÔÓÒÚ ̊ÏË‚ÓÎ̇ÏË).¬ [2] ‰Îˇ ‚ÓÎÌ(12)ÔËÏÂÌÂÌÚÂÏËÌ"·Â„Û ̆Ë‚ÓÎÌ ̊χΠ̊ı‚ÓÁÏÛ ̆ÂÌË

  5. In-text reference with the coordinate start=22378
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    ¡Û‰ÂÏ„Ó‚ÓËÚ ̧Ó ‚ÓÎÌÓ‚ÓÏÔÓ‰ıÓ‰ÂÍ ÎËÌÂÈÌÓÈÒËÒÚÂÏÂÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡(8), ÍÓÚÓ‡ˇ‡ÒÒχÚË‚‡ÂÚÒˇ‚ ͇ ̃ÂÒÚ‚Â ̃‡ÒÚÌÓ„ÓÒÎÛ ̃‡ˇÒËÒÚÂÏ ̊(1), ÔÓÌËχˇÔÓ‰ ̋ÚËÏ‡ÁÎÓÊÂÌËÂ ̄ÂÌˡÔÓn-ÏÂÌ ̊Ï·Â„Û ̆ËÏ‚ÓÎ̇Ï. 7. œ‰ÒÚ‡‚ÎÂÌËÂ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄Ë Õ‡ÔÓÏÌËÏ
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    , ̃ÚÓ ÒËÒÚÂχ E ∂ ~B(t, x) ∂t =A ∂ ~B(t, x) ∂x ,~B(t, x) = (B1(t, x), . . . , Bn(t, x)),x∈R1,(32) „ ‰ Â~B(t, x)|n-ÏÂ̇ˇ‚ÂÍÚÓ-ÙÛÌÍˆËˇ,A| ÔÓËÁ‚ÓÎ ̧̇ˇ ̃ËÒÎÓ‚‡ˇÏ‡Úˈ‡Ò ̋ÎÂÏÂÌÚ‡ÏË{aij}ni,j=1, ˇ‚ΡÂÚÒˇÒËÒÚÂÏÓÈ Ó‚‡Î‚ÒÍÓÈ.ƒÎˇ Ú‡ÍÓÈÒËÒÚÂÏ ̊ÂÒÚÂÒÚ‚ÂÌÌ ̊ÏÓ·‡ÁÓÏ ‚ÓÁÌË͇ÂÚÁ‡‰‡ ̃‡ Ó ̄Ë. œÛÒÚ ̧n-ÏÂ̇ˇ‚ÂÍÚÓ-ÙÛÌÍˆËˇ~φ(x)ˇ‚ΡÂÚÒˇ‡Ì‡ÎËÚË ̃ÂÒÍÓÈÙÛÌ͈ËÂÈx‚ ÌÂÍÓÚÓÓÈ Ó·Î‡ÒÚËG0ÔÎÓÒÍÓÒÚËt= 0, ÒÓ‰Âʇ ̆ÂÈÚÓ

  6. In-text reference with the coordinate start=23162
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    = 0, ÒÓ‰Âʇ ̆ÂÈÚÓ ̃ ÍÛx0,x0= 0, Ë Ô‰ÒÚ‡‚ÎÂ̇‚G0ˇ‰ÓÏ “ÂÈÎÓ‡ φ~(x) = ∑∞ α=0 ~cα·xα,~cα= 1 α! ∂α ∂xα ~φ(x) ∣∣∣ ∣∣∣ ∣ x=0 .(33) –‡ÒÒÏÓÚËÏ̇ ̃‡Î ̧ÌÓÂÛÒÎÓ‚ËÂÔÓ ÔÂÂÏÂÌÌÓÈt: B~(t, x) ∣∣∣ ∣ x=0 =~φ(x),~φ(x) = (φ1(x), . . . , φn(x)) Ú .(34) —ӄ·ÒÌÓÚÂÓÂÏÂ Ó ̄Ë| Ó‚‡Î‚ÒÍÓÈÁ‡‰‡ ̃‡ Ó ̄Ë(32){(34)ËÏÂÂÚ‚ ӷ·ÒÚËG,G0⊂ G, ‰ËÌÒÚ‚ÂÌÌÓ‡̇ÎËÚË ̃ÂÒÍÓÂ ̄ÂÌËÂ,ÍÓÚÓÓ‚ ̊‡Ê‡ÂÚÒˇÒÚÂÔÂÌÌ ̊Ïˇ‰ÓÏ(̇ÔËÏÂ,
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    ): Bi(t, x) = ∑∞ α1,α2=0 γiα1, α2xα1tα2.(35) ◊ËÒÎÓ‚ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊γiα1,α2Ó‰ÌÓÁ̇ ̃ÌÓÓÔ‰ÂΡ ̨ÚÒˇÔÓ Ì‡ ̃‡Î ̧Ì ̊ω‡ÌÌ ̊ÏËÁ ÒËÒÚÂÏ ̊ Û‡‚ÌÂÌËÈ(32) ‚ ÂÁÛÎ ̧Ú‡ÚÂÂÍÍÛÂÌÚÌÓÈÔӈ‰Û ̊ÔÓ Ë̉ÂÍÒÛα2Ë ˇ‚Ρ ̨ÚÒˇÔÓÎËÌÓχÏË ̋ÎÂÏÂÌÚӂχÚˈ ̊AË ÍÓ ̋ÙÙˈËÂÌÚÓ‚γiα1,l,l≤α2−1.

4
BhatnagarP.L.NonlinearWavesin One-DimensionalDispersiveSystems. Oxford,Clarendon Press,1979.(Russ.ed.:BhatnagarP.L.Nelineynyevolnyv odnomernykhdispersnykh sistemakh. Moscow,Mir Publ.,1983.136 p.).
Total in-text references: 4
  1. In-text reference with the coordinate start=2087
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    ÕÂÒÚ‡ˆËÓ̇Ì ̊ÂÛ‡‚ÌÂÌˡ„‡ÁÓ‚ÓȉË̇ÏËÍË,‚ ̊‡Ê‡ ̨ ̆ËÂÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ,ˇ‚Ρ ̨Ú Ò ˇÛ‡‚ÌÂÌˡÏË„ËÔÂ·ÓÎË ̃ÂÒÍÓ„ÓÚËÔ‡.—‚ÓÈÒÚ‚Ó„ËÔÂ·ÓÎË ̃ÌÓÒÚËÚÂÒÌÓÒ‚ˇÁ‡ÌÓÒ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌËÂÏ ̄ÂÌËÈÒÔˆˇΠ̧Ìӄӂˉ‡,ÍÓÚÓ ̊Âӷ·‰‡ ̨ÚÒ‚ÓÈÒÚ‚ÓÏËÌ‚‡ˇÌÚÌÓÒÚË. “‡ÍËÂ ̄ÂÌËˇÌ‡Á ̊‚‡ ̨Ú·Â„Û ̆ËÏË‚ÓÎ̇ÏË
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    . Õ‡ÔÓÏÌËÏ, ̃ÚÓ „ËÔÂ·ÓÎË ̃ÂÒ͇ˇÒËÒÚÂχn,n≥1, Ó‰ÌÓÓ‰Ì ̊ı‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡ÓÚÌÓÒËÚÂÎ ̧ÌÓÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı(t, x), x=xs, Ë Ó‰ÌÓÓ‰ÌÓÂÛ‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ıÔÓˇ‰Í‡m,m≥1, ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÔÂÂÏÂÌÌ ̊ı(t, x),x= (x1, . . . , xs),s≥1, ËÏ ̨Ú ̄ÂÌˡ,ÍÓÚÓ ̊ÂÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌˡ ̇ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËıÔˇÏ ̊ıË „ËÔÂÔÎÓÒÍÓÒÚˇıÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ[2, 3].

  2. In-text reference with the coordinate start=2670
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    ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı(t, x), x=xs, Ë Ó‰ÌÓÓ‰ÌÓÂÛ‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ıÔÓˇ‰Í‡m,m≥1, ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÔÂÂÏÂÌÌ ̊ı(t, x),x= (x1, . . . , xs),s≥1, ËÏ ̨Ú ̄ÂÌˡ,ÍÓÚÓ ̊ÂÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌˡ ̇ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËıÔˇÏ ̊ıË „ËÔÂÔÎÓÒÍÓÒÚˇıÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ[2, 3]. —ÎÛ ̃‡Èn= 1, s= 1Ëm= 1,s= 1ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÔÓÒÚÂÈ ̄ÂÏÛÛ‡‚ÌÂÌË ̨‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı,  ̄ÂÌËÂÏÍÓÚÓÓ„Óˇ‚ΡÂÚÒˇ ̋ÎÂÏÂÌÚ‡̇ˇ‚ÓÎ̇
    Exact
    [4, 5]
    Suffix
    . ƒÎˇ ÔÓÒÚÓÂÌˡ‚ÓÎÌ ̊ÒΉÛÂÚÓÔ‰ÂÎËÚ ̧ÍÓ ̋ÙÙˈËÂÌÚ ̊ÔË ÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı ‚ ‚ ̊‡ÊÂÌËË(ω0t+ω·x),ω·x=ω1x1+. . .+ωsxs,  ̄Ë‚‰Îˇ ̋ÚӄӇ΄·‡Ë ̃ÂÒÍÓÂÛ‡‚ÌÂÌËÂ, ÔÓÒÚÓÂÌÌÓÂÔÓ ÓÔ‰ÂÎÂÌÌÓÏÛÔ‡‚ËÎÛËÁ ̃ËÒÎÓ‚ ̊ıÍÓ ̋ÙÙˈËÂÌÚÓ‚‰ËÙÙÂÂ̈ˇΠ̧ÌÓ„Ó Û‡‚ÌÂÌˡËÎË ÒËÒÚÂÏ ̊Û‡‚ÌÂÌËÈ.

  3. In-text reference with the coordinate start=12794
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    ”‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ò ‰‚ÛÏˇÌÂÁ‡‚ËÒËÏ ̊ÏËÔÂÂÏÂÌÌ ̊ÏË(10) ÏÓ‰ÂÎËÛÂÚÓ ‰ ÌÓÏÂÌ ̊È‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ(‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ‚ Ó‰ÌÓÏÂÌÓÈÒ‰Â)Ë ˇ‚ΡÂÚÒˇÒ‡Ï ̊ÏÔÓÒÚ ̊ÏËÁ Û‡‚ÌÂÌËÈ,Í ÍÓÚÓ ̊ÏÔËÏÂÌˇÂÚÒˇÓÔ‰ÂÎÂÌËÂ<‚ÓÎÌÓ‚ÓÂ>.œÓ ̋ÚÓÏÛ‡„ÛÏÂÌÚ ̊ ÙÛÌ͈ËÈfi̇Á ̊‚‡ ̨ÚÚ‡ÍÊ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏË
    Exact
    [4]
    Suffix
    (‚ [5] | ÔÓÒÚ ̊ÏË‚ÓÎ̇ÏË).¬ [2] ‰Îˇ ‚ÓÎÌ(12)ÔËÏÂÌÂÌÚÂÏËÌ"·Â„Û ̆Ë‚ÓÎÌ ̊χΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ",ÍÓÚÓ ̊È,ÓÚ‡Ê‡ˇ Ò‚ˇÁ ̧ ̋ÚËı‚ÓÎÌÒ ËÒıÓ‰ÌÓÈÍ‚‡ÁËÎËÌÂÈÌÓÈÒËÒÚÂÏÓÈË ÏÂÚÓ‰ÓÏ ÎË̇ËÁ‡ˆËË,Û͇Á ̊‚‡ÂÚ̇ ‚‡ÊÌÓÒÚ ̧Á‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‰Îˇ ËÒÒΉӂ‡Ìˡ‚ÓÎÌÓ‚ ̊ı ÔÓˆÂÒÒÓ‚.

  4. In-text reference with the coordinate start=23739
    Prefix
    “Ó„‰‡,ÔÓÒÍÓÎ ̧ÍÛÛ‡‚ÌÂÌË ∂B(t, x) ∂t =a ∂B(t, x) ∂x ,a11=a,B1(t, x) =B(t, x),(36) ËÏÂÂÚ ̄ÂÌËÂB=f(x+at), „ ‰ Âf∈C1, ‰Îˇ  ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄Ë‚ ͇ ̃ÂÒÚ‚ÂÙÛÌ͈ËËf ÒΉÛÂÚ‚ ̊·‡Ú ̧ÙÛÌÍˆË ̨,ÍÓÚÓ‡ˇÁ‡‰‡ÂÚ̇ ̃‡Î ̧ÌÓÂÛÒÎÓ‚ËÂ
    Exact
    [4]
    Suffix
    : B(t, x) =φ(x+at),φ(x) =φ1(x).(37) Œ ̃‚ˉÌÓ(̇ÔËÏÂ,[2, 6]), ̃ÚÓ ‰Îˇ „ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏ ̊(32) ‚ˉ ÙÛÌ͈ËÈfi,i=1, n, ÍÓÚÓ ̊ÂÓÔ‰ÂΡ ̨ÚÒÓı‡Ìˇ ̨ ̆Û ̨ÒˇÙÓÏÛi-È ·Â„Û ̆ÂÈ‚ÓÎÌ ̊,ÛÒڇ̇‚ÎË‚‡ÂÚÒˇÔË ÔÓÏÓ ̆ËχÚˈ ̊ÔӉӷˡ: fi(0, x) = (S−1~φ(x))i,S−1AS=J ,J=diag(λ1, . . . , λn).(38) “Ó„‰‡ÔËÁ‡ÏÂÌ‚i-È ÍÓÏÔÓÌÂÌÚ‚ÂÍÚÓ‡~φ(x)‡„ÛÏÂÌÚ‡ẋ ‚ÓÎÌÛ(x+λit)ËÁ (9) ÔÓÎÛ ̃ËÏÒÎÂ‰Û ̨ ̆ÂÂ ̄ÂÌËÂ ̄ÂÌËÂÁ‡‰‡

5
LoytsyanskiyL.G.Mekhanikazhidkostii gaza[FluidMechanics].Moscow,NaukaPubl., 1987.840 p. (in Russian).
Total in-text references: 7
  1. In-text reference with the coordinate start=2670
    Prefix
    ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı(t, x), x=xs, Ë Ó‰ÌÓÓ‰ÌÓÂÛ‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ıÔÓˇ‰Í‡m,m≥1, ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÔÂÂÏÂÌÌ ̊ı(t, x),x= (x1, . . . , xs),s≥1, ËÏ ̨Ú ̄ÂÌˡ,ÍÓÚÓ ̊ÂÒÓı‡Ìˇ ̨ÚÒ‚ÓËÁ̇ ̃ÂÌˡ ̇ ı‡‡ÍÚÂËÒÚË ̃ÂÒÍËıÔˇÏ ̊ıË „ËÔÂÔÎÓÒÍÓÒÚˇıÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ[2, 3]. —ÎÛ ̃‡Èn= 1, s= 1Ëm= 1,s= 1ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÔÓÒÚÂÈ ̄ÂÏÛÛ‡‚ÌÂÌË ̨‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı,  ̄ÂÌËÂÏÍÓÚÓÓ„Óˇ‚ΡÂÚÒˇ ̋ÎÂÏÂÌÚ‡̇ˇ‚ÓÎ̇
    Exact
    [4, 5]
    Suffix
    . ƒÎˇ ÔÓÒÚÓÂÌˡ‚ÓÎÌ ̊ÒΉÛÂÚÓÔ‰ÂÎËÚ ̧ÍÓ ̋ÙÙˈËÂÌÚ ̊ÔË ÌÂÁ‡‚ËÒËÏ ̊ıÔÂÂÏÂÌÌ ̊ı ‚ ‚ ̊‡ÊÂÌËË(ω0t+ω·x),ω·x=ω1x1+. . .+ωsxs,  ̄Ë‚‰Îˇ ̋ÚӄӇ΄·‡Ë ̃ÂÒÍÓÂÛ‡‚ÌÂÌËÂ, ÔÓÒÚÓÂÌÌÓÂÔÓ ÓÔ‰ÂÎÂÌÌÓÏÛÔ‡‚ËÎÛËÁ ̃ËÒÎÓ‚ ̊ıÍÓ ̋ÙÙˈËÂÌÚÓ‚‰ËÙÙÂÂ̈ˇΠ̧ÌÓ„Ó Û‡‚ÌÂÌˡËÎË ÒËÒÚÂÏ ̊Û‡‚ÌÂÌËÈ.

  2. In-text reference with the coordinate start=5391
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    Á‡‰‡ ̨ ̆ËÂÒËÒÚÂÏÛ.ƒÎˇ ÔÓÎÛ ̃ÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄ˉÓÒÚ‡ÚÓ ̃ÌÓ‚ Ó· ̆ÂÏ ̄ÂÌËËÁ‡ÏÂÌËÚ ̧ ÌÂÓÔ‰ÂÎÂÌÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊̇ ‚ÂÍÚÓÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊‡ÁÎÓÊÂÌˡ‡Ì‡ÎËÚË ̃ÂÒÍÓÈ̇ ̃‡Î ̧ÌÓÈÙÛÌ͈ËË‚ ˇ‰ “ÂÈÎÓ‡[8]. ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚ [7, 8, 9] ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊
    Exact
    [1, 5, 12]
    Suffix
    .¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡[2, 5, 12], ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.¬ ͇ ̃ÂÒÚ‚ÂÁ‡Ï ͇̊ ̨ ̆ËıÛ‡‚ÌÂÌËÈ[12] ‰Îˇ Û‡ÌÂÌËÈÒÓı‡ÌÂÌˡÒÍÓÓÒÚËË ËÏÔÛÎ ̧Ò‡‡ÒÒÏÓÚËÏÛ‡‚ÌÂÌËÂÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË, ‚ ̊‡Á˂ ̃ÂÂÁ ̋ÌڇΠ̧ÔË ̨,Ë Û‡‚ÌÂÌËÂÒÓÒÚÓˇÌˡ,Ò‚ˇÁ ̊‚‡ ̨ ̆ ̋ÌڇΠ̧ÔË ̨Ò ÔÎÓÚÌÓÒÚ ̧ ̨Ë ‰‡‚ÎÂÌËÂÏ„‡Á‡.

  3. In-text reference with the coordinate start=5519
    Prefix
    ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚ [7, 8, 9] ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊[1, 5, 12].¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡
    Exact
    [2, 5, 12]
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    , ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.¬ ͇ ̃ÂÒÚ‚ÂÁ‡Ï ͇̊ ̨ ̆ËıÛ‡‚ÌÂÌËÈ[12] ‰Îˇ Û‡ÌÂÌËÈÒÓı‡ÌÂÌˡÒÍÓÓÒÚËË ËÏÔÛÎ ̧Ò‡‡ÒÒÏÓÚËÏÛ‡‚ÌÂÌËÂÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË, ‚ ̊‡Á˂ ̃ÂÂÁ ̋ÌڇΠ̧ÔË ̨,Ë Û‡‚ÌÂÌËÂÒÓÒÚÓˇÌˡ,Ò‚ˇÁ ̊‚‡ ̨ ̆ ̋ÌڇΠ̧ÔË ̨Ò ÔÎÓÚÌÓÒÚ ̧ ̨Ë ‰‡‚ÎÂÌËÂÏ„‡Á‡.

  4. In-text reference with the coordinate start=6495
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    ◊ÚÓ· ̊ËÁ Í‚‡ÁËÎËÌÂÈÌÓ„ÓÔÓˆÂÒÒ‡‚ ̊‰ÂÎËÚ ̧ÎËÌÂÈÌ ̊‚ÓÎÌ ̊,ÔÓÓʉ‡ÂÏ ̊ÂÒËÒÚÂÏÓÈ(1), ÔËÏÂÌËÏÏÂÚӉχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ(̇ÔËÏÂ,[2]).¬ÓÁÌË͇ ̨ ̆Û ̨ÔË ̋ÚÓÏ Á‡‰‡ ̃ÛÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ,ÍÓÚÓ‡ˇÔ‰ÒÚ‡‚ΡÂÚÒ‡ÏÓÒÚÓˇÚÂÎ ̧Ì ̊ÈËÌÚÂÂÒ
    Exact
    [2, 5, 13]
    Suffix
    , ‡ÒÒÏÓÚËÏÍ ‡ Í ÚÂÒÚÓ‚Û ̨‰Îˇ ÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒÚ‚ËˇÌ‡ ÔËÏÂ ‰‚ËÊÂÌˡˉ‡Π̧ÌÓ„Ó„‡Á‡ ÔË Á‡‰‡ÌÌ ̊ınËs. œÓ ÒÚÛÍÚÛÂÔÓÒÚÓÂÌÌÓÈÏÓ‰ÂÎËÓ· ̆ÂÂ ̄ÂÌËÂÒËÒÚÂÏ ̊ E ∂ ~B ∂t =Ak ∂ ~B ∂xk ,~B= (B1(t, xk), .

  5. In-text reference with the coordinate start=7929
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    Ó ÔÓˇ‰Í‡Ì‡ ÔËÏÂÂÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ „‡ÁÂÔË Ó‰ÌÓÏÂÌÓÏÚ ̃ÂÌËËÒ ÔÓÒÎÂ‰Û ̨ ̆ËÏËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏÔÓÎÛ ̃ÂÌÌÓ„Ó ̄ÂÌˡ‰Îˇ ÔÓÒÚÓÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃ËÔË Ú ̃ÂÌËË„‡Á‡ ‚ ÔÓÒÚ‡ÌÒÚ‚Â. 2. ”‡‚ÌÂÌˡ‡ÒÔÓÒÚ‡ÌÂÌËˇÏ‡Î ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ‚‡ÁËÎËÌÂÈÌ ̊ÂÛ‡‚ÌÂÌˡӉÌÓÏÂÌÓ„Ó‰‚ËÊÂÌˡˉ‡Π̧ÌÓ„Ó„‡Á‡ ÔË ÓÚÒÛÚÒÚ‚ËËÏ‡Ò ÒÓ‚ ̊ıÒËÎ ‚ ‰Â͇ÚÓ‚ÓÈÒËÒÚÂÏÂÍÓÓ‰Ë̇ÚËÏ ̨ڂˉ
    Exact
    [2, 5, 13]
    Suffix
    :          ∂u ̃ ∂ ̃t + ̃u ∂u ̃ ∂ ̃x =− 1 ρ ̃ ∂ ̃P ∂ ̃x , ∂ ̃ρ ∂ ̃t + ∂( ̃ρ ̃u) ∂ ̃x = 0, ∂I ̃ ∂ ̃t + ̃u ∂I ̃ ∂ ̃x = 1 ρ ̃ ( ∂ ̃P ∂ ̃t + ̃u ∂ ̃P ∂ ̃x ) , x ̃∈R1,(3) „ ‰  ̃u( ̃t, ̃x), ̃ρ( ̃t, ̃x), ̃P( ̃t, ̃x), ̃I( ̃t ̃x)| ÒÍÓÓÒÚ ̧‰‚ËÊÂÌˡ„‡Á‡,Â„Ó ÔÎÓÚÌÓÒÚ ̧,‰‡‚ÎÂÌËÂË ̋ÌڇΠ̧ÔˡÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ.

  6. In-text reference with the coordinate start=9710
    Prefix
    , St ∂ρ ∂t =−ρ ∂u ∂x −u ∂ρ ∂x , St ∂I ∂t =−(γ−1)I ∂u ∂x −u ∂I ∂x . (5) ¬ (5)St| ̃ËÒÎÓ—ÚÛı‡Î‡;St=L0/U0T0;M| ̃ËÒÎÓÇı‡;M=U0/a,a| ÒÍÓÓÒÚ ̧Á‚Û͇ ‚ „‡ÁÓ‚ÓÈÒ‰Â,ÔË ̋ÚÓÏa2= (γ−1)I0. œÓÒÍÓÎ ̧ÍÛÒ‰Ë‡ÁÏÂÌ ̊ı‚ÂÎË ̃ËÌ,Á‡‰‡ ̨ ̆Ëı ̃ËÒÎÓ—ÚÛı‡Î‡,‰‚ ˇ‚Ρ ̨ÚÒˇÌÂÁ‡‚ËÒËÏ ̊ÏË,ÏÓÊÌÓ‚ ̊·‡Ú ̧St= 1. –‡ÒÒÏÓÚËÏÁ‡‰‡ ̃ÛÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡ÁÂÔË Ó‰ÌÓÏÂÌÓÏÔ‡‡ÎÎÂÎ ̧ÌÓÏÓÒËOX‰‚ËÊÂÌËË
    Exact
    [2, 5]
    Suffix
    . œÛÒÚ ̧‚ ‰‚ËÊÛ ̆ÂÏÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨U0 ˉ‡Π̧ÌÓÏ„‡ÁÂ,ÍÓÚÓ ̊ÈËÏÂÂÚÔÎÓÚÌÓÒÚ ̧ρ0Ë ̋ÌڇΠ̧ÔË ̨I0, ÒÓÁ‰‡ ̨ÚÒˇÏ‡Î ̊‚ÓÁÏÛ ̆ÂÌˡ ÒÍÓÓÒÚËu∗(t, x), ÔÎÓÚÌÓÒÚËρ∗(t, x)Ë ̋ÌڇΠ̧ÔËËI∗(t, x). “Ó„‰‡‰‚ËÊÂÌË ̨‚ÓÁÏÛ ̆ÂÌÌÓ„Ó „‡Á‡ ÓÚ‚Â ̃‡ ̨ÚÙÛÌ͈ËË u(t, x) = 1 +u∗(t, x),ρ(t, x) = 1 +ρ∗(t, x),I(t, x) = 1 +I∗(t, x).(6) ◊ÚÓ· ̊ÌÂÎËÌÂÈ̇ˇÒËÒÚÂχ(5) ËÏ· ̄ÂÌË‚ˉ‡(6), ÙÛÌ͈ËË,Á‡‰‡ ̨ ̆ËÂχΠ̊‚ÓÁÏÛ ̆ÂÌˡ,‰ÓÎÊÌ ̊

  7. In-text reference with the coordinate start=12802
    Prefix
    ”‡‚ÌÂÌË‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ò ‰‚ÛÏˇÌÂÁ‡‚ËÒËÏ ̊ÏËÔÂÂÏÂÌÌ ̊ÏË(10) ÏÓ‰ÂÎËÛÂÚÓ ‰ ÌÓÏÂÌ ̊È‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ(‚ÓÎÌÓ‚ÓÈÔÓˆÂÒÒ‚ Ó‰ÌÓÏÂÌÓÈÒ‰Â)Ë ˇ‚ΡÂÚÒˇÒ‡Ï ̊ÏÔÓÒÚ ̊ÏËÁ Û‡‚ÌÂÌËÈ,Í ÍÓÚÓ ̊ÏÔËÏÂÌˇÂÚÒˇÓÔ‰ÂÎÂÌËÂ<‚ÓÎÌÓ‚ÓÂ>.œÓ ̋ÚÓÏÛ‡„ÛÏÂÌÚ ̊ ÙÛÌ͈ËÈfi̇Á ̊‚‡ ̨ÚÚ‡ÍÊ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏË[4] (‚
    Exact
    [5]
    Suffix
    | ÔÓÒÚ ̊ÏË‚ÓÎ̇ÏË).¬ [2] ‰Îˇ ‚ÓÎÌ(12)ÔËÏÂÌÂÌÚÂÏËÌ"·Â„Û ̆Ë‚ÓÎÌ ̊χΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ",ÍÓÚÓ ̊È,ÓÚ‡Ê‡ˇ Ò‚ˇÁ ̧ ̋ÚËı‚ÓÎÌÒ ËÒıÓ‰ÌÓÈÍ‚‡ÁËÎËÌÂÈÌÓÈÒËÒÚÂÏÓÈË ÏÂÚÓ‰ÓÏ ÎË̇ËÁ‡ˆËË,Û͇Á ̊‚‡ÂÚ̇ ‚‡ÊÌÓÒÚ ̧Á‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‰Îˇ ËÒÒΉӂ‡Ìˡ‚ÓÎÌÓ‚ ̊ı ÔÓˆÂÒÒÓ‚.

6
ToroE.F.RiemannSolversand NumericalMethodsfor FluidDynamics. SpringerBerlin Heidelberg,2009.724 p. DOI:10.1007/b79761
Total in-text references: 4
  1. In-text reference with the coordinate start=3136
    Prefix
    ̊ıÔÂÂÏÂÌÌ ̊ı ‚ ‚ ̊‡ÊÂÌËË(ω0t+ω·x),ω·x=ω1x1+. . .+ωsxs,  ̄Ë‚‰Îˇ ̋ÚӄӇ΄·‡Ë ̃ÂÒÍÓÂÛ‡‚ÌÂÌËÂ, ÔÓÒÚÓÂÌÌÓÂÔÓ ÓÔ‰ÂÎÂÌÌÓÏÛÔ‡‚ËÎÛËÁ ̃ËÒÎÓ‚ ̊ıÍÓ ̋ÙÙˈËÂÌÚÓ‚‰ËÙÙÂÂ̈ˇΠ̧ÌÓ„Ó Û‡‚ÌÂÌˡËÎË ÒËÒÚÂÏ ̊Û‡‚ÌÂÌËÈ. ’ÓÓ ̄ÓËÁ‚ÂÒÚÌ ̊ÏËÔËÏÂ‡Ïˡ‚Ρ ̨ÚÒˇ:ÒËÒÚÂχÛ‡‚ÌÂÌËȇÍÛÒÚËÍË(n= 2) Ë Â  ̄ÂÌË‚ ‚ˉ ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ(xs−at)Ë(xs+at), ‡ÒÔÓÒÚ‡Ìˇ ̨ ̆ËıÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨ Á‚Û͇a‚‰ÓÎ ̧ÍÓÓ‰Ë̇ÚÌÓÈÓÒËOXs
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    [1, 2, 6]
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    ; ‚ÓÎÌÓ‚ÓÂÛ‡‚ÌÂÌËÂ(m= 2) ‚ ÔÓÒÚ‡ÌÒÚ‚Âs ÔÂÂÏÂÌÌ ̊ı,ÍÓÚÓÓÂÔÓÓʉ‡ÂÚ‚ÓÎÌÛf(ω0t+ω·x)(Á‰ÂÒ ̧f| ÔÓËÁ‚ÓÎ ̧̇ˇ„·‰Í‡ˇ ÙÛÌÍˆËˇ),‡ÒÔÓÒÚ‡Ìˇ ̨ ̆‡ˇÒˇ‚ ̇Ô‡‚ÎÂÌËË‚ÂÍÚÓ‡(ω0, ω)[1, 3]. ¬ ‡·ÓÚ‡ı[7, 8, 9] ‰Îˇ ÒËÒÚÂÏ ̊ E ∂ ~B ∂t = ∑s k=1 Ak ∂ ~B ∂xk ,~B= (B1(t, x), .

  2. In-text reference with the coordinate start=4093
    Prefix
    ‡ÏËÒ Ï‡ÚË ̃Ì ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏËËÎËn-ÏÂÌ ̊ÏË·Â„Û ̆ËÏË ‚ÓÎ̇ÏË.œËs= 1‚ ÒÎÛ ̃‡Â„ËÔÂ·ÓÎË ̃ÌÓÒÚËÒËÒÚÂÏ ̊(χÚˈ‡AsÔӉӷ̇‰Ë‡„Ó̇Π̧ÌÓÈ ÊÓ‰‡ÌÓ‚ÓÈχÚˈÂ)‚‚‰ÂÌ̇ˇn-ÏÂ̇ˇ·Â„Û ̆‡ˇ‚ÓÎ̇ËÏÂÂÚÔÓÎÌÛ ̨ËÌÚÂÔÂÚ‡ˆË ̨ ̃ÂÂÁn ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ,ÍÓÚÓ ̊Â‡ÒÒÏÓÚÂÌ ̊Í ‡ Í Ò͇ΡÌ ̊Â[10]. ÕÂÒÏÓÚˇÌ‡ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌË ̃ËÒÎÂÌÌ ̊ıÏÂÚÓ‰Ó‚ ̄ÂÌˡÒËÒÚÂÏ ̊(1) ‚ ÔËÎÓÊÂÌËËÍ ÓÚ‰ÂÎ ̧Ì ̊ÏÙËÁË ̃ÂÒÍËÏÔÓˆÂÒÒ‡Ï(̇ÔËÏÂ,
    Exact
    [6]
    Suffix
    ),‡ÍÚۇΠ̧Ì ̊Ïˇ‚ΡÂÚÒˇÔÓÒÚÓÂÌˇ̇ÎËÚË ̃ÂÒÍÓ„ÓÏÂÚÓ‰‡ ̄ÂÌˡ ̋ÚÓÈÒËÒÚÂÏ ̊Ë Ò‚ˇÁ‡ÌÌÓÈÒ ÌÂÈ Á‡‰‡ ̃Ë Ó ̄ËÔË ÔÓËÁ‚ÓÎ ̧Ì ̊ın,sË ÔÓËÁ‚ÓÎ ̧Ì ̊ıχÚË ̃Ì ̊ıÍÓ ̋ÙÙˈËÂÌÚ‡ıAk,k=1, s. ƒÎˇ ÒËÒÚÂÏ ̊(1), ÍÓÚÓ‡ˇ ı‡‡ÍÚÂ̇‰Îˇ ‡ÁÎË ̃Ì ̊ıӷ·ÒÚÂÈχÚÂχÚË ̃ÂÒÍÓÈÙËÁËÍË[11],‚ ‡·ÓÚ‡ı[7, 8, 9] ‡‚ÚÓ ‡ÏËÔÓÒÚÓÂ̇ÏÓ‰ÂÎ ̧‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ.

  3. In-text reference with the coordinate start=7124
    Prefix
    s,(2) Ô‰ÒÚ‡‚ÎÂÌÌÓÂÍ ‡ Í ‡ÁÎÓÊÂÌËÂÔÓ ‚ÓÎÌÂ(Exk+Akt), ˇ‚ΡÂÚÒˇ ̃‡ÒÚÌ ̊Ï ̄ÂÌËÂÏÒËÒÚÂÏ ̊(1) Ë ÒÓÒÚ‡‚ÌÓÈ ̃‡ÒÚ ̧ ̨ ӷ ̆„Ó ̄ÂÌˡ,Ô‰ÒÚ‡‚ÎÂÌÌÓ„Ó‚ ‚ˉÂ‡ÁÎÓÊÂÌˡÔÓs ‚Á‡ËÏÓ‰ÂÈÒÚ‚Û ̨ ̆ËÏ‚ÓÎ̇Ï[7, 9]. œÛÒÚ ̧(2) ÓÚ‚Â ̃‡ÂÚ‡ÒÔÓÒÚ‡ÌÂÌË ̨χΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË Ó‰ÌÓÏÂÌÓÏ Ú ̃ÂÌËË.ՇȉÂÏ ̄ÂÌË ̋ÚÓÈÒËÒÚÂÏ ̊Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏÚ‡‰ËˆËÓÌÌÓ„ÓÔÓ‰ıÓ‰‡Í Ó‰ÌÓÏÂÌÓÈ„ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏÂ
    Exact
    [2, 6]
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    Ë Ò‰ÒÚ‚‡ÏËÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ ·Â Á ËÒÔÓÎ ̧ÁÓ‚‡ÌˡÛÒÎӂˡ„ËÔÂ·ÓÎË ̃ÌÓÒÚË[7]. œÓ͇ÊÂÏÚÓʉÂÒÚ‚ÂÌÌÓÒÚ ̧ÔÓÎÛ ̃ÂÌÌ ̊ı ̄ÂÌËÈË ÔÂËÏÛ ̆ÂÒÚ‚‡‚ÓÎÌÓ‚Ó„ÓÔÓ‰ıÓ‰‡ÔË ËÒÒΉӂ‡ÌËËÒËÒÚÂÏ ̊(2). ÷ÂÎ ̧ ̨‡·ÓÚ ̊ˇ‚ΡÂÚÒˇ‡ÒÒÏÓÚÂÌËÂÔ‰·„‡ÂÏÓ„Ó‡‚ÚÓ‡ÏËÌӂӄӇ̇ÎËÚË ̃ÂÒÍÓ„Ó ÏÂÚÓ‰‡ ̄ÂÌˡÌÓχΠ̧ÌÓÈÒËÒÚÂÏ ̊ÎËÌÂÈÌ ̊ıÓ‰ÌÓÓ‰Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ì‡ ÔËÏÂÂÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ

  4. In-text reference with the coordinate start=23795
    Prefix
    “Ó„‰‡,ÔÓÒÍÓÎ ̧ÍÛÛ‡‚ÌÂÌË ∂B(t, x) ∂t =a ∂B(t, x) ∂x ,a11=a,B1(t, x) =B(t, x),(36) ËÏÂÂÚ ̄ÂÌËÂB=f(x+at), „ ‰ Âf∈C1, ‰Îˇ  ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄Ë‚ ͇ ̃ÂÒÚ‚ÂÙÛÌ͈ËËf ÒΉÛÂÚ‚ ̊·‡Ú ̧ÙÛÌÍˆË ̨,ÍÓÚÓ‡ˇÁ‡‰‡ÂÚ̇ ̃‡Î ̧ÌÓÂÛÒÎÓ‚ËÂ[4]: B(t, x) =φ(x+at),φ(x) =φ1(x).(37) Œ ̃‚ˉÌÓ(̇ÔËÏÂ,
    Exact
    [2, 6]
    Suffix
    ), ̃ÚÓ ‰Îˇ „ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏ ̊(32) ‚ˉ ÙÛÌ͈ËÈfi,i=1, n, ÍÓÚÓ ̊ÂÓÔ‰ÂΡ ̨ÚÒÓı‡Ìˇ ̨ ̆Û ̨ÒˇÙÓÏÛi-È ·Â„Û ̆ÂÈ‚ÓÎÌ ̊,ÛÒڇ̇‚ÎË‚‡ÂÚÒˇÔË ÔÓÏÓ ̆ËχÚˈ ̊ÔӉӷˡ: fi(0, x) = (S−1~φ(x))i,S−1AS=J ,J=diag(λ1, . . . , λn).(38) “Ó„‰‡ÔËÁ‡ÏÂÌ‚i-È ÍÓÏÔÓÌÂÌÚ‚ÂÍÚÓ‡~φ(x)‡„ÛÏÂÌÚ‡ẋ ‚ÓÎÌÛ(x+λit)ËÁ (9) ÔÓÎÛ ̃ËÏÒÎÂ‰Û ̨ ̆ÂÂ ̄ÂÌËÂ ̄ÂÌËÂÁ‡‰‡ ̃Ë Ó ̄ˉΡ ÒËÒÚÂÏ ̊(8), ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ÂÈ ‡ÒÔÓÒÚ‡ÌÂÌË ̨Ï

7
FeoktistovV.V.,MyakinnikO.O.Structureof Seriesfor SolvingSystemof 1st OrderEquations withPartialDerivatives.VestnikMGTUim. N.E.Baumana.Ser. Estestvennyenauki=Herald of the BaumanMSTU.Ser. Naturalscience, 2009,no. 4, pp. 3{22(in Russian).
Total in-text references: 8
  1. In-text reference with the coordinate start=3334
    Prefix
    ’ÓÓ ̄ÓËÁ‚ÂÒÚÌ ̊ÏËÔËÏÂ‡Ïˡ‚Ρ ̨ÚÒˇ:ÒËÒÚÂχÛ‡‚ÌÂÌËȇÍÛÒÚËÍË(n= 2) Ë Â  ̄ÂÌË‚ ‚ˉ ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ(xs−at)Ë(xs+at), ‡ÒÔÓÒÚ‡Ìˇ ̨ ̆ËıÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨ Á‚Û͇a‚‰ÓÎ ̧ÍÓÓ‰Ë̇ÚÌÓÈÓÒËOXs[1, 2, 6]; ‚ÓÎÌÓ‚ÓÂÛ‡‚ÌÂÌËÂ(m= 2) ‚ ÔÓÒÚ‡ÌÒÚ‚Âs ÔÂÂÏÂÌÌ ̊ı,ÍÓÚÓÓÂÔÓÓʉ‡ÂÚ‚ÓÎÌÛf(ω0t+ω·x)(Á‰ÂÒ ̧f| ÔÓËÁ‚ÓÎ ̧̇ˇ„·‰Í‡ˇ ÙÛÌÍˆËˇ),‡ÒÔÓÒÚ‡Ìˇ ̨ ̆‡ˇÒˇ‚ ̇Ô‡‚ÎÂÌËË‚ÂÍÚÓ‡(ω0, ω)[1, 3]. ¬ ‡·ÓÚ‡ı
    Exact
    [7, 8, 9]
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    ‰Îˇ ÒËÒÚÂÏ ̊ E ∂ ~B ∂t = ∑s k=1 Ak ∂ ~B ∂xk ,~B= (B1(t, x), . . . Bn(t, x)) Ú ,x= (x1, . . . , xs),(1) „ ‰ ÂAk,k=1, s, | ̃ËÒÎÓ‚ ̊ÂÍ‚‡‰‡ÚÌ ̊ÂχÚˈ ̊ÔÓˇ‰Í‡n,E| ‰ËÌË ̃̇ˇÏ‡Úˈ‡, ·ÂÁÓÚÌÓÒËÚÂÎ ̧ÌÓÍ ÓÔ‰ÂΡÂÏÓÏÛχÚˈ‡ÏË-ÍÓ ̋ÙÙˈËÂÌÚ‡ÏËÚËÔÛ‡ÒÒÏÓÚÂÌ ̊ ̄ÂÌˡ (Exk+Akt),k=1, s, Ë ÙÛÌ͈ËËÓÚ ÌËı.œÓ ‡Ì‡ÎÓ„ËËÒ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏËÚ‡ÍËÂ ̄ÂÌˡ ̇Á‚‡Ì ̊·Â„Û ̆ËÏË‚ÓÎ̇ÏËÒ Ï‡ÚË ̃Ì ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏË

  2. In-text reference with the coordinate start=4368
    Prefix
    ÕÂÒÏÓÚˇÌ‡ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌË ̃ËÒÎÂÌÌ ̊ıÏÂÚÓ‰Ó‚ ̄ÂÌˡÒËÒÚÂÏ ̊(1) ‚ ÔËÎÓÊÂÌËËÍ ÓÚ‰ÂÎ ̧Ì ̊ÏÙËÁË ̃ÂÒÍËÏÔÓˆÂÒÒ‡Ï(̇ÔËÏÂ,[6]),‡ÍÚۇΠ̧Ì ̊Ïˇ‚ΡÂÚÒˇÔÓÒÚÓÂÌˇ̇ÎËÚË ̃ÂÒÍÓ„ÓÏÂÚÓ‰‡ ̄ÂÌˡ ̋ÚÓÈÒËÒÚÂÏ ̊Ë Ò‚ˇÁ‡ÌÌÓÈÒ ÌÂÈ Á‡‰‡ ̃Ë Ó ̄ËÔË ÔÓËÁ‚ÓÎ ̧Ì ̊ın,sË ÔÓËÁ‚ÓÎ ̧Ì ̊ıχÚË ̃Ì ̊ıÍÓ ̋ÙÙˈËÂÌÚ‡ıAk,k=1, s. ƒÎˇ ÒËÒÚÂÏ ̊(1), ÍÓÚÓ‡ˇ ı‡‡ÍÚÂ̇‰Îˇ ‡ÁÎË ̃Ì ̊ıӷ·ÒÚÂÈχÚÂχÚË ̃ÂÒÍÓÈÙËÁËÍË[11],‚ ‡·ÓÚ‡ı
    Exact
    [7, 8, 9]
    Suffix
    ‡‚ÚÓ ‡ÏËÔÓÒÚÓÂ̇ÏÓ‰ÂÎ ̧‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ.›ÎÂÏÂÌÚ‡ÏËÏÓ‰ÂÎˡ‚Ρ ̨ÚÒˇ:n-ÏÂÌ ̊Â·Â„Û ̆Ë‚ÓÎÌ ̊,ÒÓ‰Âʇ ̆ËÂÍÓ ̋ÙÙˈËÂÌÚ ̊ÒËÒÚÂÏ ̊Ak,k=1, s, ÍÓÚÓ ̊Â‡ÒÔÓÒÚ‡Ìˇ ̨ÚÒˇÔÓ ÍÓÓ‰Ë̇ÚÌ ̊ÏÓÒˇÏË Ì‡ıÓ‰ˇÚÒˇ‚Ó ‚Á‡ËÏÓ‰ÂÈÒÚ‚ËË;ÓÔÂ‡ÚÓ‚ÓÎÌÓ‚Ó„Ó ‚Á‡ËÏÓ‰ÂÈÒڂˡ,ı‡‡ÍÚÂËÁÛ ̨ ̆ËÈÒˇ‡ÁÏÂÌÓÒÚ ̧ ̨s; ÓÚ‚Â ̃‡ ̨ ̆‡ˇÒËÒÚÂÏÂÓÔÂ‡ÚÓ̇ˇ ̋ÍÒÔÓÌÂÌÚ‡.

  3. In-text reference with the coordinate start=5257
    Prefix
    ՇȉÂÌÌÓÂ ̄ÂÌË‚ ˇ‚ÌÓςˉÂÒÓ‰ÂÊËÚχÚˈ ̊-ÍÓ ̋ÙÙˈËÂÌÚ ̊Ak,k=1, s, Á‡‰‡ ̨ ̆ËÂÒËÒÚÂÏÛ.ƒÎˇ ÔÓÎÛ ̃ÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄ˉÓÒÚ‡ÚÓ ̃ÌÓ‚ Ó· ̆ÂÏ ̄ÂÌËËÁ‡ÏÂÌËÚ ̧ ÌÂÓÔ‰ÂÎÂÌÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊̇ ‚ÂÍÚÓÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊‡ÁÎÓÊÂÌˡ‡Ì‡ÎËÚË ̃ÂÒÍÓÈ̇ ̃‡Î ̧ÌÓÈÙÛÌ͈ËË‚ ˇ‰ “ÂÈÎÓ‡[8]. ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚
    Exact
    [7, 8, 9]
    Suffix
    ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊[1, 5, 12].¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡[2, 5, 12], ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.

  4. In-text reference with the coordinate start=6921
    Prefix
    Bn(t, xk)) Ú ,k=1, s,(2) Ô‰ÒÚ‡‚ÎÂÌÌÓÂÍ ‡ Í ‡ÁÎÓÊÂÌËÂÔÓ ‚ÓÎÌÂ(Exk+Akt), ˇ‚ΡÂÚÒˇ ̃‡ÒÚÌ ̊Ï ̄ÂÌËÂÏÒËÒÚÂÏ ̊(1) Ë ÒÓÒÚ‡‚ÌÓÈ ̃‡ÒÚ ̧ ̨ ӷ ̆„Ó ̄ÂÌˡ,Ô‰ÒÚ‡‚ÎÂÌÌÓ„Ó‚ ‚ˉÂ‡ÁÎÓÊÂÌˡÔÓs ‚Á‡ËÏÓ‰ÂÈÒÚ‚Û ̨ ̆ËÏ‚ÓÎ̇Ï
    Exact
    [7, 9]
    Suffix
    . œÛÒÚ ̧(2) ÓÚ‚Â ̃‡ÂÚ‡ÒÔÓÒÚ‡ÌÂÌË ̨χΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË Ó‰ÌÓÏÂÌÓÏ Ú ̃ÂÌËË.ՇȉÂÏ ̄ÂÌË ̋ÚÓÈÒËÒÚÂÏ ̊Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏÚ‡‰ËˆËÓÌÌÓ„ÓÔÓ‰ıÓ‰‡Í Ó‰ÌÓÏÂÌÓÈ„ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏÂ[2, 6] Ë Ò‰ÒÚ‚‡ÏËÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ ·Â Á ËÒÔÓÎ ̧ÁÓ‚‡ÌˡÛÒÎӂˡ„ËÔÂ·ÓÎË ̃ÌÓÒÚË[7]. œÓ͇ÊÂÏÚÓʉÂÒÚ‚ÂÌÌÓÒÚ ̧ÔÓÎÛ ̃ÂÌÌ ̊ı ̄ÂÌËÈË ÔÂËÏÛ ̆ÂÒÚ‚‡‚ÓÎÌÓ‚Ó„ÓÔÓ‰ıÓ‰‡ÔË ËÒÒΉӂ‡ÌËËÒËÒÚÂÏ ̊(2).

  5. In-text reference with the coordinate start=7217
    Prefix
    Ë ÒÓÒÚ‡‚ÌÓÈ ̃‡ÒÚ ̧ ̨ ӷ ̆„Ó ̄ÂÌˡ,Ô‰ÒÚ‡‚ÎÂÌÌÓ„Ó‚ ‚ˉÂ‡ÁÎÓÊÂÌˡÔÓs ‚Á‡ËÏÓ‰ÂÈÒÚ‚Û ̨ ̆ËÏ‚ÓÎ̇Ï[7, 9]. œÛÒÚ ̧(2) ÓÚ‚Â ̃‡ÂÚ‡ÒÔÓÒÚ‡ÌÂÌË ̨χΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË Ó‰ÌÓÏÂÌÓÏ Ú ̃ÂÌËË.ՇȉÂÏ ̄ÂÌË ̋ÚÓÈÒËÒÚÂÏ ̊Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏÚ‡‰ËˆËÓÌÌÓ„ÓÔÓ‰ıÓ‰‡Í Ó‰ÌÓÏÂÌÓÈ„ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏÂ[2, 6] Ë Ò‰ÒÚ‚‡ÏËÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ ·Â Á ËÒÔÓÎ ̧ÁÓ‚‡ÌˡÛÒÎӂˡ„ËÔÂ·ÓÎË ̃ÌÓÒÚË
    Exact
    [7]
    Suffix
    . œÓ͇ÊÂÏÚÓʉÂÒÚ‚ÂÌÌÓÒÚ ̧ÔÓÎÛ ̃ÂÌÌ ̊ı ̄ÂÌËÈË ÔÂËÏÛ ̆ÂÒÚ‚‡‚ÓÎÌÓ‚Ó„ÓÔÓ‰ıÓ‰‡ÔË ËÒÒΉӂ‡ÌËËÒËÒÚÂÏ ̊(2). ÷ÂÎ ̧ ̨‡·ÓÚ ̊ˇ‚ΡÂÚÒˇ‡ÒÒÏÓÚÂÌËÂÔ‰·„‡ÂÏÓ„Ó‡‚ÚÓ‡ÏËÌӂӄӇ̇ÎËÚË ̃ÂÒÍÓ„Ó ÏÂÚÓ‰‡ ̄ÂÌˡÌÓχΠ̧ÌÓÈÒËÒÚÂÏ ̊ÎËÌÂÈÌ ̊ıÓ‰ÌÓÓ‰Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡Ì‡ ÔËÏÂÂÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ „‡ÁÂÔË Ó‰ÌÓÏÂÌÓÏÚ ̃ÂÌËËÒ ÔÓÒÎÂ‰Û ̨ ̆ËÏËÒÔÓÎ ̧ÁÓ‚‡Ì

  6. In-text reference with the coordinate start=18630
    Prefix
    Ex+J tËEx+A t,(25) ÍÓÚÓ ̊ÂÓÚ‚Â ̃‡ ̨Ú‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ÏÓÔÂ‡ÚÓ‡Ï ∂J≡E ∂ ∂t −J ∂ ∂x Ë∂A≡E ∂ ∂t −A ∂ ∂x .(26) –‡ÁÏÂÌÓÒÚ ̧ÙÛÌ͈ËÈ,̇ ÍÓÚÓ ̊‰ÂÈÒÚ‚Û ̨ÚÓÔÂ‡ÚÓ ̊,Á‡‰‡Ì‡‡ÁÏÂÌÓÒÚ ̧ ̨ÍÓ ̋ÙÙˈËÂÌÚ‡ÔË ÔÓËÁ‚Ó‰Ì ̊ı:ÓÔÂ‡ÚÓ(24) ‰ÂÈÒÚ‚ÛÂÚ̇ Ò͇ΡÌÛ ̨ÙÛÌÍˆË ̨Vi, ÓÔÂ‡ÚÓ ̊(26) ‰ÂÈÒÚ‚Û ̨Ú̇n-ÏÂÌ ̊‚ÂÍÚÓ-ÙÛÌ͈ËË~VË~BÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ. œÓ ‡Ì‡ÎÓ„ËËÒ ·Â„Û ̆ËÏË‚ÓÎ̇ÏË(23),‚ ̊‡ÊÂÌˡ(25) ‡ÒÒÏÓÚÂÌ ̊‚ ‡·ÓÚ‡ı
    Exact
    [7, 8, 9]
    Suffix
    Í ‡ Í n-ÏÂÌ ̊Â·Â„Û ̆Ë‚ÓÎÌ ̊(ËÎË·Â„Û ̆Ë‚ÓÎÌ ̊Ò Ï‡ÚË ̃Ì ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏË),ÓÚ‚Â ̃‡ ̨ ̆Ë ÒËÒÚÂχωËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡(8) Ë (10). ‚ÓÎ̇Ï(23) ‚ ÌÂÍÓÚÓ ̊ıÒÎÛ ̃‡ˇı·Û‰Âωӷ‡‚ÎˇÚ ̧ÓÔ‰ÂÎÂÌËÂ"Ò͇ΡÌ ̊Â".¬ ̋ÚÓÏÒÏ ̊ÒÎÂ Í ËÌ‚‡ˇÌڇϖËχ̇(S−1~B)i·Û‰ÂÏÔËÏÂÌˇÚ ̧ÓÔ‰ÂÎÂÌËÂ<Ò͇ΡÌ ̊È>,‡ ‚ ̊‡ÊÂÌËÂ~V, V~=S−1~B, ‡ÒÒχÚË‚‡Ú ̧Í ‡ Ín-ÏÂÌ ̊ÈËÌ‚‡ˇÌÚ. œÓ

  7. In-text reference with the coordinate start=20819
    Prefix
    βα= (β1α, β2α, β3α) Ú .(29) ≈ÒÎË‚ ̊‡ÊÂÌˡ(28) Ë (29) ˇ‚Ρ ̨ÚÒˇ ̄ÂÌˡÏËÛ‡‚ÌÂÌËÈ(11) Ë (10),ÚÓ ‚ ͇ ̃ÂÒÚ‚Â ̄ÂÌˡ ÒËÒÚÂÏ ̊(8) ÂÒÚÂÒÚ‚ÂÌÌÓ‡ÒÒÏÓÚÂÚ ̧‚ ̊‡ÊÂÌË B~≡~B(t, x) = ∑∞ α=0 (Ex+A t)α~γα,~γα= (γ1α, γ2α, γ3α) Ú ,(30) „ ‰ Â~γα| ̃ËÒÎÓ‚ÓÈ‚ÂÍÚÓ-ÍÓ ̋ÙÙˈËÂÌÚ.œÓ ÔÓÒÚÓÂÌË ̨͇ʉ ̊È ̃ÎÂÌˇ‰‡(28),(29) Ë (30), ‡ Ú‡ÍÊÂÎ ̨·‡ˇ ̃‡ÒÚË ̃̇ˇÒÛÏχ ̋ÚËıˇ‰Ó‚ˇ‚Ρ ̨ÚÒˇ ̄ÂÌËÂÏÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËıÛ‡‚ÌÂÌËÈ
    Exact
    [7]
    Suffix
    . ŒÒÓ·ÂÌÌÓÒÚ ̧ ̨ÔÓÒÚÓÂÌÌ ̊ı ̄ÂÌËÈ(28){(30)ˇ‚ΡÂÚÒˇÚÓ , ̃ÚÓ Ó‰ÌÓÁ̇ ̃ÌÓÁ‡‰‡ ̨ ̆ËÈ Û‡‚ÌÂÌËÂËÎË ÒËÒÚÂÏÛÍÓ ̋ÙÙˈËÂÌÚÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ÂÈ‡ÁÏÂÌÓÒÚËÔË ÔÓËÁ‚Ó‰ÌÓÈÔÓ ÔÓÒÚ‡ÌÒÚ‚ÂÌÌÓÈÔÂÂÏÂÌÌÓÈx‚ıÓ‰ËÚ‚  ̄ÂÌË‚ ͇ ̃ÂÒÚ‚ÂÍÓ ̋ÙÙˈËÂÌÚ‡ÔË ÌÂÁ‡‚ËÒËÏÓÈ ÔÂÂÏÂÌÌÓÈt. œÓ ÓÚÌÓ ̄ÂÌË ̨Í ÒËÒÚÂχÏ(8) Ë (20) ̋ÚÓ, ‚ ̃‡ÒÚÌÓÒÚË,ÓÁ̇ ̃‡ÂÚ, ̃ÚÓ ̃ÚÓ  ̄ÂÌËÂÓ‰ÌÓÏÂÌÓÈÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚

  8. In-text reference with the coordinate start=31182
    Prefix
    ¬ ÔÓ‰ÓÎÊÂÌËÂËÒÒΉӂ‡ÌËÈ‚ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓÈÏÓ‰ÂÎˉΡ ÒËÒÚÂÏ ̊(1) ÔËs >1 Ô‰·„‡ÂÚÒˇ‡ÒÒÏÓÚÂÚ ̧Á‡‰‡ ̃ÛÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË ‰‚ÛÏÂÌÓÏ(ÔÎÓÒÍÓÔ‡‡ÎÎÂÎ ̧ÌÓÏ)‰‚ËÊÂÌËËË ‰‚ËÊÂÌËË‚ ÔÓÒÚ‡ÌÒÚ‚Â.›ÚÓ ÔÓÚ·ÛÂÚ ÔËÏÂÌÂÌˡÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆„ÓÓÔÂ‡ÚÓ‡‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ
    Exact
    [7, 8]
    Suffix
    . ¬Á‡ËÏÓ‰ÂÈÒÚ‚Ë ÓÒÛ ̆ÂÒڂΡÂÚÒˇÏÂʉۂÓÎ̇ÏË(Ò͇ΡÌ ̊ÏËËÎËn-ÏÂÌ ̊ÏË),3< n≤5, ÔÓÒÚÓÂÌÌ ̊ÏË ‰Îˇ ͇ʉÓÈχÚˈ ̊-ÍÓ ̋ÙÙˈËÂÌÚ‡Ak,s=1,3, ÒËÒÚÂÏ ̊(1).œÓÎÛ ̃ÂÌÌÓÂ ̄ÂÌËÂ(30), Ò‚ˇÁ‡ÌÌÓÂÒ ‚ÓÎ̇ÏË,ÍÓÚÓ ̊Â‡ÒÔÓÒÚ‡Ìˇ ̨ÚÒˇ‚‰ÓÎ ̧‚ ̊·‡ÌÌÓÈÓÒËOXk,k=1,3, ˇ‚ΡÂÚÒˇ ̃‡ÒÚÌ ̊Ï ̄ÂÌËÂÏÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË ‰‚ËÊÂÌËË‚ ÔÓÒÚ‡ÌÒÚ‚ÂË ÒÓÒÚ‡‚ÌÓÈ ̃‡ÒÚ ̧ ̨ ӷ ̆„Ó ̄ÂÌˡ.

8
FeoktistovV.V.,MyakinnikO.O.Operatorof waveinteractionand normalformof systemof linearequationsin partialderivativesof the 1st order.Voronezhskayavesennyayamatematicheskayashkola\Sovremennyemetodyteoriikraevykhzadach.Pontryaginskiechteniya| 21": mater. [VoronezhSpringSchoolin Mathematics\Modernmethodsof the theoryof boundary valueproblems.Pontryaginreading| 21":proc.].Voronezh,VSUPubl.,2010,pp. 232{233 (in Russian).
Total in-text references: 7
  1. In-text reference with the coordinate start=3334
    Prefix
    ’ÓÓ ̄ÓËÁ‚ÂÒÚÌ ̊ÏËÔËÏÂ‡Ïˡ‚Ρ ̨ÚÒˇ:ÒËÒÚÂχÛ‡‚ÌÂÌËȇÍÛÒÚËÍË(n= 2) Ë Â  ̄ÂÌË‚ ‚ˉ ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ(xs−at)Ë(xs+at), ‡ÒÔÓÒÚ‡Ìˇ ̨ ̆ËıÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨ Á‚Û͇a‚‰ÓÎ ̧ÍÓÓ‰Ë̇ÚÌÓÈÓÒËOXs[1, 2, 6]; ‚ÓÎÌÓ‚ÓÂÛ‡‚ÌÂÌËÂ(m= 2) ‚ ÔÓÒÚ‡ÌÒÚ‚Âs ÔÂÂÏÂÌÌ ̊ı,ÍÓÚÓÓÂÔÓÓʉ‡ÂÚ‚ÓÎÌÛf(ω0t+ω·x)(Á‰ÂÒ ̧f| ÔÓËÁ‚ÓÎ ̧̇ˇ„·‰Í‡ˇ ÙÛÌÍˆËˇ),‡ÒÔÓÒÚ‡Ìˇ ̨ ̆‡ˇÒˇ‚ ̇Ô‡‚ÎÂÌËË‚ÂÍÚÓ‡(ω0, ω)[1, 3]. ¬ ‡·ÓÚ‡ı
    Exact
    [7, 8, 9]
    Suffix
    ‰Îˇ ÒËÒÚÂÏ ̊ E ∂ ~B ∂t = ∑s k=1 Ak ∂ ~B ∂xk ,~B= (B1(t, x), . . . Bn(t, x)) Ú ,x= (x1, . . . , xs),(1) „ ‰ ÂAk,k=1, s, | ̃ËÒÎÓ‚ ̊ÂÍ‚‡‰‡ÚÌ ̊ÂχÚˈ ̊ÔÓˇ‰Í‡n,E| ‰ËÌË ̃̇ˇÏ‡Úˈ‡, ·ÂÁÓÚÌÓÒËÚÂÎ ̧ÌÓÍ ÓÔ‰ÂΡÂÏÓÏÛχÚˈ‡ÏË-ÍÓ ̋ÙÙˈËÂÌÚ‡ÏËÚËÔÛ‡ÒÒÏÓÚÂÌ ̊ ̄ÂÌˡ (Exk+Akt),k=1, s, Ë ÙÛÌ͈ËËÓÚ ÌËı.œÓ ‡Ì‡ÎÓ„ËËÒ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏËÚ‡ÍËÂ ̄ÂÌˡ ̇Á‚‡Ì ̊·Â„Û ̆ËÏË‚ÓÎ̇ÏËÒ Ï‡ÚË ̃Ì ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏË

  2. In-text reference with the coordinate start=4368
    Prefix
    ÕÂÒÏÓÚˇÌ‡ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌË ̃ËÒÎÂÌÌ ̊ıÏÂÚÓ‰Ó‚ ̄ÂÌˡÒËÒÚÂÏ ̊(1) ‚ ÔËÎÓÊÂÌËËÍ ÓÚ‰ÂÎ ̧Ì ̊ÏÙËÁË ̃ÂÒÍËÏÔÓˆÂÒÒ‡Ï(̇ÔËÏÂ,[6]),‡ÍÚۇΠ̧Ì ̊Ïˇ‚ΡÂÚÒˇÔÓÒÚÓÂÌˇ̇ÎËÚË ̃ÂÒÍÓ„ÓÏÂÚÓ‰‡ ̄ÂÌˡ ̋ÚÓÈÒËÒÚÂÏ ̊Ë Ò‚ˇÁ‡ÌÌÓÈÒ ÌÂÈ Á‡‰‡ ̃Ë Ó ̄ËÔË ÔÓËÁ‚ÓÎ ̧Ì ̊ın,sË ÔÓËÁ‚ÓÎ ̧Ì ̊ıχÚË ̃Ì ̊ıÍÓ ̋ÙÙˈËÂÌÚ‡ıAk,k=1, s. ƒÎˇ ÒËÒÚÂÏ ̊(1), ÍÓÚÓ‡ˇ ı‡‡ÍÚÂ̇‰Îˇ ‡ÁÎË ̃Ì ̊ıӷ·ÒÚÂÈχÚÂχÚË ̃ÂÒÍÓÈÙËÁËÍË[11],‚ ‡·ÓÚ‡ı
    Exact
    [7, 8, 9]
    Suffix
    ‡‚ÚÓ ‡ÏËÔÓÒÚÓÂ̇ÏÓ‰ÂÎ ̧‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ.›ÎÂÏÂÌÚ‡ÏËÏÓ‰ÂÎˡ‚Ρ ̨ÚÒˇ:n-ÏÂÌ ̊Â·Â„Û ̆Ë‚ÓÎÌ ̊,ÒÓ‰Âʇ ̆ËÂÍÓ ̋ÙÙˈËÂÌÚ ̊ÒËÒÚÂÏ ̊Ak,k=1, s, ÍÓÚÓ ̊Â‡ÒÔÓÒÚ‡Ìˇ ̨ÚÒˇÔÓ ÍÓÓ‰Ë̇ÚÌ ̊ÏÓÒˇÏË Ì‡ıÓ‰ˇÚÒˇ‚Ó ‚Á‡ËÏÓ‰ÂÈÒÚ‚ËË;ÓÔÂ‡ÚÓ‚ÓÎÌÓ‚Ó„Ó ‚Á‡ËÏÓ‰ÂÈÒڂˡ,ı‡‡ÍÚÂËÁÛ ̨ ̆ËÈÒˇ‡ÁÏÂÌÓÒÚ ̧ ̨s; ÓÚ‚Â ̃‡ ̨ ̆‡ˇÒËÒÚÂÏÂÓÔÂ‡ÚÓ̇ˇ ̋ÍÒÔÓÌÂÌÚ‡.

  3. In-text reference with the coordinate start=5193
    Prefix
    ՇȉÂÌÌÓÂ ̄ÂÌË‚ ˇ‚ÌÓςˉÂÒÓ‰ÂÊËÚχÚˈ ̊-ÍÓ ̋ÙÙˈËÂÌÚ ̊Ak,k=1, s, Á‡‰‡ ̨ ̆ËÂÒËÒÚÂÏÛ.ƒÎˇ ÔÓÎÛ ̃ÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄ˉÓÒÚ‡ÚÓ ̃ÌÓ‚ Ó· ̆ÂÏ ̄ÂÌËËÁ‡ÏÂÌËÚ ̧ ÌÂÓÔ‰ÂÎÂÌÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊̇ ‚ÂÍÚÓÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊‡ÁÎÓÊÂÌˡ‡Ì‡ÎËÚË ̃ÂÒÍÓÈ̇ ̃‡Î ̧ÌÓÈÙÛÌ͈ËË‚ ˇ‰ “ÂÈÎÓ‡
    Exact
    [8]
    Suffix
    . ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚ [7, 8, 9] ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊[1, 5, 12].¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡[2, 5, 12], ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.

  4. In-text reference with the coordinate start=5257
    Prefix
    ՇȉÂÌÌÓÂ ̄ÂÌË‚ ˇ‚ÌÓςˉÂÒÓ‰ÂÊËÚχÚˈ ̊-ÍÓ ̋ÙÙˈËÂÌÚ ̊Ak,k=1, s, Á‡‰‡ ̨ ̆ËÂÒËÒÚÂÏÛ.ƒÎˇ ÔÓÎÛ ̃ÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄ˉÓÒÚ‡ÚÓ ̃ÌÓ‚ Ó· ̆ÂÏ ̄ÂÌËËÁ‡ÏÂÌËÚ ̧ ÌÂÓÔ‰ÂÎÂÌÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊̇ ‚ÂÍÚÓÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊‡ÁÎÓÊÂÌˡ‡Ì‡ÎËÚË ̃ÂÒÍÓÈ̇ ̃‡Î ̧ÌÓÈÙÛÌ͈ËË‚ ˇ‰ “ÂÈÎÓ‡[8]. ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚
    Exact
    [7, 8, 9]
    Suffix
    ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊[1, 5, 12].¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡[2, 5, 12], ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.

  5. In-text reference with the coordinate start=18630
    Prefix
    Ex+J tËEx+A t,(25) ÍÓÚÓ ̊ÂÓÚ‚Â ̃‡ ̨Ú‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ÏÓÔÂ‡ÚÓ‡Ï ∂J≡E ∂ ∂t −J ∂ ∂x Ë∂A≡E ∂ ∂t −A ∂ ∂x .(26) –‡ÁÏÂÌÓÒÚ ̧ÙÛÌ͈ËÈ,̇ ÍÓÚÓ ̊‰ÂÈÒÚ‚Û ̨ÚÓÔÂ‡ÚÓ ̊,Á‡‰‡Ì‡‡ÁÏÂÌÓÒÚ ̧ ̨ÍÓ ̋ÙÙˈËÂÌÚ‡ÔË ÔÓËÁ‚Ó‰Ì ̊ı:ÓÔÂ‡ÚÓ(24) ‰ÂÈÒÚ‚ÛÂÚ̇ Ò͇ΡÌÛ ̨ÙÛÌÍˆË ̨Vi, ÓÔÂ‡ÚÓ ̊(26) ‰ÂÈÒÚ‚Û ̨Ú̇n-ÏÂÌ ̊‚ÂÍÚÓ-ÙÛÌ͈ËË~VË~BÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ. œÓ ‡Ì‡ÎÓ„ËËÒ ·Â„Û ̆ËÏË‚ÓÎ̇ÏË(23),‚ ̊‡ÊÂÌˡ(25) ‡ÒÒÏÓÚÂÌ ̊‚ ‡·ÓÚ‡ı
    Exact
    [7, 8, 9]
    Suffix
    Í ‡ Í n-ÏÂÌ ̊Â·Â„Û ̆Ë‚ÓÎÌ ̊(ËÎË·Â„Û ̆Ë‚ÓÎÌ ̊Ò Ï‡ÚË ̃Ì ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏË),ÓÚ‚Â ̃‡ ̨ ̆Ë ÒËÒÚÂχωËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡(8) Ë (10). ‚ÓÎ̇Ï(23) ‚ ÌÂÍÓÚÓ ̊ıÒÎÛ ̃‡ˇı·Û‰Âωӷ‡‚ÎˇÚ ̧ÓÔ‰ÂÎÂÌËÂ"Ò͇ΡÌ ̊Â".¬ ̋ÚÓÏÒÏ ̊ÒÎÂ Í ËÌ‚‡ˇÌڇϖËχ̇(S−1~B)i·Û‰ÂÏÔËÏÂÌˇÚ ̧ÓÔ‰ÂÎÂÌËÂ<Ò͇ΡÌ ̊È>,‡ ‚ ̊‡ÊÂÌËÂ~V, V~=S−1~B, ‡ÒÒχÚË‚‡Ú ̧Í ‡ Ín-ÏÂÌ ̊ÈËÌ‚‡ˇÌÚ. œÓ

  6. In-text reference with the coordinate start=26406
    Prefix
    ¬ ÚÓ Ê ‚ÂÏˇ,‡ÒÒÏÓÚ‚ÔËt= 0Ó· ̆ÂÂ ̄ÂÌËÂÒËÒÚÂÏ ̊(32) ‚ ‚ˉÂ(30),ÔËıÓ‰ËÏ Í ‚ ̊‚Ó‰Û, ̃ÚÓ ‰Îˇ ÔÓÎÛ ̃ÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄Ë(32){(33)‚ ͇ ̃ÂÒÚ‚ÂÍÓ ̋ÙÙˈËÂÌÚÓ‚~γα ÔË(Ex+At)αÒΉÛÂÚ‚ ̊·‡Ú ̧ÍÓ ̋ÙÙˈËÂÌÚ ̊ˇ‰‡(33)ÔËxα. “Ó„‰‡ ̄ÂÌËÂÁ‡‰‡ ̃Ë Ó ̄ËÔËÌËχÂڂˉ
    Exact
    [8]
    Suffix
    B~(t, x) = ∑∞ α=0 (Ex+At)α 1 α! ∂α ∂xα φ~(x) ∣∣∣ ∣∣∣ x=0 .(43) – ̄ÂÌËÂ(43),ÍÓÚÓÓÂÁ‡ÔËÒ‡ÌÓ·Â Á ‰ÓÔÓÎÌËÚÂÎ ̧Ì ̊ı‚ ̊ ̃ËÒÎÂÌËÈ,ÒÓ‰ÂÊËÚχÚË ̃Ì ̊ÈÍ Ó ̋ÙÙˈËÂÌÚÒËÒÚÂÏ ̊‚ ˇ‚ÌÓςˉÂ,‡ ̇ ̃‡Î ̧Ì ̊‰‡ÌÌ ̊Â| ‚ ‚ˉÂÍÓ ̋ÙÙˈËÂÌÚÓ‚‡ÁÎÓÊÂÌˡ ‡Ì‡ÎËÚË ̃ÂÒÍÓÈ̇ ̃‡Î ̧ÌÓÈÙÛÌ͈ËË‚ ˇ‰ “ÂÈÎÓ‡.– ˇ ‰ (43) Ó„‡ÌËÁÓ‚‡ÌÔÓ ‚ÓÎÌÓ‚ÓÏÛÔË̈ËÔÛ,ÓÚÎË ̃ÌÓÏÛÓÚ ÔË̈ËÔ‡Ó„‡ÌËÁ‡ˆËËˇ‰‡(35).

  7. In-text reference with the coordinate start=31182
    Prefix
    ¬ ÔÓ‰ÓÎÊÂÌËÂËÒÒΉӂ‡ÌËÈ‚ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓÈÏÓ‰ÂÎˉΡ ÒËÒÚÂÏ ̊(1) ÔËs >1 Ô‰·„‡ÂÚÒˇ‡ÒÒÏÓÚÂÚ ̧Á‡‰‡ ̃ÛÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË ‰‚ÛÏÂÌÓÏ(ÔÎÓÒÍÓÔ‡‡ÎÎÂÎ ̧ÌÓÏ)‰‚ËÊÂÌËËË ‰‚ËÊÂÌËË‚ ÔÓÒÚ‡ÌÒÚ‚Â.›ÚÓ ÔÓÚ·ÛÂÚ ÔËÏÂÌÂÌˡÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆„ÓÓÔÂ‡ÚÓ‡‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ
    Exact
    [7, 8]
    Suffix
    . ¬Á‡ËÏÓ‰ÂÈÒÚ‚Ë ÓÒÛ ̆ÂÒڂΡÂÚÒˇÏÂʉۂÓÎ̇ÏË(Ò͇ΡÌ ̊ÏËËÎËn-ÏÂÌ ̊ÏË),3< n≤5, ÔÓÒÚÓÂÌÌ ̊ÏË ‰Îˇ ͇ʉÓÈχÚˈ ̊-ÍÓ ̋ÙÙˈËÂÌÚ‡Ak,s=1,3, ÒËÒÚÂÏ ̊(1).œÓÎÛ ̃ÂÌÌÓÂ ̄ÂÌËÂ(30), Ò‚ˇÁ‡ÌÌÓÂÒ ‚ÓÎ̇ÏË,ÍÓÚÓ ̊Â‡ÒÔÓÒÚ‡Ìˇ ̨ÚÒˇ‚‰ÓÎ ̧‚ ̊·‡ÌÌÓÈÓÒËOXk,k=1,3, ˇ‚ΡÂÚÒˇ ̃‡ÒÚÌ ̊Ï ̄ÂÌËÂÏÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË ‰‚ËÊÂÌËË‚ ÔÓÒÚ‡ÌÒÚ‚ÂË ÒÓÒÚ‡‚ÌÓÈ ̃‡ÒÚ ̧ ̨ ӷ ̆„Ó ̄ÂÌˡ.

9
FeoktistovV.V.,MyakinnikO.O.Travelingwaveswithmatrixcoefficients.Voronezhskaya zimnyayamatematicheskayashkola\Sovremennyemetodyteoriifunktsiyi smezhnyeproblem":mater. [VoronezhWinterSchoolin Mathematics\Modernmethodsof functiontheory and relatedproblems"].Voronezh,VSUPubl.,2013,pp. 255{256(in Russian).
Total in-text references: 5
  1. In-text reference with the coordinate start=3334
    Prefix
    ’ÓÓ ̄ÓËÁ‚ÂÒÚÌ ̊ÏËÔËÏÂ‡Ïˡ‚Ρ ̨ÚÒˇ:ÒËÒÚÂχÛ‡‚ÌÂÌËȇÍÛÒÚËÍË(n= 2) Ë Â  ̄ÂÌË‚ ‚ˉ ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ(xs−at)Ë(xs+at), ‡ÒÔÓÒÚ‡Ìˇ ̨ ̆ËıÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨ Á‚Û͇a‚‰ÓÎ ̧ÍÓÓ‰Ë̇ÚÌÓÈÓÒËOXs[1, 2, 6]; ‚ÓÎÌÓ‚ÓÂÛ‡‚ÌÂÌËÂ(m= 2) ‚ ÔÓÒÚ‡ÌÒÚ‚Âs ÔÂÂÏÂÌÌ ̊ı,ÍÓÚÓÓÂÔÓÓʉ‡ÂÚ‚ÓÎÌÛf(ω0t+ω·x)(Á‰ÂÒ ̧f| ÔÓËÁ‚ÓÎ ̧̇ˇ„·‰Í‡ˇ ÙÛÌÍˆËˇ),‡ÒÔÓÒÚ‡Ìˇ ̨ ̆‡ˇÒˇ‚ ̇Ô‡‚ÎÂÌËË‚ÂÍÚÓ‡(ω0, ω)[1, 3]. ¬ ‡·ÓÚ‡ı
    Exact
    [7, 8, 9]
    Suffix
    ‰Îˇ ÒËÒÚÂÏ ̊ E ∂ ~B ∂t = ∑s k=1 Ak ∂ ~B ∂xk ,~B= (B1(t, x), . . . Bn(t, x)) Ú ,x= (x1, . . . , xs),(1) „ ‰ ÂAk,k=1, s, | ̃ËÒÎÓ‚ ̊ÂÍ‚‡‰‡ÚÌ ̊ÂχÚˈ ̊ÔÓˇ‰Í‡n,E| ‰ËÌË ̃̇ˇÏ‡Úˈ‡, ·ÂÁÓÚÌÓÒËÚÂÎ ̧ÌÓÍ ÓÔ‰ÂΡÂÏÓÏÛχÚˈ‡ÏË-ÍÓ ̋ÙÙˈËÂÌÚ‡ÏËÚËÔÛ‡ÒÒÏÓÚÂÌ ̊ ̄ÂÌˡ (Exk+Akt),k=1, s, Ë ÙÛÌ͈ËËÓÚ ÌËı.œÓ ‡Ì‡ÎÓ„ËËÒ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏËÚ‡ÍËÂ ̄ÂÌˡ ̇Á‚‡Ì ̊·Â„Û ̆ËÏË‚ÓÎ̇ÏËÒ Ï‡ÚË ̃Ì ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏË

  2. In-text reference with the coordinate start=4368
    Prefix
    ÕÂÒÏÓÚˇÌ‡ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌË ̃ËÒÎÂÌÌ ̊ıÏÂÚÓ‰Ó‚ ̄ÂÌˡÒËÒÚÂÏ ̊(1) ‚ ÔËÎÓÊÂÌËËÍ ÓÚ‰ÂÎ ̧Ì ̊ÏÙËÁË ̃ÂÒÍËÏÔÓˆÂÒÒ‡Ï(̇ÔËÏÂ,[6]),‡ÍÚۇΠ̧Ì ̊Ïˇ‚ΡÂÚÒˇÔÓÒÚÓÂÌˇ̇ÎËÚË ̃ÂÒÍÓ„ÓÏÂÚÓ‰‡ ̄ÂÌˡ ̋ÚÓÈÒËÒÚÂÏ ̊Ë Ò‚ˇÁ‡ÌÌÓÈÒ ÌÂÈ Á‡‰‡ ̃Ë Ó ̄ËÔË ÔÓËÁ‚ÓÎ ̧Ì ̊ın,sË ÔÓËÁ‚ÓÎ ̧Ì ̊ıχÚË ̃Ì ̊ıÍÓ ̋ÙÙˈËÂÌÚ‡ıAk,k=1, s. ƒÎˇ ÒËÒÚÂÏ ̊(1), ÍÓÚÓ‡ˇ ı‡‡ÍÚÂ̇‰Îˇ ‡ÁÎË ̃Ì ̊ıӷ·ÒÚÂÈχÚÂχÚË ̃ÂÒÍÓÈÙËÁËÍË[11],‚ ‡·ÓÚ‡ı
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    ‡‚ÚÓ ‡ÏËÔÓÒÚÓÂ̇ÏÓ‰ÂÎ ̧‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ.›ÎÂÏÂÌÚ‡ÏËÏÓ‰ÂÎˡ‚Ρ ̨ÚÒˇ:n-ÏÂÌ ̊Â·Â„Û ̆Ë‚ÓÎÌ ̊,ÒÓ‰Âʇ ̆ËÂÍÓ ̋ÙÙˈËÂÌÚ ̊ÒËÒÚÂÏ ̊Ak,k=1, s, ÍÓÚÓ ̊Â‡ÒÔÓÒÚ‡Ìˇ ̨ÚÒˇÔÓ ÍÓÓ‰Ë̇ÚÌ ̊ÏÓÒˇÏË Ì‡ıÓ‰ˇÚÒˇ‚Ó ‚Á‡ËÏÓ‰ÂÈÒÚ‚ËË;ÓÔÂ‡ÚÓ‚ÓÎÌÓ‚Ó„Ó ‚Á‡ËÏÓ‰ÂÈÒڂˡ,ı‡‡ÍÚÂËÁÛ ̨ ̆ËÈÒˇ‡ÁÏÂÌÓÒÚ ̧ ̨s; ÓÚ‚Â ̃‡ ̨ ̆‡ˇÒËÒÚÂÏÂÓÔÂ‡ÚÓ̇ˇ ̋ÍÒÔÓÌÂÌÚ‡.

  3. In-text reference with the coordinate start=5257
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    ՇȉÂÌÌÓÂ ̄ÂÌË‚ ˇ‚ÌÓςˉÂÒÓ‰ÂÊËÚχÚˈ ̊-ÍÓ ̋ÙÙˈËÂÌÚ ̊Ak,k=1, s, Á‡‰‡ ̨ ̆ËÂÒËÒÚÂÏÛ.ƒÎˇ ÔÓÎÛ ̃ÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄ˉÓÒÚ‡ÚÓ ̃ÌÓ‚ Ó· ̆ÂÏ ̄ÂÌËËÁ‡ÏÂÌËÚ ̧ ÌÂÓÔ‰ÂÎÂÌÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊̇ ‚ÂÍÚÓÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊‡ÁÎÓÊÂÌˡ‡Ì‡ÎËÚË ̃ÂÒÍÓÈ̇ ̃‡Î ̧ÌÓÈÙÛÌ͈ËË‚ ˇ‰ “ÂÈÎÓ‡[8]. ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚
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    ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊[1, 5, 12].¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡[2, 5, 12], ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.

  4. In-text reference with the coordinate start=6921
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    Bn(t, xk)) Ú ,k=1, s,(2) Ô‰ÒÚ‡‚ÎÂÌÌÓÂÍ ‡ Í ‡ÁÎÓÊÂÌËÂÔÓ ‚ÓÎÌÂ(Exk+Akt), ˇ‚ΡÂÚÒˇ ̃‡ÒÚÌ ̊Ï ̄ÂÌËÂÏÒËÒÚÂÏ ̊(1) Ë ÒÓÒÚ‡‚ÌÓÈ ̃‡ÒÚ ̧ ̨ ӷ ̆„Ó ̄ÂÌˡ,Ô‰ÒÚ‡‚ÎÂÌÌÓ„Ó‚ ‚ˉÂ‡ÁÎÓÊÂÌˡÔÓs ‚Á‡ËÏÓ‰ÂÈÒÚ‚Û ̨ ̆ËÏ‚ÓÎ̇Ï
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    . œÛÒÚ ̧(2) ÓÚ‚Â ̃‡ÂÚ‡ÒÔÓÒÚ‡ÌÂÌË ̨χΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ÔË Ó‰ÌÓÏÂÌÓÏ Ú ̃ÂÌËË.ՇȉÂÏ ̄ÂÌË ̋ÚÓÈÒËÒÚÂÏ ̊Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏÚ‡‰ËˆËÓÌÌÓ„ÓÔÓ‰ıÓ‰‡Í Ó‰ÌÓÏÂÌÓÈ„ËÔÂ·ÓÎË ̃ÂÒÍÓÈÒËÒÚÂÏÂ[2, 6] Ë Ò‰ÒÚ‚‡ÏËÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ ·Â Á ËÒÔÓÎ ̧ÁÓ‚‡ÌˡÛÒÎӂˡ„ËÔÂ·ÓÎË ̃ÌÓÒÚË[7]. œÓ͇ÊÂÏÚÓʉÂÒÚ‚ÂÌÌÓÒÚ ̧ÔÓÎÛ ̃ÂÌÌ ̊ı ̄ÂÌËÈË ÔÂËÏÛ ̆ÂÒÚ‚‡‚ÓÎÌÓ‚Ó„ÓÔÓ‰ıÓ‰‡ÔË ËÒÒΉӂ‡ÌËËÒËÒÚÂÏ ̊(2).

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    Ex+J tËEx+A t,(25) ÍÓÚÓ ̊ÂÓÚ‚Â ̃‡ ̨Ú‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ÏÓÔÂ‡ÚÓ‡Ï ∂J≡E ∂ ∂t −J ∂ ∂x Ë∂A≡E ∂ ∂t −A ∂ ∂x .(26) –‡ÁÏÂÌÓÒÚ ̧ÙÛÌ͈ËÈ,̇ ÍÓÚÓ ̊‰ÂÈÒÚ‚Û ̨ÚÓÔÂ‡ÚÓ ̊,Á‡‰‡Ì‡‡ÁÏÂÌÓÒÚ ̧ ̨ÍÓ ̋ÙÙˈËÂÌÚ‡ÔË ÔÓËÁ‚Ó‰Ì ̊ı:ÓÔÂ‡ÚÓ(24) ‰ÂÈÒÚ‚ÛÂÚ̇ Ò͇ΡÌÛ ̨ÙÛÌÍˆË ̨Vi, ÓÔÂ‡ÚÓ ̊(26) ‰ÂÈÒÚ‚Û ̨Ú̇n-ÏÂÌ ̊‚ÂÍÚÓ-ÙÛÌ͈ËË~VË~BÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ. œÓ ‡Ì‡ÎÓ„ËËÒ ·Â„Û ̆ËÏË‚ÓÎ̇ÏË(23),‚ ̊‡ÊÂÌˡ(25) ‡ÒÒÏÓÚÂÌ ̊‚ ‡·ÓÚ‡ı
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    Í ‡ Í n-ÏÂÌ ̊Â·Â„Û ̆Ë‚ÓÎÌ ̊(ËÎË·Â„Û ̆Ë‚ÓÎÌ ̊Ò Ï‡ÚË ̃Ì ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏË),ÓÚ‚Â ̃‡ ̨ ̆Ë ÒËÒÚÂχωËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„Ó ÔÓˇ‰Í‡(8) Ë (10). ‚ÓÎ̇Ï(23) ‚ ÌÂÍÓÚÓ ̊ıÒÎÛ ̃‡ˇı·Û‰Âωӷ‡‚ÎˇÚ ̧ÓÔ‰ÂÎÂÌËÂ"Ò͇ΡÌ ̊Â".¬ ̋ÚÓÏÒÏ ̊ÒÎÂ Í ËÌ‚‡ˇÌڇϖËχ̇(S−1~B)i·Û‰ÂÏÔËÏÂÌˇÚ ̧ÓÔ‰ÂÎÂÌËÂ<Ò͇ΡÌ ̊È>,‡ ‚ ̊‡ÊÂÌËÂ~V, V~=S−1~B, ‡ÒÒχÚË‚‡Ú ̧Í ‡ Ín-ÏÂÌ ̊ÈËÌ‚‡ˇÌÚ. œÓ

10
FeoktistovV.V.,MyakinnikO.O.N-dimensionaltravelingwavesrepresentationin terms scalarwaves.Voronezhskayavesennyayamatematicheskayashkola\Sovremennyemetody teoriikraevykhzadach.Pontryaginskiechteniya| 25":mater. [VoronezhSpringSchool in Mathematics\Modernmethodsof the theoryof boundaryvalueproblems.Pontryagin reading| 25":proc.].Voronezh,VSUPubl.,2014,pp. 178{180(in Russian).
Total in-text references: 2
  1. In-text reference with the coordinate start=3970
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    ‡ÒÒÏÓÚÂÌ ̊ ̄ÂÌˡ (Exk+Akt),k=1, s, Ë ÙÛÌ͈ËËÓÚ ÌËı.œÓ ‡Ì‡ÎÓ„ËËÒ ̋ÎÂÏÂÌÚ‡Ì ̊ÏË‚ÓÎ̇ÏËÚ‡ÍËÂ ̄ÂÌˡ ̇Á‚‡Ì ̊·Â„Û ̆ËÏË‚ÓÎ̇ÏËÒ Ï‡ÚË ̃Ì ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏËËÎËn-ÏÂÌ ̊ÏË·Â„Û ̆ËÏË ‚ÓÎ̇ÏË.œËs= 1‚ ÒÎÛ ̃‡Â„ËÔÂ·ÓÎË ̃ÌÓÒÚËÒËÒÚÂÏ ̊(χÚˈ‡AsÔӉӷ̇‰Ë‡„Ó̇Π̧ÌÓÈ ÊÓ‰‡ÌÓ‚ÓÈχÚˈÂ)‚‚‰ÂÌ̇ˇn-ÏÂ̇ˇ·Â„Û ̆‡ˇ‚ÓÎ̇ËÏÂÂÚÔÓÎÌÛ ̨ËÌÚÂÔÂÚ‡ˆË ̨ ̃ÂÂÁn ̋ÎÂÏÂÌÚ‡Ì ̊ı‚ÓÎÌ,ÍÓÚÓ ̊Â‡ÒÒÏÓÚÂÌ ̊Í ‡ Í Ò͇ΡÌ ̊Â
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    . ÕÂÒÏÓÚˇÌ‡ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌË ̃ËÒÎÂÌÌ ̊ıÏÂÚÓ‰Ó‚ ̄ÂÌˡÒËÒÚÂÏ ̊(1) ‚ ÔËÎÓÊÂÌËËÍ ÓÚ‰ÂÎ ̧Ì ̊ÏÙËÁË ̃ÂÒÍËÏÔÓˆÂÒÒ‡Ï(̇ÔËÏÂ,[6]),‡ÍÚۇΠ̧Ì ̊Ïˇ‚ΡÂÚÒˇÔÓÒÚÓÂÌˇ̇ÎËÚË ̃ÂÒÍÓ„ÓÏÂÚÓ‰‡ ̄ÂÌˡ ̋ÚÓÈÒËÒÚÂÏ ̊Ë Ò‚ˇÁ‡ÌÌÓÈÒ ÌÂÈ Á‡‰‡ ̃Ë Ó ̄ËÔË ÔÓËÁ‚ÓÎ ̧Ì ̊ın,sË ÔÓËÁ‚ÓÎ ̧Ì ̊ıχÚË ̃Ì ̊ıÍÓ ̋ÙÙˈËÂÌÚ‡ıAk,k=1, s. ƒÎˇ ÒËÒÚÂÏ ̊(1), ÍÓÚÓ‡ˇ ı‡‡ÍÚÂ̇‰Îˇ ‡ÁÎË ̃Ì ̊ıӷ·ÒÚÂÈχÚÂχÚË ̃ÂÒÍÓÈÙËÁËÍË[11],‚ ‡·ÓÚ‡ı[7,

  2. In-text reference with the coordinate start=19576
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    ‡ÚˈÂJ Ò Ï‡ÚˈÂÈÔӉӷˡS, ÚÓn-ÏÂ̇ˇ·Â„Û ̆‡ˇ‚ÓÎ̇(Ex+At), „‰ÂE| ‰ËÌË ̃̇ˇÏ‡Úˈ‡ ÔÓˇ‰Í‡n, ÔӉӷ̇·Â„Û ̆ÂÈ‚ÓÎÌÂ(Ex+Jt)Ò Ï‡ÚˈÂÈÔӉӷˡS−1: J=S−1AS=⇒(Ex+At) =S(Ex+Jt)S−1.(27) ≈ÒÎËχÚˈ‡Jˇ‚ΡÂÚÒˇ‰Ë‡„Ó̇Π̧ÌÓÈÊÓ‰‡ÌÓ‚ÓÈÙÓÏÓÈχÚˈ ̊A, ÚÓ ËÁ ÒÓÓÚÌÓ ̄ÂÌˡ (27) ÒΉÛÂÚÔ‰ÒÚ‡‚ÎÂÌËÂn-ÏÂÌÓÈ·Â„Û ̆ÂÈ‚ÓÎÌ ̊ ̃ÂÂÁÒ͇ΡÌ ̊‚ÓÎÌ ̊(x+λit), ÔÓÒÚÓÂÌÌÌ ̊ÂÔË ÔÓÏÓ ̆ËÒÓ·ÒÚ‚ÂÌÌ ̊ıÁ̇ ̃ÂÌËÈχÚˈ ̊A,i=1, n,
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    . ŒÚÏÂÚËÏ, ̃ÚÓ ÏÓ‰ÂÎ ̧Ó‰ÌÓÏÂÌÓ„Ó‰‚ËÊÂÌˡˉ‡Π̧ÌÓ„Ó„‡Á‡ ˇ‚ΡÂÚÒˇÔËÏÂÓÏÏÓ‰ÂÎË, ‚ ÍÓÚÓÓÈËÏÂÂÚÏÂÒÚÓÔÓÎ̇ˇËÌÚÂÔÂÚ‡ˆËˇ·Â„Û ̆ÂÈ‚ÓÎÌ ̊Ò Ï‡ÚË ̃Ì ̊ÏÍÓ ̋ÙÙˈËÂÌÚÓÏ ÔÓÒ‰ÒÚ‚ÓÏÒ͇ΡÌ ̊ı‚ÓÎÌ.ƒ‡ÌÌ ̊ÈÙ‡ÍÚÓÚ‡Ê‡ÂÚ„ËÔÂ·ÓÎË ̃ÌÓÒÚ ̧ÒËÒÚÂÏ ̊Û‡‚ÌÂÌËÈ (8) (ÒËÒÚÂÏ ̊(7)). 6.

11
GerochR. PartialDifferentialEquationsof Physics.In:GeneralRelativity:Proceedingsof the 46thScottishUniversitiesSummerSchoolin Physics, Aberdeen,July1995.Edinburgh, SUSSPPubl.,1996,pp. 19{60.
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  1. In-text reference with the coordinate start=4354
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    ÕÂÒÏÓÚˇÌ‡ ÒÛ ̆ÂÒÚ‚Ó‚‡ÌË ̃ËÒÎÂÌÌ ̊ıÏÂÚÓ‰Ó‚ ̄ÂÌˡÒËÒÚÂÏ ̊(1) ‚ ÔËÎÓÊÂÌËËÍ ÓÚ‰ÂÎ ̧Ì ̊ÏÙËÁË ̃ÂÒÍËÏÔÓˆÂÒÒ‡Ï(̇ÔËÏÂ,[6]),‡ÍÚۇΠ̧Ì ̊Ïˇ‚ΡÂÚÒˇÔÓÒÚÓÂÌˇ̇ÎËÚË ̃ÂÒÍÓ„ÓÏÂÚÓ‰‡ ̄ÂÌˡ ̋ÚÓÈÒËÒÚÂÏ ̊Ë Ò‚ˇÁ‡ÌÌÓÈÒ ÌÂÈ Á‡‰‡ ̃Ë Ó ̄ËÔË ÔÓËÁ‚ÓÎ ̧Ì ̊ın,sË ÔÓËÁ‚ÓÎ ̧Ì ̊ıχÚË ̃Ì ̊ıÍÓ ̋ÙÙˈËÂÌÚ‡ıAk,k=1, s. ƒÎˇ ÒËÒÚÂÏ ̊(1), ÍÓÚÓ‡ˇ ı‡‡ÍÚÂ̇‰Îˇ ‡ÁÎË ̃Ì ̊ıӷ·ÒÚÂÈχÚÂχÚË ̃ÂÒÍÓÈÙËÁËÍË
    Exact
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    ,‚ ‡·ÓÚ‡ı[7, 8, 9] ‡‚ÚÓ ‡ÏËÔÓÒÚÓÂ̇ÏÓ‰ÂÎ ̧‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒڂˡ.›ÎÂÏÂÌÚ‡ÏËÏÓ‰ÂÎˡ‚Ρ ̨ÚÒˇ:n-ÏÂÌ ̊Â·Â„Û ̆Ë‚ÓÎÌ ̊,ÒÓ‰Âʇ ̆ËÂÍÓ ̋ÙÙˈËÂÌÚ ̊ÒËÒÚÂÏ ̊Ak,k=1, s, ÍÓÚÓ ̊Â‡ÒÔÓÒÚ‡Ìˇ ̨ÚÒˇÔÓ ÍÓÓ‰Ë̇ÚÌ ̊ÏÓÒˇÏË Ì‡ıÓ‰ˇÚÒˇ‚Ó ‚Á‡ËÏÓ‰ÂÈÒÚ‚ËË;ÓÔÂ‡ÚÓ‚ÓÎÌÓ‚Ó„Ó ‚Á‡ËÏÓ‰ÂÈÒڂˡ,ı‡‡ÍÚÂËÁÛ ̨ ̆ËÈÒˇ‡ÁÏÂÌÓÒÚ ̧ ̨s; ÓÚ‚Â ̃‡ ̨ ̆‡ˇÒËÒÚÂÏÂÓÔÂ‡ÚÓ̇ˇ ̋ÍÒÔÓÌÂÌÚ‡.

12
KovenyaV.M.,YanenkoN.N.Metodrasshchepleniyav zadachakhgazovoydinamiki[Splitting Methodsfor Gas DynamicsProblems].Novosibirsk,NaukaPubl.,1981.304 p. (in Russian).
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  1. In-text reference with the coordinate start=5391
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    Á‡‰‡ ̨ ̆ËÂÒËÒÚÂÏÛ.ƒÎˇ ÔÓÎÛ ̃ÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃Ë Ó ̄ˉÓÒÚ‡ÚÓ ̃ÌÓ‚ Ó· ̆ÂÏ ̄ÂÌËËÁ‡ÏÂÌËÚ ̧ ÌÂÓÔ‰ÂÎÂÌÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊̇ ‚ÂÍÚÓÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊‡ÁÎÓÊÂÌˡ‡Ì‡ÎËÚË ̃ÂÒÍÓÈ̇ ̃‡Î ̧ÌÓÈÙÛÌ͈ËË‚ ˇ‰ “ÂÈÎÓ‡[8]. ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚ [7, 8, 9] ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊
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    [1, 5, 12]
    Suffix
    .¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡[2, 5, 12], ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.¬ ͇ ̃ÂÒÚ‚ÂÁ‡Ï ͇̊ ̨ ̆ËıÛ‡‚ÌÂÌËÈ[12] ‰Îˇ Û‡ÌÂÌËÈÒÓı‡ÌÂÌˡÒÍÓÓÒÚËË ËÏÔÛÎ ̧Ò‡‡ÒÒÏÓÚËÏÛ‡‚ÌÂÌËÂÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË, ‚ ̊‡Á˂ ̃ÂÂÁ ̋ÌڇΠ̧ÔË ̨,Ë Û‡‚ÌÂÌËÂÒÓÒÚÓˇÌˡ,Ò‚ˇÁ ̊‚‡ ̨ ̆ ̋ÌڇΠ̧ÔË ̨Ò ÔÎÓÚÌÓÒÚ ̧ ̨Ë ‰‡‚ÎÂÌËÂÏ„‡Á‡.

  2. In-text reference with the coordinate start=5519
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    ¬ ͇ ̃ÂÒÚ‚ÂÙËÁË ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚ [7, 8, 9] ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊[1, 5, 12].¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡
    Exact
    [2, 5, 12]
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    , ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.¬ ͇ ̃ÂÒÚ‚ÂÁ‡Ï ͇̊ ̨ ̆ËıÛ‡‚ÌÂÌËÈ[12] ‰Îˇ Û‡ÌÂÌËÈÒÓı‡ÌÂÌˡÒÍÓÓÒÚËË ËÏÔÛÎ ̧Ò‡‡ÒÒÏÓÚËÏÛ‡‚ÌÂÌËÂÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË, ‚ ̊‡Á˂ ̃ÂÂÁ ̋ÌڇΠ̧ÔË ̨,Ë Û‡‚ÌÂÌËÂÒÓÒÚÓˇÌˡ,Ò‚ˇÁ ̊‚‡ ̨ ̆ ̋ÌڇΠ̧ÔË ̨Ò ÔÎÓÚÌÓÒÚ ̧ ̨Ë ‰‡‚ÎÂÌËÂÏ„‡Á‡.

  3. In-text reference with the coordinate start=5608
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    ̃ÂÒÍÓ„ÓÔÓˆÂÒÒ‡‰Îˇ ËÌÚÂÔÂÚ‡ˆËË‚‚‰ÂÌÌ ̊ı‚ [7, 8, 9] ÔÓÌˇÚËÈË ‚ ̊ÔÓÎÌÂÌÌ ̊ıÔÓÒÚÓÂÌËÈ‚ ̊·ÂÂÏÚ ̃ÂÌËÂˉ‡Π̧ÌÓ„Ó„‡Á‡,ÎËÌÂÈÌ ̊ÂË ÌÂÎËÌÂÈÌ ̊‚ÓÎÌÓ‚ ̊ ÔÓˆÂÒÒ ̊‚ ÍÓÚÓÓÏËÁÛ ̃ÂÌ ̊[1, 5, 12].¬ ̊·‡ÌÌÓÏÛÔÓˆÂÒÒÛÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚÍ‚‡ÁËÎËÌÂÈ̇ˇ ÒËÒÚÂχ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈ‚ ̃‡ÒÚÌ ̊ıÔÓËÁ‚Ó‰Ì ̊ı1-„ÓÔÓˇ‰Í‡[2, 5, 12], ÍÓÚÓ ̊ÂÔ‰ÒÚ‡‚Ρ ̨ÚÒÓ·ÓÈÁ‡ÍÓÌ ̊ÒÓı‡ÌÂÌˡ.¬ ͇ ̃ÂÒÚ‚ÂÁ‡Ï ͇̊ ̨ ̆ËıÛ‡‚ÌÂÌËÈ
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    [12]
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    ‰Îˇ Û‡ÌÂÌËÈÒÓı‡ÌÂÌˡÒÍÓÓÒÚËË ËÏÔÛÎ ̧Ò‡‡ÒÒÏÓÚËÏÛ‡‚ÌÂÌËÂÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË, ‚ ̊‡Á˂ ̃ÂÂÁ ̋ÌڇΠ̧ÔË ̨,Ë Û‡‚ÌÂÌËÂÒÓÒÚÓˇÌˡ,Ò‚ˇÁ ̊‚‡ ̨ ̆ ̋ÌڇΠ̧ÔË ̨Ò ÔÎÓÚÌÓÒÚ ̧ ̨Ë ‰‡‚ÎÂÌËÂÏ„‡Á‡.“‡ÍËÏÓ·‡ÁÓÏ,ÂÁÛÎ ̧ÚËÛ ̨ ̆‡ˇÍ‚‡ÁËÎËÌÂÈ̇ˇÒËÒÚÂχÁ‡ÔË҇̇ ÓÚÌÓÒËÚÂÎ ̧ÌÓn-ÏÂÌÓ„Ó‚ÂÍÚÓ‡,3≤n≤5, ‰ËÌÓÓ·‡ÁÌÓÒÓ‰Âʇ ̆„ӂ ͇ ̃ÂÒÚ‚ÂÒ‚ÓËı ÍÓÏÔÓÌÂÌÚÍÓÏÔÓÌÂÌÚ ̊ÒÍÓÓÒÚË„‡Á‡u(t, x),v(t, x),w(t, x), Â„Ó ÔÎÓÚÌÓÒÚËρ(t,

  4. In-text reference with the coordinate start=8354
    Prefix
    1 ρ ̃ ∂ ̃P ∂ ̃x , ∂ ̃ρ ∂ ̃t + ∂( ̃ρ ̃u) ∂ ̃x = 0, ∂I ̃ ∂ ̃t + ̃u ∂I ̃ ∂ ̃x = 1 ρ ̃ ( ∂ ̃P ∂ ̃t + ̃u ∂ ̃P ∂ ̃x ) , x ̃∈R1,(3) „ ‰  ̃u( ̃t, ̃x), ̃ρ( ̃t, ̃x), ̃P( ̃t, ̃x), ̃I( ̃t ̃x)| ÒÍÓÓÒÚ ̧‰‚ËÊÂÌˡ„‡Á‡,Â„Ó ÔÎÓÚÌÓÒÚ ̧,‰‡‚ÎÂÌËÂË ̋ÌڇΠ̧ÔˡÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ.◊ÚÓ· ̊Ò‰ÂÎ‡Ú ̧ÒËÒÚÂÏÛÚÂı‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ıÛ‡‚ÌÂÌËÈÓÚ ÌÓÒËÚÂÎ ̧ÌÓ ̃ÂÚ ̊ÂıÌÂËÁ‚ÂÒÚ̇ıÙÛÌ͈ËÈÓÔ‰ÂÎÂÌÌÓÈ,‰ÓÔÓÎÌËÏ Û‡‚ÌÂÌËÂÏÒÓÒÚÓˇÌˡ
    Exact
    [12]
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    P ̃=γ−1 γ ρI, ̃ ̃ ̃I=cP ̃T,γ=cP cV ,(4) ̃ ÍÓÚÓÓÂÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ·‡ÓÚÓÔÌÓÏÛ‰‚ËÊÂÌË ̨ˉ‡Π̧ÌÓ„Ó„‡Á‡ Ë Ò‚ˇÁ ̊‚‡ÂÚ‰‡‚ÎÂÌË„‡Á‡ Ò Â„Ó ÔÎÓÚÌÓÒÚ ̧ ̨Ë ̋ÌڇΠ̧ÔËÂÈ( ̃T( ̃t, ̃x)| ÚÂÏÔÂ‡ÚÛ‡„‡Á‡,cP| Û‰ÂÎ ̧̇ˇÚÂÔÎÓÂÏÍÓÒÚ ̧„‡Á‡ ÔË ÔÓÒÚÓˇÌÌÓω‡‚ÎÂÌËË,cV| Û‰ÂÎ ̧̇ˇÚÂÔÎÓÂÏÍÓÒÚ ̧„‡Á‡ ÔË ÔÓÒÚÓˇÌÌÓÏÓ· ̇ÂÏÂ).

13
SobolevS.L.Uravneniyamatematicheskoyfiziki[Equationsof MathematicalPhysics]. Moscow,NaukaPubl.,1992.432 p. (in Russian).
Total in-text references: 3
  1. In-text reference with the coordinate start=6495
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    ◊ÚÓ· ̊ËÁ Í‚‡ÁËÎËÌÂÈÌÓ„ÓÔÓˆÂÒÒ‡‚ ̊‰ÂÎËÚ ̧ÎËÌÂÈÌ ̊‚ÓÎÌ ̊,ÔÓÓʉ‡ÂÏ ̊ÂÒËÒÚÂÏÓÈ(1), ÔËÏÂÌËÏÏÂÚӉχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ(̇ÔËÏÂ,[2]).¬ÓÁÌË͇ ̨ ̆Û ̨ÔË ̋ÚÓÏ Á‡‰‡ ̃ÛÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ,ÍÓÚÓ‡ˇÔ‰ÒÚ‡‚ΡÂÚÒ‡ÏÓÒÚÓˇÚÂÎ ̧Ì ̊ÈËÌÚÂÂÒ
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    , ‡ÒÒÏÓÚËÏÍ ‡ Í ÚÂÒÚÓ‚Û ̨‰Îˇ ÏÓ‰ÂÎË‚ÓÎÌÓ‚Ó„Ó‚Á‡ËÏÓ‰ÂÈÒÚ‚ËˇÌ‡ ÔËÏÂ ‰‚ËÊÂÌˡˉ‡Π̧ÌÓ„Ó„‡Á‡ ÔË Á‡‰‡ÌÌ ̊ınËs. œÓ ÒÚÛÍÚÛÂÔÓÒÚÓÂÌÌÓÈÏÓ‰ÂÎËÓ· ̆ÂÂ ̄ÂÌËÂÒËÒÚÂÏ ̊ E ∂ ~B ∂t =Ak ∂ ~B ∂xk ,~B= (B1(t, xk), .

  2. In-text reference with the coordinate start=7929
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    Ó ÔÓˇ‰Í‡Ì‡ ÔËÏÂÂÁ‡‰‡ ̃ËÓ ‡ÒÔÓÒÚ‡ÌÂÌËËχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ „‡ÁÂÔË Ó‰ÌÓÏÂÌÓÏÚ ̃ÂÌËËÒ ÔÓÒÎÂ‰Û ̨ ̆ËÏËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏÔÓÎÛ ̃ÂÌÌÓ„Ó ̄ÂÌˡ‰Îˇ ÔÓÒÚÓÂÌˡ ̄ÂÌˡÁ‡‰‡ ̃ËÔË Ú ̃ÂÌËË„‡Á‡ ‚ ÔÓÒÚ‡ÌÒÚ‚Â. 2. ”‡‚ÌÂÌˡ‡ÒÔÓÒÚ‡ÌÂÌËˇÏ‡Î ̊ı‚ÓÁÏÛ ̆ÂÌËÈ‚ ˉ‡Π̧ÌÓÏ„‡Á ‚‡ÁËÎËÌÂÈÌ ̊ÂÛ‡‚ÌÂÌˡӉÌÓÏÂÌÓ„Ó‰‚ËÊÂÌˡˉ‡Π̧ÌÓ„Ó„‡Á‡ ÔË ÓÚÒÛÚÒÚ‚ËËÏ‡Ò ÒÓ‚ ̊ıÒËÎ ‚ ‰Â͇ÚÓ‚ÓÈÒËÒÚÂÏÂÍÓÓ‰Ë̇ÚËÏ ̨ڂˉ
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    :          ∂u ̃ ∂ ̃t + ̃u ∂u ̃ ∂ ̃x =− 1 ρ ̃ ∂ ̃P ∂ ̃x , ∂ ̃ρ ∂ ̃t + ∂( ̃ρ ̃u) ∂ ̃x = 0, ∂I ̃ ∂ ̃t + ̃u ∂I ̃ ∂ ̃x = 1 ρ ̃ ( ∂ ̃P ∂ ̃t + ̃u ∂ ̃P ∂ ̃x ) , x ̃∈R1,(3) „ ‰  ̃u( ̃t, ̃x), ̃ρ( ̃t, ̃x), ̃P( ̃t, ̃x), ̃I( ̃t ̃x)| ÒÍÓÓÒÚ ̧‰‚ËÊÂÌˡ„‡Á‡,Â„Ó ÔÎÓÚÌÓÒÚ ̧,‰‡‚ÎÂÌËÂË ̋ÌڇΠ̧ÔˡÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ.

  3. In-text reference with the coordinate start=15587
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    ÓÒÚ‡ÌÂÌËÂχΠ̊ı‚ÓÁÏÛ ̆ÂÌËÈÔËÓÚÌÓÒËÚÂÎ ̧ÌÓω‚ËÊÂÌËË ¬ ‡Á‰ÂÎÂ3 ÔÓÎÛ ̃ÂÌÓ ̄ÂÌËÂÁ‡‰‡ ̃ˉΡ ‡·ÒÓÎ ̨ÚÌÓ„Ó‰‚ËÊÂÌˡ„‡Á‡.œÓÒÍÓÎ ̧ÍÛ‡·ÒÓÎ ̨ÚÌÓÂÔÓ ÓÚÌÓ ̄ÂÌË ̨Í ÌÂÔÓ‰‚ËÊÌÓÈÒËÒÚÂÏÂÍÓÓ‰Ë̇ډ‚ËÊÂÌË„‡Á‡ÒÓÒÚÓËÚËÁ ÔÂÂÌÓÒÌÓ„ÓË ÓÚÌÓÒËÚÂÎ ̧ÌÓ„Ó‰‚ËÊÂÌˡ,ÒΉÛÂÚ‡ÒÒÏÓÚÂÚ ̧·Â„Û ̆Ë‚ÓÎÌ ̊,‚ ̊Á‚‡ÌÌ ̊ χΠ̊ÏË‚ÓÁÏÛ ̆ÂÌˡÏËÔË ÓÚÌÓÒËÚÂÎ ̧ÌÓω‚ËÊÂÌËË„‡Á‡. œÛÒÚ ̧‚ÓÁÏÛ ̆ÂÌˡÔË‚ÌÓÒˇÚÒˇ‚ „‡Á Ò ̋ÎÂÏÂÌÚ‡ÏË(u−U0, ρ0, I0)
    Exact
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    .›ÚÓÚÒÎÛ ̃‡È ÒÓÓÚ‚ÂÚÒÚ‚ÛÂÚ‡ÒÔÓÒÚ‡ÌÂÌË ̨‚ÓÁÏÛ ̆ÂÌËÈ‚ ÒËÒÚÂÏÂÍÓÓ‰Ë̇Ú,ÍÓÚÓ‡ˇÒ‚ˇÁ‡Ì‡Ò ̃‡ÒÚˈÂÈ „‡Á‡,‰‚ËÊÛ ̆ÂÈÒˇÒÓ ÒÍÓÓÒÚ ̧ ̨U0. “Ó„‰‡‚ ÛÒÎÓ‚ËË(6) ÒÍÓÓÒÚ ̧‚ÓÁÏÛ ̆ÂÌÌÓ„Ó„‡Á‡ ÒÓ‚Ô‡‰‡ÂÚÒ ‚ÓÁÏÛ ̆ÂÌËÂÏÂ„Ó ÒÍÓÓÒÚË: u(t, x) =u∗(t, x).(19) ¬ ̊ÔÓÎÌË‚ÎË̇ËÁ‡ˆË ̨ÒËÒÚÂÏ ̊(5) ÓÍÓÎÓÒÓÒÚÓˇÌˡ(0, ρ0, I0), ÔÓ ‡Ì‡ÎÓ„ËËÒ ÒËÒÚÂÏÓÈ(8) ÔÓÎÛ ̃ËÏÒËÒÚÂÏÛ E ∂ ~B ∂t =A∗ ∂ ~B ∂x ,A∗=     0− 1 γM2 − 1 γM2 −1

14
KaptsovO.V.Invariantsof characteristicsof somesystemsof partialdifferentialequations.Sibirskiymatematicheskiyzhurnal, 2004,vol. 45, no. 3, pp. 577{591.(English translation:SiberianMathematicalJournal, 2004,vol. 45, is. 3, pp. 475{487.DOI: 10.1023/B:SIMJ.0000028612.51060.32).
Total in-text references: 1
  1. In-text reference with the coordinate start=13918
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    ̊–Ëχ̇ÒËÒÚÂÏ ̊(8): (γ−1) γ ρ(t, x)− 1 γ I(t, x) =f1(x−t),(14) 1 2 u(t, x)− 1 2γM ρ(t, x)− 1 2γM I(t, x) =f2 ( x+ (1−M) M t ) ,(15) 1 2 u(t, x) + 1 2γM ρ(t, x) + 1 2γM I(t, x) =f3 ( x− (1 +M) M t ) .(16) ‡Ê‰ ̊ÈËÁ ÚÂıËÌ‚‡ˇÌÚÓ‚(14){(16)‡ÒÔÓÒÚ‡ÌˇÂÚÒˇ·Â Á ËÒ͇ÊÂÌˡÙÓÏ ̊Ò ı‡‡ÍÚÂËÒÚË ̃ÂÒÍÓÈÒÍÓÓÒÚ ̧ ̨λi,i=1,3. ¬ Ò‚ˇÁËÒ ÔÓÒÚ‡ÌÓ‚ÍÓÈÁ‡‰‡ ̃Ë(3){(4)Ë Ì‡ıÓʉÂÌËÂÏËÌ‚‡ˇÌÚÓ‚(14){(16)ÓÚÏÂÚËÏ ‡·ÓÚÛ
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    [14]
    Suffix
    ,‚ ÍÓÚÓÓȉÓ͇Á‡ÌÓ, ̃ÚÓ ÒËÒÚÂχӉÌÓÏÂÌ ̊ıÛ‡‚ÌÂÌËÈ„‡ÁÓ‚ÓȉË̇ÏËÍË ËÏÂÂÚÚË Ë‚‡ˇÌÚ‡–Ëχ̇,ÂÒÎË‚ ͇ ̃ÂÒÚ‚ÂÁ‡Ï ͇̊ ̨ ̆Ëı‚ ̊·‡ÌÓÛ‡‚ÌÂÌËÂÒÓı‡ÌÂÌˡ ̋ÌÂ„ËË,‚ ̊‡ÊÂÌÌÓ ̃ÂÂÁ ̋ÌÚÓÔË ̨s, Ë Û‡‚ÌÂÌËÂÒÓÒÚÓˇÌˡ‚ ‚ˉ V=G(P) +A(s),V= 1 ρ ,G′(P) =− 1 g2(P) ,c= g(P) ρ ,(17) „ ‰ Âa| ÒÍÓÓÒÚ ̧Á‚Û͇;gËA| ÔÓËÁ‚ÓÎ ̧Ì ̊ÂÙÛÌ͈Ëˉ‡‚ÎÂÌËˇË ̋ÌÚÓÔËË.