We propose an algorithm for the construction of asymptotic expansions for solutions of the Dirichlet problem for the heat equation with impulses.Запропоновано алгоритм побудови асимптотичних розвинень для розв'язків задачі Діріхле для рівняння теплопровідності з імпульсною дією

Matarazzo, G.

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 517.9G. Matarazzo (Salerno Univ., Italy)ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEMFOR THE HEAT EQUATION WITH IMPULSESASYMPTOTYÇNI ROZV’QZKY ZADAÇI DIRIXLE DLQRIVNQNNQ TEPLOPROVIDNOSTI Z IMPUL|SNOG DI{GWe propose an algorithm for the construction of asymptotic expansions for solutions of the Dirichletproblem for the heat equation with impulses.Zaproponovano alhorytm pobudovy asymptotyçnyx rozvynen\ dlq rozv’qzkiv zadaçi Dirixle dlqrivnqnnq teploprovidnosti z impul\snog di[g.1.  Introduction.  The theory of impulsive differential equations [1] is an importantpart of the modern theory of differential equations which has many applications inpractice.  Till now, a lot of different problems connected with impulsive differentialequations are studied.  It has been found that the solutions of differential equations withimpulses can demonstrate very complicated behaviour [1 – 5].  In the present paper, westudy the problem of the construction of an asymptotic solution of the Dirichletproblem for the heat equation with impulses.2.  Formulation of the problem.  Let us consider a differential heat equation withsmall parameter  ε ∈ ( 0; ε0 )  of the form∂∂ =∂∂+ ( )utuxf x t uε ε222 , , , ,      ( x, t ) ∈ Ω  =  ( 0; 1 ) × ( 0; + ∞ ), (1)under initial conditionsu ( x, 0, ε )  =  ϕ ( x, ε ), (2)boundary conditionsu ( 0, t, ε )  =  0,      u ( 1, t, ε )  =  0,    t ∈ [ 0, + ∞ ), (3)and impulsive conditions at a fixed moment of time∆ u ( x, t, ε ) | t = ti  =  u ( x, ti + 0, ε ) – u ( x, ti – 0, ε )  =  Ii  ( x, ε ),      i ∈ N. (4)We suppose the fulfillment of the following assumptions:P1 .  Functions  f (  x, t, u, ε ),  ϕ  ( x, ε ),  Ii  ( x, ε ),  i ∈  N,  are infinitely differentiablewith respect to their variables and can be represented as regular asymptotic expansionswith respect to a small parameter  ε ∈ ( 0; ε0 ).  In particular,ϕ ( x, ε )  =  ε ϕk kkx( )=∞∑0,      Ii  ( x, ε )  =  εk ikkxT ( )=∞∑0.P2 .  The nonperturbed problem∂∂ = ( )utf x t u, , , 0 ,      u ( x, 0, 0 )  =  ϕ ( x, 0 ),∆ u ( x, t, 0 ) | t = ti  =  u ( x, ti + 0, 0 ) – u ( x, ti – 0, 0 )  =  Ii  ( x, 0 ),      i ∈ N,for any  x ∈ ( 0; 1 )  ( x  is considered as a parameter) possesses a solution  u = u x t0( ),defined for any  ( x, t ) ∈  [ 0; 1 ] ×  [  0; + T )  (here, the case  T = + ∞  is not excluded)which is infinitely differentiable for  t ≠ ti ,  i ∈ N.©  G. MATARAZZO, 2006ISSN  1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3 427428 G. MATARAZZOP3 .  The condition  ′ ( )( )f x t u x tu , , , ,0 0  ≠ 0  takes place.P4 .  The agreement condition  ϕ0 ( 0, ε ) = 0  takes place.3.  Algorithm of asymptotic expansion.  We seek a solution of problem (1) – (4)in the form of the asymptotic seriesu ( x, t, ε )  =  u x t( ), , ε  + Q u ( ξ, t, ε ) + Q* u ( ξ* , t, ε ), (5)whereu x t u x t u x t u x t( ) = ( ) + ( ) + ( ) + …, , , , ,ε ε ε0 1 2 2is the regular part of asymtotics andQ u ( ξ, t, ε )  =  Q0 u ( ξ, t ) + ε Q1 u ( ξ, t ) + ε2 Q2 u ( ξ, t ) + … ,Q* u ( ξ* , t, ε )  =  Q*0 u ( ξ* , t ) + ε Q*1 u ( ξ* , t ) + ε2 Q*2 u ( ξ* , t ) + …are singular parts of asymptotics.Here, we denote  ξ = x / ε,  ξ* = ( 1 – x ) / ε.  The functions  Qk u ( ξ, t ),  k ≥ 0,  aresupposed to be defined for  ξ ∈  [ 0; ε– 1 ],  t ≥ 0,  and, at the same time, the functionsQ*k u ( ξ* , t ),  k ≥ 0,  are supposed to be defined for  ξ* ∈ [ 0; ε– 1 ],  t ≥ 0.By the usual way, we obtain relations for defining terms of asymptotics (5).  Theterms of the regular part of asymptotics (5) may be found as solutions of the followingproblems:dudt f x t u00 0= ( ), , , ,u x0 0( ),   =  ϕ0 ( x ),      x ∈ [ 0; 1 ],∆u x t t ti0( ) =,   =  Ii  ( x, 0 ),      i ∈ N,dudt f x t u u f x t u u uku k k k= ( ) + ( … )−, , , , , , , ,0 0 1 10 ,u xk ( ), 0   =  ϕk ( x ),      x ∈ [ 0; 1 ],∆u x tk t ti( ) =,   =  Tik ( x ),      i ∈ N,where functions  f x t u u uk k( … )−, , , , ,0 1 1 ,  k ≥ 1,  are recurrently defined by values ofu x t0( ), , u x t1( ), , … , u x tk − ( )1 , .  Here,  x ∈ [ 0; 1 ]  is considered as a parameter.After definition of the regular part  u x t( ), , ε   of asymptotics (5), we can find thesingular part  Q  u ( ξ, t, ε  )  of asymptotic (5) which is defined as solutions of thefollowing boundary-value problems:∂∂ =∂∂+ ( ) + − ( )( ) ( )Q utQ u f t u t Q u f t u t0202 0 0 00 0 0 0 0 0ξ , , , , , , , , ,Q0 u ( ξ, 0 )  =  0,      ξ ∈ [ 0; ∞ ],(6)Q0 u ( 0, t )  =  − ( )u t0 0, ,      t ∈ [ 0; ∞ ],limξ→∞ Q0 u ( ξ, t )  =  0    for any    t ∈ [ 0; ∞ ];ISSN  1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM … 429∂∂ =∂∂+ ( ) + ( )( )Q utQ u f t u t Q u Q f tk k u k k22 00 0 0ξ ξ, , , , , ,Qk u ( 0, ξ )  =  0,      ξ ∈ [ 0; ∞ ],(7)Qk u ( 0, t )  =  − ( )u t0 0, ,      t ∈ [ 0; ∞ ],limξ→∞ Qk u ( ξ, t )  =  0    for any    t ∈ [ 0; ∞ ],where the functions  Qk f ( ξ, t  ),  k  ≥ 1,  are recurrently defined by the standardprocedure by values of the regular part  u x t0( ), , u x t1( ), , … , u x tk − ( )1 , ,  x  = ε ξ,  ofasymptotics (5).We can now proceed to the calculation of the singular part  Q* u ( ξ, t, ε  )  ofasymptotics (5) which is defined as solutions of the following boundary-valueproblems:∂∂ =∂∂+ ( ) + + − ( ) +( ) ( )Q utQ u f t u t Q u Q u f t u t Q u* ***, , , , , , , ,0202 0 0 0 0 01 1 0 1 1 0ξ ,Q*0 u ( ξ* , 0 )  =  0,      ξ* ∈ [ 0; ∞ ],(8)Q*0 u ( 0, t )  =  − ( )u t0 1, ,      t ∈ [ 0; ∞ ],lim*ξ →∞ Q*0 u ( ξ* , t )  =  0    for any    t ∈ [ 0; ∞ ];∂∂ =∂∂+ ( ) + ( )( )Q utQ u f t u t Q u Q f tk k u k k* *** * *, , , , ,22 01 1 0ξ ξ ,Q*k u ( 0, ξ* )  =  0,      ξ* ∈ [ 0; ∞ ],(9)Q*k u ( 0, t )  =  − ( )u t0 1, ,      t ∈ [ 0; ∞ ],lim*ξ →∞ Q*k u ( ξ* , t )  =  0    for any    t ∈ [ 0; ∞ ],where the functions  Q*k f ( ξ* , t  ),  k  ≥ 1,  are recurrently defined by the standardprocedure by values of the regular part  u x t0( ), , u x t1( ), , …  , u x tk − ( )1 ,   and thesingular part  Q0 u ( ξ, t  ), Q *0 u ( ξ* , t  ), Q 1 u ( ξ, t  ), Q *1 u ( ξ* , t  ), …  , Qk – 1 u ( ξ, t  ),Q*k – 1 u ( ξ* , t ),  x = 1 + ε ξ* ,  of asymptotics (5).4.  Main result.  The following statements are true:Lemma 1.  Additionally to assumptions P1 – P4, let us assume the fulfillment ofthe following conditions:1)  there exist positive values   C0    and   γ0  such that a solution   Q0 u ( ξ, t )   o fproblem (6) satisfies the inequality| Q0 u ( ξ, t ) |  ≤  C0 e– γ0 ξ;2)  the derivative  f t u t0 00 0 0( )( ), , , ,   is negative for all  t ∈ [ 0; T ).Then, for any  k ∈ N,  there exist solutions  Q k u ( ξ, t  )  of problem (7) such thatinequalities| Qk u ( ξ, t ) |  ≤  Ck e– γk ξare true for some  Ck > 0,  γk > 0.ISSN  1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3430 G. MATARAZZOLemma 2.  Additionally to assumptions P1 – P4, let us assume the fulfillment ofthe following conditions:1)  there exist positive values  C*0  and   γ*0  such that a solution  Q*0 u ( ξ* , t  )of problem (8) satisfies the inequality| Q*0 u ( ξ* , t ) |  ≤  C*0 e– γ*0 ξ*;2)  the derivative  f t u tu( )( )1 1 00, , , ,   is negative for all  t ∈ [ 0; T ).Then, for any  k ∈  N,  there exist solutions  Q*k u ( ξ* , t  )  of problem (9) suchthat inequalities| Q*k u ( ξ* , t ) |  ≤  C*k e– γ*k ξ*are true for some  C*k > 0,  γ*k > 0.Theorem.  Let the conditions of Lemmas 1, 2 be fulfilled.  Then the series (5) isan asymptotic solution of problem (1) – (4), i.e.,maxΩ1| u ( x, t, ε ) – uN ( x, t, ε ) |  =  O ( εN + 1 ),where  Ω1 ⊂  (  0; 1 ) ×  (  0; T )  is a compact set and a functions  uN  ( x, t, ε )  is definedwith the relationuN ( x, t, ε )  =  ε ε εk k k kkNu x t Q u x t Q u x t[ ]( ) + ( ) + ( + )=∑ , , ,* 10.1. Samoilenko A. M., Perestyuk N. A.  Impulsive differential equations // World Sci. Ser. NonlinearSci. Ser. A. – Singapore etc.: World Sci., 1995. – 14. – 468 p.2. Samilenko A. M., Kaplun Yu. I., Samoylenko V. Hr. Singularly perturbed differential equations withimpulses // Ukr. Math. J. – 2002. – 54, # 8. – P. 1089 – 1099.3. Samoylenko V. Hr., Elgondyev K. K. On periodic solutions of linear differential equations withimpulses // Ibid. – 1997. – 49, # 1. – P. 141 – 148.4. Elgondyev K. K.  On periodic solutions of impulsive systems // Uzbek Math. J. – 1999. – # 4. –P. 62 – 67.5. Samoylenko V. Hr., Khomchenko L. V.  Neumann boundary-value problem for singularly perturbedheat equation with impulses // Nonlinear Oscillations. – 2005. – 8, # 1. – P. 89 – 123.Received 06.09.2005ISSN  1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3

Asymptotic solutions of the Dirichlet problem for the heat equation with impulses

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Asymptotic solutions of the Dirichlet problem for the heat equation with impulses

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)