Asymptotic solutions of the Dirichlet problem for the heat equation with impulses
We propose an algorithm for the construction of asymptotic expansions for solutions of the Dirichlet problem for the heat equation with impulses.
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| author | Matarazzo, G. Матаразо, Г. |
| author_facet | Matarazzo, G. Матаразо, Г. |
| author_sort | Matarazzo, G. |
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| description | We propose an algorithm for the construction of asymptotic expansions for solutions of the Dirichlet problem for the heat equation with impulses. |
| first_indexed | 2026-03-24T02:43:03Z |
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UDC 517.9
G. Matarazzo (Salerno Univ., Italy)
ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM
FOR THE HEAT EQUATION WITH IMPULSES
ASYMPTOTYÇNI ROZV’QZKY ZADAÇI DIRIXLE DLQ
RIVNQNNQ TEPLOPROVIDNOSTI Z IMPUL|SNOG DI{G
We propose an algorithm for the construction of asymptotic expansions for solutions of the Dirichlet
problem for the heat equation with impulses.
Zaproponovano alhorytm pobudovy asymptotyçnyx rozvynen\ dlq rozv’qzkiv zadaçi Dirixle dlq
rivnqnnq teploprovidnosti z impul\snog di[g.
1. Introduction. The theory of impulsive differential equations [1] is an important
part of the modern theory of differential equations which has many applications in
practice. Till now, a lot of different problems connected with impulsive differential
equations are studied. It has been found that the solutions of differential equations with
impulses can demonstrate very complicated behaviour [1 – 5]. In the present paper, we
study the problem of the construction of an asymptotic solution of the Dirichlet
problem for the heat equation with impulses.
2. Formulation of the problem. Let us consider a differential heat equation with
small parameter ε ∈ ( 0; ε0 ) of the form
∂
∂ = ∂
∂
+ ( )u
t
u
x
f x t uε ε2
2
2 , , , , ( x, t ) ∈ Ω = ( 0; 1 ) × ( 0; + ∞ ), (1)
under initial conditions
u ( x, 0, ε ) = ϕ ( x, ε ), (2)
boundary conditions
u ( 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 differentiable
with respect to their variables and can be represented as regular asymptotic expansions
with respect to a small parameter ε ∈ ( 0; ε0 ). In particular,
ϕ ( x, ε ) = ε ϕk
k
k
x( )
=
∞
∑
0
, Ii ( x, ε ) = εk
ik
k
xT ( )
=
∞
∑
0
.
P2 . The nonperturbed problem
∂
∂ = ( )u
t
f 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, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3 427
428 G. MATARAZZO
P3 . 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 series
u ( x, t, ε ) = u x t( ), , ε + Q u ( ξ, t, ε ) + Q* u ( ξ* , t, ε ), (5)
where
u x t u x t u x t u x t( ) = ( ) + ( ) + ( ) + …, , , , ,ε ε ε0 1
2
2
is the regular part of asymtotics and
Q 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, are
supposed to be defined for ξ ∈ [ 0; ε–
1
], t ≥ 0, and, at the same time, the functions
Q*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). The
terms of the regular part of asymptotics (5) may be found as solutions of the following
problems:
du
dt f x t u0
0 0= ( ), , , ,
u x0 0( ), = ϕ0 ( x ), x ∈ [ 0; 1 ],
∆u x t t ti0( ) =, = Ii ( x, 0 ), i ∈ N,
du
dt f x t u u f x t u u uk
u 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 of
u 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 the
singular part Q u ( ξ, t, ε ) of asymptotic (5) which is defined as solutions of the
following boundary-value problems:
∂
∂ = ∂
∂
+ ( ) + − ( )( ) ( )Q u
t
Q u
f t u t Q u f t u t0
2
0
2 0 0 00 0 0 0 0 0
ξ
, , , , , , , , ,
Q0 u ( ξ, 0 ) = 0, ξ ∈ [ 0; ∞ ],
(6)
Q0 u ( 0, t ) = − ( )u t0 0, , t ∈ [ 0; ∞ ],
lim
ξ→∞
Q0 u ( ξ, t ) = 0 for any t ∈ [ 0; ∞ ];
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
ASYMPTOTIC SOLUTIONS OF THE DIRICHLET PROBLEM … 429
∂
∂ = ∂
∂
+ ( ) + ( )( )Q u
t
Q u
f t u t Q u Q f tk k
u k k
2
2 00 0 0
ξ
ξ, , , , , ,
Qk u ( 0, ξ ) = 0, ξ ∈ [ 0; ∞ ],
(7)
Qk u ( 0, t ) = − ( )u t0 0, , t ∈ [ 0; ∞ ],
lim
ξ→∞
Qk u ( ξ, t ) = 0 for any t ∈ [ 0; ∞ ],
where the functions Qk f ( ξ, t ), k ≥ 1, are recurrently defined by the standard
procedure by values of the regular part u x t0( ), , u x t1( ), , … , u x tk − ( )1 , , x = ε ξ, of
asymptotics (5).
We can now proceed to the calculation of the singular part Q* u ( ξ, t, ε ) of
asymptotics (5) which is defined as solutions of the following boundary-value
problems:
∂
∂ = ∂
∂
+ ( ) + + − ( ) +( ) ( )Q u
t
Q u
f t u t Q u Q u f t u t Q u* *
*
*, , , , , , , ,0
2
0
2 0 0 0 0 01 1 0 1 1 0
ξ
,
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 u
t
Q u
f t u t Q u Q f tk k
u k k
* *
*
* * *, , , , ,
2
2 01 1 0
ξ
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 standard
procedure by values of the regular part u x t0( ), , u x t1( ), , … , u x tk − ( )1 , and the
singular 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 of
the following conditions:
1) there exist positive values C0 and γ0 such that a solution Q0 u ( ξ, t ) o f
problem (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 that
inequalities
| Qk u ( ξ, t ) | ≤ Ck e–
γk
ξ
are true for some Ck > 0, γk > 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
430 G. MATARAZZO
Lemma 2. Additionally to assumptions P1 – P4, let us assume the fulfillment of
the 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) such
that 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) is
an asymptotic solution of problem (1) – (4), i.e.,
max
Ω1
| 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 defined
with the relation
uN ( x, t, ε ) = ε ε εk
k k k
k
N
u x t Q u x t Q u x t[ ]( ) + ( ) + ( + )
=
∑ , , ,* 1
0
.
1. Samoilenko A. M., Perestyuk N. A. Impulsive differential equations // World Sci. Ser. Nonlinear
Sci. Ser. A. – Singapore etc.: World Sci., 1995. – 14. – 468 p.
2. Samilenko A. M., Kaplun Yu. I., Samoylenko V. Hr. Singularly perturbed differential equations with
impulses // Ukr. Math. J. – 2002. – 54, # 8. – P. 1089 – 1099.
3. Samoylenko V. Hr., Elgondyev K. K. On periodic solutions of linear differential equations with
impulses // 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 perturbed
heat equation with impulses // Nonlinear Oscillations. – 2005. – 8, # 1. – P. 89 – 123.
Received 06.09.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
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| id | umjimathkievua-article-3465 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:43:03Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/df/61e39e4cb456cac67ff9786740d1d8df.pdf |
| spelling | umjimathkievua-article-34652020-03-18T19:55:07Z Asymptotic solutions of the Dirichlet problem for the heat equation with impulses Асимптотичні розв'язки задачі Діріхле для рівняння теплопровідності з імпульсною дією Matarazzo, G. Матаразо, Г. 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. Institute of Mathematics, NAS of Ukraine 2006-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3465 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 3 (2006); 427–430 Український математичний журнал; Том 58 № 3 (2006); 427–430 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3465/3667 https://umj.imath.kiev.ua/index.php/umj/article/view/3465/3668 Copyright (c) 2006 Matarazzo G. |
| spellingShingle | Matarazzo, G. Матаразо, Г. Asymptotic solutions of the Dirichlet problem for the heat equation with impulses |
| title | Asymptotic solutions of the Dirichlet problem for the heat equation with impulses |
| title_alt | Асимптотичні розв'язки задачі Діріхле для рівняння теплопровідності з імпульсною дією |
| title_full | Asymptotic solutions of the Dirichlet problem for the heat equation with impulses |
| title_fullStr | Asymptotic solutions of the Dirichlet problem for the heat equation with impulses |
| title_full_unstemmed | Asymptotic solutions of the Dirichlet problem for the heat equation with impulses |
| title_short | Asymptotic solutions of the Dirichlet problem for the heat equation with impulses |
| title_sort | asymptotic solutions of the dirichlet problem for the heat equation with impulses |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3465 |
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