Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$
A method for construction of exact solutions to nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ which is based on ansatz $p(x)=w_1(t)\varphi(u)$ is proposed. Here the function $p(x)$ is a solution to one of the equations $(p')^2 = Ap^2 + B$, $(p')^2 = Ap^4 + Bp^2 +C$, and the functi...
Saved in:
| Date: | 2019 |
|---|---|
| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Інститут математики НАН України
2019
|
| Online Access: | https://trim.imath.kiev.ua/index.php/trim/article/view/363 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Transactions of Institute of Mathematics of NAS of Ukraine |
| Download file: |  |
Institution
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552857974079488 |
|---|---|
| author | Barannyk, A. Barannyk, T. Yuryk, I. Баранник, А. Баранник, Т. Юрик, І. |
| author_facet | Barannyk, A. Barannyk, T. Yuryk, I. Баранник, А. Баранник, Т. Юрик, І. |
| author_institution_txt_mv | [
{
"author": "А. Баранник",
"institution": "Поморська академія, Слупськ"
},
{
"author": "Т. Баранник",
"institution": "Полтавський національний педагогічний університет імені В.Г.Короленка"
},
{
"author": "І. Юрик",
"institution": "Національний університет харчових технологій"
}
] |
| author_sort | Barannyk, A. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | A method for construction of exact solutions to nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ which is based on ansatz $p(x)=w_1(t)\varphi(u)$ is proposed. Here the function $p(x)$ is a solution to one of the equations $(p')^2 = Ap^2 + B$, $(p')^2 = Ap^4 + Bp^2 +C$, and the functions $w_1(t)$ and $\varphi(u)$ can be found from the condition that this ansatz reduces the equation to an ordinary differential equation with unknown function $w_1(t)$. |
| first_indexed | 2026-08-04T01:06:06Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 6�15
ÓÄÊ 517.912:512.816
Òî÷íi ðîçâ'ÿçêè íåëiíiéíîãî
ðiâíÿííÿ òåïëîïðîâiäíîñòi
ut = (F (u)ux)x +H(u)
À.Ô. Áàðàííèê †, Ò.À. Áàðàííèê ‡, I.I. Þðèê §
† Ïîìîðñüêà àêàäåìiÿ, Ñëóïñüê, Ïîëüùà
‡ Ïîëòàâñüêèé íàöiîíàëüíèé ïåäàãîãi÷íèé óíiâåðñèòåò
iìåíi Â.Ã. Êîðîëåíêà
§ Íàöiîíàëüíèé óíiâåðñèòåò õàð÷îâèõ òåõíîëîãié, Êè¨â
E-mail: i.yu@ukr.net
Çàïðîïîíîâàíî ìåòîä ïîáóäîâè òî÷íèõ ðîçâ'ÿçêiâ íåëiíiéíîãî ðiâíÿí-
íÿ òåïëîïðîâiäíîñòi ut = (F (u)ux)x +H(u), ÿêèé ðóíòó¹òüñÿ íà âèêî-
ðèñòàííi ïiäñòàíîâêè p(x) = w1(t)ϕ(u), äå ôóíêöiÿ p(x) ¹ ðîçâ'ÿçêîì
îäíîãî ç ðiâíÿíü (p′)2 = Ap2 +B, (p′)2 = Ap4 +Bp2 +C, à ôóíêöi¨ w1(t)
i ϕ(u) çíàõîäÿòüñÿ ç óìîâè, ùî öÿ ïiäñòàíîâêà ðåäóêó¹ ðiâíÿííÿ äî
çâè÷àéíîãî äèôåðåíöiàëüíîãî ðiâíÿííÿ ç íåâiäîìîþ ôóíêöi¹þ w1(t).
A method for construction of exact solutions to nonlinear heat equation
ut = (F (u)ux)x + H(u) which is based on ans�atz p(x) = w1(t)ϕ(u) is
proposed. Here the function p(x) is a solution to one of the equations
(p′)2 = Ap2 +B, (p′)2 = Ap4 +Bp2 +C, and the functions w1(t) and ϕ(u)
can be found from the condition that this ansatz reduces the equation to
an ordinary di�erential equation with unknown function w1(t).
1. Âñòóï. Ðîáîòà ïðèñâÿ÷åíà ïîáóäîâi òî÷íèõ ðîçâ'ÿçêiâ íåëiíié-
íîãî ðiâíÿííÿ òåïëîïðîâiäíîñòi
ut = (F (u)ux)x +H(u), (1)
ÿêå îïèñó¹ íåñòàöiîíàðíó òåïëîïðîâiäíiñòü â íåðóõîìîìó ñåðåäîâè-
ùi, ÿêùî êîåôiöi¹íò òåïëîïðîâiäíîñòi i øâèäêiñòü ðåàêöi¨ ¹ äîâiëü-
íèìè ôóíêöiÿìè òåìïåðàòóðè. Ãðóïîâà êëàñèôiêàöiÿ ðiâíÿíü öüîãî
âèäó, à òàêîæ òî÷íi ðîçâ'ÿçêè äëÿ ðiçíèõ ôóíêöié F (u) i H(u) îïè-
ñàíî â ðîáîòàõ (äèâ. [1, 2, 3] i öèòîâàíó òàì ëiòåðàòóðó).
Òî÷íi ðîçâ'ÿçêè íåëiíiéíîãî ðiâíÿííÿ òåïëîïðîâiäíîñòi 7
Ó öié ñòàòòi ìè âèêîðèñòîâó¹ìî ìåòîä ïîáóäîâè òî÷íèõ ðîçâ'ÿçêiâ
ðiâíÿííÿ (1), ÿêèé ðóíòó¹òüñÿ íà êëàñè÷íîìó ìåòîäi âiäîêðåìëåííÿ
çìiííèõ òà éîãî óçàãàëüíåííi, à òàêîæ ìåòîäi ðåäóêöi¨, ùî ëåæèòü â
îñíîâi ñèìåòðiéíîãî ìåòîäó Ñ. Ëi. Äëÿ ïîáóäîâè òî÷íèõ ðîçâ'ÿçêiâ
ðiâíÿííÿ (1) çàñòîñîâó¹òüñÿ ïiäñòàíîâêà
p(x) = w1(t)ϕ(u), (2)
ÿêà ìiñòèòü äâi íåâiäîìi ôóíêöi¨ w1(t) i ϕ(u), à òàêîæ ôóíêöiþ p(x),
ÿêà çàäà¹òüñÿ àïðiîðíî. Äåòàëüíî ðîçãëÿäàþòüñÿ âèïàäêè, êîëè p(x)
¹ ðîçâ'ÿçêîì îäíîãî ç òàêèõ ðiâíÿíü:
(p′)2 = Ap2 +B,
(p′)2 = Ap4 +Bp2 + C,
äå A, B, C � ñòàëi. Ïðè òàêîìó âèáîði ôóíêöi¨ p(x) íåâiäîìi ôóíê-
öi¨ w1(t) i ϕ(u) âèçíà÷àþòüñÿ ç óìîâè, ùî ïiäñòàíîâêà (2) ðåäóêó¹
ðiâíÿííÿ (1) äî çâè÷àéíîãî äèôåðåíöiàëüíîãî ðiâíÿííÿ ç íåâiäîìîþ
ôóíêöi¹þ w1(t).
Âiäìiòèìî, ùî òàêèé ïiäõiä áóâ âèêîðèñòàíèé äëÿ ïîáóäîâè òî÷-
íèõ ðîçâ'ÿçêiâ ðiâíÿííÿ òèïó Êîðòåâåãà�äå Ôðiçà â [4, 5] i íåëiíiéíîãî
ðiâíÿííÿ
utt = F (u)uxx + F ′(u)u2
x.
2. Ðîçâ'ÿçêè ðiâíÿííÿ (1), ùî âèðàæàþòüñÿ ÷åðåç òðèãî-
íîìåòðè÷íi ôóíêöi¨. Ââåäåìî îçíà÷åííÿ
Îçíà÷åííÿ 1. Áóäåìî ãîâîðèòè, ùî ðiâíÿííÿ (1) äîïóñê๠ïiäñòà-
íîâêó (2), ÿêùî âîíà ðåäóêó¹ ðiâíÿííÿ (1) äî çâè÷àéíîãî äèôåðåí-
öiàëüíîãî ðiâíÿííÿ íà ôóíêöiþ ω1(t).
Äëÿ ïîáóäîâè òî÷íèõ ðîçâ'ÿçêiâ ðiâíÿííÿ (1) âèêîðèñòîâó¹òüñÿ
ïiäñòàíîâêà
p(x) = w1(t)ϕ(u), (3)
äå p(x) ¹ ðîçâ'ÿçêîì ðiâíÿííÿ
(p′)2 = Ap2 +B, A 6= 0, B 6= 0.
8 À.Ô. Áàðàííèê, Ò.À. Áàðàííèê, I.I. Þðèê
Ïiäñòàâèìî (3) â ðiâíÿííÿ (1):
−w
′
1
w1
ϕ
ϕ′
=
1
w2
1
(
−FB ϕ′′
(ϕ′)3
+ F ′B
1
(ϕ′)2
)
+
(
−FAϕ
2ϕ′′
(ϕ′)3
+ F ′A
ϕ2
(ϕ′)2
+ FA
ϕ
ϕ′
+H
)
. (4)
Äëÿ âèçíà÷åííÿ ôóíêöié F (u) i ϕ(u) îòðèìà¹ìî òàêó ñèñòåìó ðiâ-
íÿíü:
−F ϕ′′
(ϕ′)3
+ F ′
1
(ϕ′)2
= λ1
ϕ
ϕ′
, (5)
−FAϕ
2ϕ′′
(ϕ′)3
+ F ′A
ϕ2
(ϕ′)2
+ FA
ϕ
ϕ′
+H = λ2
ϕ
ϕ′
, (6)
äå λ1, λ2 ∈ R. Íåõàé F ′(u) 6= 0. Iíòåãðóþ÷è ðiâíÿííÿ (5), ÿêå ¹ ëiíié-
íèì âiäíîñíî ôóíêöi¨ F = F (u), çíàõîäèìî
F =
(
λ1
∫
ϕdu+ C1
)
ϕ, (7)
äå òóò i äàëi C,C1, C2, . . . � äîâiëüíi ñòàëi iíòåãðóâàííÿ. Ïiäñòàâèâ-
øè (5), (6) â ðiâíÿííÿ (4), îòðèìó¹ìî ðiâíÿííÿ äëÿ âèçíà÷åííÿ ôóí-
êöi¨ w1(t):
w′1
w1
+ λ1B
1
w2
1
+ λ2 = 0. (8)
Ç ðiâíÿíü (5), (6) çíàõîäèìî
H =
1
ϕ′
(−λ1Aϕ
3 −AFϕ+ λ2ϕ). (9)
Ó ïiäñóìêó îòðèìà¹ìî òàêó òåîðåìó:
Òåîðåìà 1. ßêùî ðiâíÿííÿ (1) äîïóñê๠ïiäñòàíîâêó âèãëÿäó (3)
i F ′(u) 6= 0, òî ôóíêöi¨ F (u) i H(u) âèçíà÷àþòüñÿ ôîðìóëàìè (7)
i (9) âiäïîâiäíî, à ôóíêöiÿ w1(t) ¹ ðîçâ'ÿçêîì ðiâíÿííÿ (8).
Îòðèìàíi ðîçâ'ÿçêè ðiâíÿííÿ (1) ìîæíà óçàãàëüíèòè, âèêîðèñòî-
âóþ÷è ïiäñòàíîâêè:
ϕ(u) = w1(t) ch(k (x+ C3)) + w2(t) sh(k (x+ C3)), (10)
Òî÷íi ðîçâ'ÿçêè íåëiíiéíîãî ðiâíÿííÿ òåïëîïðîâiäíîñòi 9
ÿêùî A = k2 > 0,
ϕ(u) = w1(t) cos(k (x+ C3)) + w2(t) sin(k (x+ C3)), (11)
ÿêùî A = −k2 < 0.
Ðîçãëÿíåìî, íàïðèêëàä, ïiäñòàíîâêó (10). ßêùî ôóíêöi¨ F (u) i
H(u) âèçíà÷àþòüñÿ çà ôîðìóëàìè (7) i (9) âiäïîâiäíî i A = k2 > 0,
òî ïiäñòàíîâêà (10) ðåäóêó¹ ðiâíÿííÿ (1) äî ñèñòåìè
w′1 =
(
−λ1k
2w2
1 + λ1k
2w2
2
)
w1 + λ2w1, (12)
w′2 =
(
−λ1k
2w2
1 + λ1k
2w2
2
)
w2 + λ2w2. (13)
Íåõàé w1 6= 0. Ç ðiâíÿíü (12), (13) âèïëèâà¹, ùî w2 = Cw1. Ðiâíÿííÿ
(12) íàáóâ๠âèãëÿäó
w′1 = λ1k
2
(
C2 − 1
)
w3
1 + λ2w1. (14)
ßêùî λ2 6= 0, òî ðîçâ'ÿçêîì ðiâíÿííÿ (14) ¹ ôóíêöiÿ
w2
1 =
(
C2
λ2
exp(−2λ2t)−
λ1
λ2
k2
(
C2 − 1
))−1
,
äå C2 6= 0. Ìà¹ìî òàêèé ðîçâ'ÿçîê ðiâíÿííÿ (1):
ϕ(u) = ±
(
C2
λ2
exp(−2λ2t)−
λ1
λ2
k2
(
C2 − 1
))−1/2
× [ch(k (x+ C3)) + w2(t) sh(k (x+ C3))].
ßêùî λ2 = 0, òî ðîçâ'ÿçêîì ðiâíÿííÿ (14) ¹ ôóíêöiÿ
w2
1 = [−2λ1k
2
(
C2 − 1
)
t+ C2]−1, λ2 6= 0.
Ó ïiäñóìêó îòðèìó¹ìî òàêèé ðîçâ'ÿçîê ðiâíÿííÿ (1):
ϕ(u) = [−2λ1k
2
(
C2 − 1
)
t+ C2]−1/2
× [ch(k(x+ C3)) + w2(t) sh(k(x+ C3))].
Âèïàäîê w1 = 0 çâîäèòüñÿ äî iíòåãðóâàííÿ ðiâíÿííÿ
w′2 = λ1k
2w3
2 + λ2w2.
10 À.Ô. Áàðàííèê, Ò.À. Áàðàííèê, I.I. Þðèê
Îòæå, ÿêùî λ2 6= 0, òî ìà¹ìî òàêèé ðîçâ'ÿçîê ðiâíÿííÿ (1):
ϕ(u) =
(
C2
λ2
exp(−2λ2t)−
λ1
λ2
k2
)−1/2
sh(k(x+ C3)),
äå C2 6= 0, à ó âèïàäêó λ2 = 0 � ðîçâ'ÿçîê
ϕ(u) =
(
−2λ1k
2
(
C2 − 1
)
t+ C2
)−1/2
sh(k (x+ C3)).
Àíàëîãi÷íî, ïiäñòàíîâêà (11) ðåäóêó¹ ðiâíÿííÿ (1) äî ñèñòåìè
w′1 = (λ2
1k
2w2
1 + λ2
1k
2w2
2)w1 + λ2w1, (15)
w′2 = (λ2
1k
2w2
1 + λ2
1k
2w2
2)w2 + λ2w2. (16)
Ïðîiíòåãðóâàâøè (15), (16), îòðèìó¹ìî òàêi ðîçâ'ÿçêè ðiâíÿííÿ (1):
ϕ(u) =
(
C2
λ2
exp(−2λ2t)−
λ1
λ2
k2
(
1 + C2
))−1/2
× [cos(k (x+ C3)) + C sin(k (x+ C3))],
äå C2 6= 0, λ2 6= 0;
ϕ(u) = (−2λ1k
2
(
C2 + 1
)
t+ C2)−1/2
× [cos(k(x+ C3)) + C sin(k(x+ C3))], λ1 6= 0,
äå λ1 6= 0, λ2 = 0;
ϕ(u) =
(
C2
λ2
exp(−2λ2t)−
λ1
λ2
k2
)−1/2
sin(k (x+ C3)),
äå C2 6= 0, λ2 6= 0;
ϕ(u) = (−2λ1k
2t+ C2)−1/2 sin(k(x+ C3)),
äå λ1 6= 0, λ2 = 0.
3. Ðîçâ'ÿçêè ðiâíÿííÿ (1), ùî âèðàæàþòüñÿ ÷åðåç åëiï-
òè÷íi ôóíêöi¨ ßêîái. Îïèøåìî ðiâíÿííÿ âèäó (1) i ¨õ òî÷íi ðîç-
â'ÿçêè, ÿêi äîïóñêàþòü ïiäñòàíîâêó
p(x) = w1(t)ϕ(u), (17)
Òî÷íi ðîçâ'ÿçêè íåëiíiéíîãî ðiâíÿííÿ òåïëîïðîâiäíîñòi 11
äå p(x) ¹ ðîçâ'ÿçêîì ðiâíÿííÿ
(p′)2 = Ap4 +Bp2 + C, A 6= 0, C 6= 0. (18)
Ïiäñòàâèâøè â ðiâíÿííÿ (1), îòðèìó¹ìî
−w
′
1
w1
ϕ
ϕ′
= w2
1
(
2AF
ϕ3
ϕ′
−AF ϕ
4ϕ′′
(ϕ′)3
+AF ′
ϕ4
(ϕ′)2
)
+
1
w2
1
(
−CF ϕ′′
(ϕ′)3
+ CF ′
1
(ϕ′)2
)
(19)
+
(
−BF ϕ
2ϕ′′
ϕ′3
+BF ′
ϕ2
ϕ′2
+BF
ϕ
ϕ′
+H(u)
)
.
Ç ðiâíÿííÿ (19) îòðèìó¹ìî ñèñòåìó
−F ϕ′′
(ϕ′)3
+ F ′
1
(ϕ′)2
= λ1
ϕ
ϕ′
, (20)
2AF
ϕ3
ϕ′
+Aϕ4
(
−F ϕ′′
ϕ′3
+ F ′
1
(ϕ′)2
)
= λ2
ϕ
ϕ′
, (21)
−BF ϕ
2ϕ′′
(ϕ′)3
+BF ′
ϕ2
(ϕ′)2
+BF
ϕ
ϕ′
+H(u) = λ3
ϕ
ϕ′
, (22)
äå λ1, λ2, λ3 ∈ R. Ïiäñòàâèâøè (20) â (21), çíàõîäèìî
F =
λ2
2A
1
ϕ2
− λ1
2
ϕ2. (23)
Ç ðiâíÿííÿ (22)
H = −Bϕ2
(
−F ϕ′′
ϕ′3
+ F ′
1
(ϕ′)2
)
−BF ϕ
ϕ′
+ λ3
ϕ
ϕ′
,
à òîìó íà ïiäñòàâi (20) i (23):
H(u) = −λ1B
2
ϕ3
ϕ′
+ λ3
ϕ
ϕ′
− λ2B
2A
1
ϕϕ′
. (24)
Ïiäñòàâèâøè (23) â (20), çíàõîäèìî ðiâíÿííÿ äëÿ âèçíà÷åííÿ ôóíêöi¨
ϕ = ϕ(u):
ϕ′′ =
(
λ2
A
+ 2λ1ϕ
4
)(
λ1
2
ϕ5 − λ2
2A
ϕ
)−1
(ϕ′)2. (25)
12 À.Ô. Áàðàííèê, Ò.À. Áàðàííèê, I.I. Þðèê
Ïiäñòàâèâøè (20)�(22) â (19), îòðèìó¹ìî ðiâíÿííÿ äëÿ âèçíà÷åííÿ
ôóíêöi¨ w1 = w1(t):
w′1
w1
+ λ2w
2
1 +
λ1C
w2
1
+ λ3 = 0. (26)
Ó ïiäñóìêó îòðèìà¹ìî òàêó òåîðåìó:
Òåîðåìà 2. ßêùî ðiâíÿííÿ (1) äîïóñê๠ïiäñòàíîâêó (17), òî ôóíê-
öi¨ F (u) i H(u) âèçíà÷àþòüñÿ ôîðìóëàìè (23) i (24) âiäïîâiäíî,
à ôóíêöi¨ ϕ òà w1(t) ¹ ðîçâ'ÿçêàìè çâè÷àéíèõ äèôåðåíöiàëüíèõ ðiâ-
íÿíü (25) òà (26).
Òàêèì ÷èíîì, ïîáóäîâó òî÷íèõ ðîçâ'ÿçêiâ âèäó (17) ðiâíÿííÿ (1)
çâåäåíî äî iíòåãðóâàííÿ ðiâíÿíü (25), (26).
Ðîçãëÿíåìî äâà âèïàäêè.
I) Âèïàäîê λ2 = 0. Ðiâíÿííÿ (25) íàáóâ๠âèãëÿäó
ϕ′′ =
4
ϕ
(ϕ′)2. (27)
Iíòåãðóþ÷è ðiâíÿííÿ (27), çíàõîäèìî
ϕ = (C1u+ C2)−1/3,
C1 6= 0, i íà ïiäñòàâi (23), (25)
F = −λ1
2
(C1u+ C2)−2/3,
H =
3λ1B
2C1
(C1u+ C2)1/3 − 3λ3
C1
(C1u+ C2).
Ðiâíÿííÿ (1) íàáóâ๠âèãëÿäó
ut =
(
−λ1
2
(C1u+ C2)−2/3ux
)
x
+
3λ1B
2C1
(C1u+ C2)1/3
− 3λ3
C1
(C1u+ C2), (28)
i ïiäñòàíîâêîþ
v = ϕ(u) = (C1u+ C2)−1/3
Òî÷íi ðîçâ'ÿçêè íåëiíiéíîãî ðiâíÿííÿ òåïëîïðîâiäíîñòi 13
çâîäèòüñÿ äî âèäó
vt = −λ1
2
v2vxx + λ1v(vx)2 − λ1
2
Bv3 + λ3v. (29)
Iíòåãðóþ÷è ðiâíÿííÿ (26) ó âèïàäêó λ2 = 0, çíàõîäèìî
w2
1 = C3 exp(−2λ3t)−
λ1
λ3
C, C3 6= 0, ÿêùî λ3 6= 0,
w2
1 = −2λ1Ct+ C3, ÿêùî λ3 = 0.
Ó ïiäñóìêó îòðèìó¹ìî òàêi ðîçâ'ÿçêè ðiâíÿíü (28), (29):
à) ßêùî A = k2, B = −
(
1 + k2
)
, C = 1, òî
v = ϕ(u) =
(
C3 exp(−2λ3t)−
λ1
λ3
)−1/2
sn(x; k), λ3 6= 0,
v = ϕ(u) = (−2λ1t+ C3)
−1/2
sn(x; k), λ3 = 0.
á) ßêùî A = −k2, B = 2k2 − 1, C = 1− k2, òî
v = ϕ(u) =
(
C3 exp(−2λ3t)− (1− k2)
λ1
λ3
)−1/2
cn(x; k), λ3 6= 0,
v = ϕ(u) =
(
−2λ1
(
1− k2
)
t+ C3
)−1/2
cn(x; k), λ3 = 0.
â) ßêùî A = −1, B = 2− k2, C = −1 + k2, òî
v = ϕ(u) =
(
C3 exp(−2λ3t)− (−1 + k2)
λ1
λ3
)−1/2
dn(x; k), λ3 6= 0,
v = ϕ(u) =
(
−2λ1(−1 + k2)t+ C3
)−1/2
dn(x; k), λ3 = 0.
II) Âèïàäîê λ1 = 0. Ðiâíÿííÿ (25) íàáóâ๠âèãëÿäó
ϕ′′ = − 2
ϕ
(ϕ′)2. (30)
Iíòåãðóþ÷è ðiâíÿííÿ (30), çíàõîäèìî
ϕ = (C1u+ C2)1/3,
äå C1 6= 0, i íà ïiäñòàâi (23), (25)
F =
λ2
2A
(C1u+ C2)−2/3,
14 À.Ô. Áàðàííèê, Ò.À. Áàðàííèê, I.I. Þðèê
H =
3λ3
2A
(C1u+ C2)− 3λ2B
2AC1
(C1u+ C2)1/3.
Ðiâíÿííÿ (1) íàáóâ๠âèãëÿäó
ut =
(
λ2
2A
(C1u+ C2)−2/3ux
)
x
+
3λ3
C1
(C1u+ C2)
− 3λ2B
2AC1
(C1u+ C2)1/3, (31)
i ïiäñòàíîâêîþ
v = ϕ(u) = (C1u+ C2)1/3
çâîäèòüñÿ äî âèäó
vt =
λ2
2A
v−2vxx + λ3v −
λ2B
2A
1
v
. (32)
Ïiäñòàâèâøè (20)�(22) â (19), îòðèìó¹ìî ðiâíÿííÿ äëÿ âèçíà÷åííÿ
ôóíêöi¨ w1 = w1(u):
w′1
w1
+ λ2w
2
1 + λ3 = 0. (33)
Iíòåãðóþ÷è ðiâíÿííÿ (33), çíàõîäèìî
w−2
1 = C3 exp(2λ3t)−
λ2
λ3
, C3 6= 0, ÿêùî λ3 6= 0;
w−2
1 = 2λ2t+ C3, ÿêùî λ3 = 0.
Ó ïiäñóìêó îòðèìó¹ìî òàêi ðîçâ'ÿçêè ðiâíÿíü (31), (32):
à) ßêùî A = k2, B = −
(
1 + k2
)
, C = 1, òî ðiâíÿííÿ (32) ìà¹
âèãëÿä
vt =
λ2
2k2
v−2vxx + λ3v +
λ2
(
1 + k2
)
2k2
1
v
. (34)
Ðîçâ'ÿçêè ðiâíÿííÿ (34):
v =
(
C3 exp(2λ3t)−
λ2
λ3
)1/2
sn(x; k), ÿêùî λ3 6= 0;
Òî÷íi ðîçâ'ÿçêè íåëiíiéíîãî ðiâíÿííÿ òåïëîïðîâiäíîñòi 15
v = (2λ2t+ C3)1/2 sn(x; k), ÿêùî λ3 = 0.
á) ßêùî A = −k2, B = 2k2 − 1, C = 1− k2, òî ðiâíÿííÿ (32) ìà¹
âèãëÿä
vt = − λ2
2k2
v−2vxx + λ3v +
λ2(2k2 − 1)
2k2
1
v
. (35)
Ðîçâ'ÿçêè ðiâíÿííÿ (35):
v =
(
C3 exp(2λ3t)−
λ2
λ3
)1/2
cn(x; k), ÿêùî λ3 6= 0;
v = (2λ2t+ C3)1/2 cn(x; k), ÿêùî λ3 = 0.
â) ßêùî A = −1, B = 2 − k2, C = −1 + k2, òî ðiâíÿííÿ (32) ìà¹
âèãëÿä
vt = −λ2
2
v−2vxx + λ3v +
λ2(2− k2)
2
1
v
. (36)
Ðîçâ'ÿçêè ðiâíÿííÿ (36):
v =
(
C3 exp(2λ3t)−
λ2
λ3
)1/2
dn(x; k), ÿêùî λ3 6= 0;
v = (2λ2t+ C3)1/2 dn(x; k), ÿêùî λ3 = 0.
[1] Polyanin A.D., Zaitsev V.F., Handbook of nonlinear partial di�erential equati-
ons, Chapman & Hall/CRC, Boca Raton, FL, 2004.
[2] Galaktionov V.A., Svirshchevskii S.R., Exact solutions and invariant subspaces
of nonlinear partial di�erential equations in mechanics and physics, Chapman
& Hall/CRC Applied Mathematics and Nonlinear Science Series, Boca Raton,
FL, 2007.
[3] Nikitin A.G., Barannyk T.A., Solitary wave and other solutions for nonlinear
heat equations. Cent. Eur. J. Math. 2 (2004), no. 5, 840�858.
[4] Barannyk A.F., Barannyk T.A., Yuryk I.I., Separation of variables for nonli-
near equations of hyperbolic and Korteweg�de Vries type, Rep. Math. Phys. 68
(2011), no. 1, 92�105.
[5] Barannyk A.F., Barannyk T.A., Yuryk I.I., Generalized separation of variables
for nonlinear equation utt = F (u)uxx +aF ′(u)u2
x, Rep. Math. Phys. 71 (2013),
no. 1, 1�13.
|
| id | oai:trim.imath.kiev.ua:article-363 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-08-04T01:06:06Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/1e/dc16fc1479b148d5372ec5712420aa1e.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3632020-08-13T08:52:53Z Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ Точні розв'язки нелінійного рівняння теплопровідності $u_t=(F(u)u_x)_x+H(u)$ Barannyk, A. Barannyk, T. Yuryk, I. Баранник, А. Баранник, Т. Юрик, І. A method for construction of exact solutions to nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ which is based on ansatz $p(x)=w_1(t)\varphi(u)$ is proposed. Here the function $p(x)$ is a solution to one of the equations $(p')^2 = Ap^2 + B$, $(p')^2 = Ap^4 + Bp^2 +C$, and the functions $w_1(t)$ and $\varphi(u)$ can be found from the condition that this ansatz reduces the equation to an ordinary differential equation with unknown function $w_1(t)$. Запропоновано метод побудови точних розв’язків нелінійного рівняння теплопровідності $u_t=(F(u)u_x)_x+H(u)$, який ґрунтується на використанні підстановки $p(x)=w_1(t)\varphi(u)$, де функція $p(x)$ є розв’язком одного з рівнянь $(p')^2 = Ap^2 + B$, $(p')^2 = Ap^4 + Bp^2 +C$, а функції $w_1(t)$ і $\varphi(u)$ знаходяться з умови, що ця підстановка редукує рівняння до звичайного диференціального рівняння з невідомою функцією $w_1(t)$. Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/363 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 6-15 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 6-15 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 6-15 3083-7529 1815-2910 uk https://trim.imath.kiev.ua/index.php/trim/article/view/363/357 Авторське право (c) 2019 А. Баранник, Т. Баранник, І. Юрик |
| spellingShingle | Barannyk, A. Barannyk, T. Yuryk, I. Баранник, А. Баранник, Т. Юрик, І. Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ |
| title | Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ |
| title_alt | Точні розв'язки нелінійного рівняння теплопровідності $u_t=(F(u)u_x)_x+H(u)$ |
| title_full | Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ |
| title_fullStr | Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ |
| title_full_unstemmed | Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ |
| title_short | Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ |
| title_sort | exact solutions of nonlinear heat equation $u_t=(f(u)u_x)_x+h(u)$ |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/363 |
| work_keys_str_mv | AT barannyka exactsolutionsofnonlinearheatequationutfuuxxhu AT barannykt exactsolutionsofnonlinearheatequationutfuuxxhu AT yuryki exactsolutionsofnonlinearheatequationutfuuxxhu AT barannika exactsolutionsofnonlinearheatequationutfuuxxhu AT barannikt exactsolutionsofnonlinearheatequationutfuuxxhu AT ûrikí exactsolutionsofnonlinearheatequationutfuuxxhu AT barannyka točnírozv039âzkinelíníjnogorívnânnâteploprovídnostíutfuuxxhu AT barannykt točnírozv039âzkinelíníjnogorívnânnâteploprovídnostíutfuuxxhu AT yuryki točnírozv039âzkinelíníjnogorívnânnâteploprovídnostíutfuuxxhu AT barannika točnírozv039âzkinelíníjnogorívnânnâteploprovídnostíutfuuxxhu AT barannikt točnírozv039âzkinelíníjnogorívnânnâteploprovídnostíutfuuxxhu AT ûrikí točnírozv039âzkinelíníjnogorívnânnâteploprovídnostíutfuuxxhu |