Exact solutions of Fisher equations with time-dependent coefficients
The equivalence method and the method of mapping between classes of differential equations proposed in [O.Vaneeva et al. Acta Appl. Math. 106 (2009), 1-46] are used for construction of exact solutions for Fisher equations with time-dependent coefficients.  
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/366 |
| 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_ | 1872552862379147264 |
|---|---|
| author | Vaneeva, O. Ванєєва, О. |
| author_facet | Vaneeva, O. Ванєєва, О. |
| author_institution_txt_mv | [
{
"author": "О. Ванєєва",
"institution": "Інститут математики НАН України"
}
] |
| author_sort | Vaneeva, O. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | The equivalence method and the method of mapping between classes of differential equations proposed in [O.Vaneeva et al. Acta Appl. Math. 106 (2009), 1-46] are used for construction of exact solutions for Fisher equations with time-dependent coefficients.
  |
| first_indexed | 2026-08-04T01:06:10Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 44�49
ÓÄÊ 517.958
Òî÷íi ðîçâ'ÿçêè ðiâíÿíü Ôiøåðà
ç êîåôiöi¹íòàìè, ùî çàëåæàòü
âiä ÷àñîâî¨ çìiííî¨
Î.Î. Âàí¹¹âà
Iíñòèòóò ìàòåìàòèêè ÍÀÍ Óêðà¨íè, Êè¨â
E-mail: vaneeva@imath.kiev.ua
Ìåòîä åêâiâàëåíòíîñòi, à òàêîæ ìåòîä ïåðåòâîðåíü ìiæ êëàñàìè äèôå-
ðåíöiàëüíèõ ðiâíÿíü, çàïðîïîíîâàíèé ó ðîáîòi [O. Vaneeva et al. Acta
Appl. Math. 106 (2009), 1�46], çàñòîñîâàíî äëÿ ïîáóäîâè òî÷íèõ ðîçâ'ÿç-
êiâ ðiâíÿíü Ôiøåðà ç êîåôiöi¹íòàìè, ùî çàëåæàòü âiä ÷àñîâî¨ çìiííî¨.
The equivalence method and the method of mapping between classes of
di�erential equations proposed in [O. Vaneeva et al. Acta Appl. Math. 106
(2009), 1�46] are used for construction of exact solutions for Fisher equa-
tions with time-dependent coe�cients.
Ðiâíÿííÿ Ôiøåðà,
ut = kuxx +mu(1− u), km 6= 0, (1)
çàïðîïîíîâàíå Ð.Å. Ôiøåðîì ó 1937 ðîöi [5], ¹ êëàñè÷íîþ äåòåðìiíiñ-
òè÷íîþ ìîäåëëþ ïîïóëÿöiéíî¨ ãåíåòèêè, ùî îïèñó¹ äèíàìiêó ÷àñòîòè
ïîÿâè ìóòàíòíîãî ãåíó ó ïîïóëÿöi¨, ÿêèé âîëîäi¹ ñåëåêòèâíîþ ïåðå-
âàãîþ. Çàëåæíà çìiííà u � ÷àñòîòà ïîÿâè ìóòàíòíîãî ãåíó ó ïîïóëÿ-
öi¨, ùî îäíîðiäíî ðîçòàøîâàíà ó ëiíiéíîìó ñåðåäîâèùi ïðîæèâàííÿ,
íàïðèêëàä, íà áåðåãîâié ëiíi¨, ñòàëà m � iíòåíñèâíiñòü ñåëåêöi¨ íà
ïåðåâàãó ìóòàíòíîãî ãåíó, k � êîåôiöi¹íò äèôóçi¨. Ìàêñèìàëüíà àë-
ãåáðà ëi¨âñüêî¨ iíâàðiàíòíîñòi ðiâíÿííÿ (1) ¹ äâîâèìiðíîþ. Áàçèñíè-
ìè îïåðàòîðàìè öi¹¨ àëãåáðè ¹ îïåðàòîðè çñóâiâ çà ÷àñîâîþ òà ïðî-
ñòîðîâîþ çìiííîþ ∂t òà ∂x, ùî äîçâîëÿ¹ ïîáóäóâàòè äëÿ öüîãî ðiâ-
íÿííÿ ðîçâ'ÿçêè òèïó áiæó÷î¨ õâèëi. Òàêi ðîçâ'ÿçêè áóëî ïîáóäîâàíî
ó ðîáîòàõ [1, 3, 4, 8, 9]. Òåîðåìè iñíóâàííÿ òà ¹äèíîñòi îáìåæåíèõ
ðîçâ'ÿçêiâ áiëüø çàãàëüíîãî êëàñó ðiâíÿíü ut = uxx +F (t, x, u) äîâå-
äåíî À.Ì. Êîìîãîðîâèì, I.Ã. Ïåòðîâñüêèì òà Ì.Ñ. Ïiñêóíîâèì [7].
Òî÷íi ðîçâ'ÿçêè ðiâíÿíü Ôiøåðà çi çìiííèìè êîåôiöi¹íàìè 45
Ïiçíiøå áóëî çàïðîïîíîâàíî ðîçãëÿíóòè óçàãàëüíåíó ìîäåëü âè-
ãëÿäó
ut = g(t)uxx + f(t)u(1− u), gf 6= 0, (2)
äå äèôóçiéíèé êîåôiöi¹íò g i êîåôiöi¹íò ñåëåêòèâíî¨ ïåðåâàãè f çàëå-
æàòü âiä ÷àñîâî¨ çìiííî¨ [6, 11]. Çàâäÿêè òàêèì êîåôiöi¹íòàì ìîæíà
âçÿòè äî óâàãè âïëèâ äîâãîòåðìiíîâî¨ çìiíè êëiìàòó àáî êîðîòêî-
ñòðîêîâî¨ ñåçîííîñòi.
Ãðóïîâó êëàñèôiêàöiþ ðiâíÿíü (2) áóëî âèêîíàíî ó ðîáîòi [17],
îäíàê çàäà÷à ïîøóêó òî÷íèõ ðîçâ'ÿçêiâ òàêèõ ðiâíÿíü òàì íå ðîç-
ãëÿäàëàñÿ. Ó öié ðîáîòi äëÿ ïîáóäîâè òî÷íèõ ðîçâ'ÿçêiâ ðiâíÿíü Ôi-
øåðà çi çìiííèìè êîåôiöi¹íòàìè çàñòîñîâàíî ìåòîäè, ùî áàçóþòüñÿ
íà âèêîðèñòàííi íåâèðîäæåíèõ òî÷êîâèõ ïåðåòâîðåíü, à ñàìå ìåòîä
åêâiâàëåíòíîñòi òà ìåòîä ïåðåòâîðåíü ìiæ êëàñàìè äèôåðåíöiàëüíèõ
ðiâíÿíü. Ó ðåçóëüòàòi ïîáóäîâàíî äåêiëüêà ñiìåé òî÷íèõ ðîçâ'ÿçêiâ
äëÿ ïåâíèõ ïiäêëàñiâ êëàñó (2).
Ìåòîä åêâiâàëåíòíîñòi. Ïiä ìåòîäîì åêâiâàëåíòíîñòi äëÿ ïî-
áóäîâè òî÷íèõ ðîçâ'ÿçêiâ ìè ðîçóìi¹ìî âèêîðèñòàííÿ íåâèðîäæåíèõ
òî÷êîâèõ ïåðåòâîðåíü ç ãðóïè åêâiâàëåíòíîñòi çàäàíîãî êëàñó òà òî÷-
íèõ ðîçâ'ÿçêiâ, ùî ¹ âiäîìèìè äëÿ äåÿêèõ ðiâíÿíü ç öüîãî êëàñó.
ßêùî äâà ðiâíÿííÿ ïîâ'ÿçàíi ìiæ ñîáîþ íåâèðîäæåíèì òî÷êîâèì
ïåðåòâîðåííÿì, òî, çà òåðìiíîëîãi¹þ Ë.Â. Îâñÿííiêîâà, âîíè íàçèâà-
þòüñÿ ïîäiáíèìè [10]. Òîäi ïîäiáíèìè âiäíîñíî öüîãî æ ïåðåòâîðåííÿ
¹ i âiäïîâiäíi íàáîðè òî÷íèõ ðîçâ'ÿçêiâ, ñèìåòðié, çàêîíiâ çáåðåæåííÿ
öèõ ðiâíÿíü. Äëÿ êëàñiâ çi çìiííèìè êîåôiöi¹íòàìè íàéáiëüø åôåê-
òèâíå âèêîðèñòàííÿ ìåòîäó åêâiâàëåíòíîñòi ïîëÿã๠ó çâåäåííi ïåâ-
íîãî ðiâíÿííÿ çi çìiííèìè êîåôiöi¹íòàìè ç äîñëiäæóâàíîãî êëàñó äî
ðiâíÿííÿ çi ñòàëèìè êîåôiöi¹íòàìè ç òîãî æ êëàñó. Íàñòóïíèì êðî-
êîì ¹ ïîáóäîâà òî÷íèõ ðîçâ'ÿçêiâ äëÿ ïåðøîãî ç öèõ ðiâíÿíü øëÿõîì
ðîçìíîæåííÿ âiäîìèõ ðîçâ'ÿçêiâ äðóãîãî ðiâíÿííÿ ïåðåòâîðåííÿìè
åêâiâàëåíòíîñòi.
Ó ðîáîòi [17] îòðèìàíî êðèòåðié çâiäíîñòi ðiâíÿíü çi çìiííèìè
êîåôiöi¹íòàìè ç êëàñó (2) äî ðiâíÿííÿ Ôiøåðà çi ñòàëèìè êîåôiöi¹í-
òàìè (1). Ðiâíÿííÿ ç êëàñó (2) ìîæíà çâåñòè äî ðiâíÿííÿ âèãëÿäó (1)
òîäi i òiëüêè òîäi, êîëè äëÿ äåÿêî¨ äîäàòíî¨ ñòàëî¨ λ êîåôiöi¹íòè f
i g çàäîâîëüíÿþòü óìîâó
λg2 − 2
gtt
g
+ 3
g2
t
g2
= f2 − 2
ftt
f
+ 3
f2
t
f2
. (3)
46 Î.Î. Âàí¹¹âà
Óìîâà (3) âèêîíó¹òüñÿ òîäi i òiëüêè òîäi, êîëè ôóíêöiþ g ìîæíà
âèðàçèòè ÷åðåç ôóíêöiþ f çà ôîðìóëîþ:
g(t) =
λ∆f(t)e
∫
f(t) dt(
αe
∫
f(t) dt + β
)(
γe
∫
f(t) dt + δ
) ,
äå λ � äîäàòíà ñòàëà, à ïàðè ñòàëèõ (α, β) i (γ, δ) âèçíà÷åíî ç òî÷íiñ-
òþ äî íåíóëüîâîãî ñòàëîãî ìíîæíèêà, ïðè öüîìó ∆ = αδ − βγ 6= 0.
Äëÿ êîìïàêòíîñòi çàïèñó ââåäåìî ïîçíà÷åííÿ h(t) = e
∫
f(t) dt.
Îòæå, êëàñ ðiâíÿíü Ôiøåðà çi çìiííèìè êîåôiöi¹íòàìè âèãëÿäó
ut =
λ∆f(t)h(t)
(αh(t) + β)(γh(t) + δ)
uxx + f(t)u(1− u), (4)
ùî ¹ ïiäêëàñîì êëàñó (2), çâîäèòüñÿ òî÷êîâèìè ïåðåòâîðåííÿìè äî
êëàñè÷íîãî ðiâíÿííÿ Ôiøåðà çi ñòàëèìè êîåôiöi¹íòàìè,
ut = uxx + u(1− u). (5)
Äëÿ òîãî, ùîá çíàéòè òî÷êîâi ïåðåòâîðåííÿ, ùî ðåàëiçóþòü ïî-
äiáíiñòü ðiâíÿíü (4) òà (5), çíàéäåìî ñïî÷àòêó ãðóïó åêâiâàëåíòíîñòi.
Òåîðåìà 1. Ðåïàðàìåòðèçîâàíèé êëàñ (2) ç íîâèì äîâiëüíèì åëå-
ìåíòîì h(t), ùî çàäîâîëüíÿ¹ ðiâíÿííÿ ht = fh, ¹ íîðìàëiçîâàíèì
âiäíîñíî ñâ óçàãàëüíåíî¨ ãðóïè åêâiâàëåíòíîñòi Ĝ∼. Ãðóïà Ĝ∼
ñêëàäà¹òüñÿ ç ïåðåòâîðåíü
t̃ = T (t), x̃ = δ1x+ δ2, ũ =
(αh+ β)(γh+ δ)
h∆
u− γ αh+ β
∆
,
f̃ =
h∆
Tt(αh+ β)(γh+ δ)
f, g̃ =
δ1
2
Tt
g, h̃ =
αh+ β
γh+ δ
,
äå T (t) � äîâiëüíà ãëàäêà ôóíêöiÿ, ùî çàäîâîëüíÿ¹ óìîâó Tt 6= 0,
δ1 i δ2 � äîâiëüíi ñòàëi, ïðè÷îìó δ1 6= 0, ïàðè ñòàëèõ (α, β) i (γ, δ)
¹ âèçíà÷åíèìè ç òî÷íiñòþ äî íåíóëüîâîãî ñòàëîãî ìíîæíèêà i ∆ =
αδ − βγ 6= 0.
Óçàãàëüíåíà ãðóïà åêâiâàëåíòíîñòi Ĝ∼ äëÿ ðåïàðàìåòðèçîâàíî-
ãî êëàñó (2), íàáið äîâiëüíèõ åëåìåíòiâ ÿêîãî ôîðìàëüíî ìiñòèòü
ôóíêöiþ h(t), ¹ ðîçøèðåíîþ óçàãàëüíåíîþ ãðóïîþ åêâiâàëåíòíîñòi
Òî÷íi ðîçâ'ÿçêè ðiâíÿíü Ôiøåðà çi çìiííèìè êîåôiöi¹íàìè 47
äëÿ âèõiäíîãî êëàñó (2). Îçíà÷åííÿ óçàãàëüíåíî¨ òà ðîçøèðåíî¨ óçà-
ãàëüíåíî¨ ãðóï åêâiâàëåíòíîñòi i íîðìàëiçîâàíîñòi êëàñó íàâåäåíî,
çîêðåìà, ó [13, 14].
Ç òåîðåìè 1 çíàõîäèìî ïåðåòâîðåííÿ, ùî âiäîáðàæàþòü ðiâíÿí-
íÿ (4) ó ðiâíÿííÿ (5). Òàêi ïåðåòâîðåííÿ ìàþòü âèãëÿä
t̃ = ln
αh(t) + β
γh(t) + δ
+ c1, x̃ =
x√
λ
+ c2, (6)
ũ =
(αh(t) + β)(γh(t) + δ)
h(t)∆
u− γ αh(t) + β
∆
,
äå c1, c2 � äîâiëüíi ñòàëi. Ç äîïîìîãîþ öèõ ïåðåòâîðåíü îòðèìó¹-
ìî ðîçâ'ÿçêè ðiâíÿííÿ (4) ç âiäîìèõ ðîçâ'ÿçêiâ êëàñè÷íîãî ðiâíÿííÿ
Ôiøåðà (5). Ïîáóäîâàíî ñiì'þ òî÷íèõ ðîçâ'ÿçêiâ ðiâíÿíü (4):
u =
h∆ exp
(
5
3 t̃+
√
6
3 x̃
)
℘
(
exp
(
5
6 t̃+
√
6
6 x̃
)
+ C̃, 0, Ĉ
)
(αh+ β)(γh+ δ)
+
γh
γh+ δ
,
÷àñòèííèé âèïàäêîì ÿêî¨ â åëåìåíòàðíèõ ôóíêöiÿõ ¹
u =
h∆
(αh+ β)(γh+ δ)
1(
C exp
(√
6
6 x̃−
5
6 t̃
)
± 1
)2 +
γh
γh+ δ
.
 îòðèìàíèõ ðîçâ'ÿçêàõ t̃ òà x̃ âèçíà÷åíî ó (6), ℘(z, k1, k2) � åëiï-
òè÷íà ôóíêöiÿ Âåé¹ðøòðàñà, c1, c2, C, C̃, Ĉ � äîâiëüíi ñòàëi, C 6= 0.
Îñêiëüêè ðiâíÿííÿ Ôiøåðà äîïóñêàþòü äèñêðåòíå ïåðåòâîðåííÿ
ñèìåòði¨ x 7→ −x, âñi îòðèìàíi ðîçâ'ÿçêè ç ïðîòèëåæíèìè çíàêàìè x
òàêîæ çàäîâîëüíÿþòü ðiâíÿííÿ (4). Ùå îäíå ïåðåòâîðåííÿ ñèìåòði¨
u 7→ 1−u òàêîæ äîçâîëÿ¹ äîäàòêîâî ðîçìíîæèòè çíàéäåíi ðîçâ'ÿçêè.
Ìåòîä ïåðåòâîðåíü ìiæ êëàñàìè äèôåðåíöiàëüíèõ ðiâ-
íÿíü. Îêðiì ïåðåòâîðåíü åêâiâàëåíòíîñòi, ùî íå çìiíþþòü ñòðóêòó-
ðó êëàñó äèôåðåíöiàëüíèõ ðiâíÿíü, à ëèøå ïåðåâîäÿòü îäíå ðiâíÿííÿ
ç êëàñó â iíøå ðiâíÿííÿ ç öüîãî æ êëàñó, ìîæëèâî òàêîæ ðîçãëÿíó-
òè íåâèðîäæåíi òî÷êîâi ïåðåòâîðåííÿ ìiæ êëàñàìè äèôåðåíöiàëüíèõ
ðiâíÿíü. Öåé ìåòîä áóëî çàïðîïîíîâàíî ó ðîáîòi [16] äëÿ âèêîíàííÿ
ãðóïîâî¨ êëàñèôiêàöi¨ êâàçiëiíiéíèõ ðiâíÿíü ðåàêöi¨-äèôóçi¨ çi çìií-
íèìè êîåôiöi¹íòàìè òà ñòåïåíåâîþ íåëiíiéíiñòþ. Ïiçíiøå öèì ìåòî-
äîì áóëî äîñëiäæåíî ç ñèìåòðiéíî¨ òî÷êè çîðó é iíøi êëàñè ðiâíÿíü
48 Î.Î. Âàí¹¹âà
(äèâ., [15], à òàêîæ [18] òà íàâåäåíi òàì ïîñèëàííÿ). Ó öié ðîáîòi ìå-
òîä ïåðåòâîðåíü ìiæ êëàñàìè äèôåðåíöiàëüíèõ ðiâíÿíü çàñòîñîâàíî
äëÿ ïîáóäîâè òî÷íèõ ðîçâ'ÿçêiâ.
Äîâåäåíî, ùî ñiì'ÿ òî÷êîâèõ ïåðåòâîðåíü, ïàðàìåòðèçîâàíèõ äî-
âiëüíèì åëåìåíòîì f(t) êëàñó (2),
t̃ =
∫
f(t)e
∫
f(t)dtdt, x̃ = x, ũ = −e−
∫
f(t)dtu, (7)
âiäîáðàæ๠êëàñ (2) ó êëàñ êâàçiëiíiéíèõ ðiâíÿíü ðåàêöi¨�äèôóçi¨
ç êâàäðàòè÷íîþ íåëiíiéíiñòþ òà îäíèì äîâiëüíèì åëåìåíòîì, ùî çà-
ëåæèòü âiä çìiííî¨ ÷àñó:
ũt̃ = g̃(t̃)ũx̃x̃ + ũ2, g̃ 6= 0. (8)
Äîâiëüíi åëåìåíòè êëàñiâ (2) òà (8) ïîâ'ÿçàíi ôîðìóëîþ
g̃ =
g(t)
f(t)
e−
∫
f(t)dt.
Äëÿ ðiâíÿííÿ ut = uxx + u2 âiäîìi äåêiëüêà òî÷íèõ ðîçâ'ÿçêiâ
(äèâ. [2, 9] òà [12, ñ. 157]). Âèêîðèñòîâóþ÷è ¨õ òà ïåðåòâîðåííÿ (7),
çíàõîäèìî íîâi òî÷íi ðîçâ'ÿçêè ðiâíÿííÿ Ôiøåðà çi çìiííèìè êîåôi-
öi¹íòàìè
ut = f(t)e
∫
f(t)dtuxx + f(t)u(1− u) :
u =
12
(
4±
√
6
)
x(x+ c1) + 120(12± 5
√
6)Θ + 12
(
2±
√
6
)
c2 + 6c21
e−
∫
f(t)dt
(
x2 + c1x+ 10(3±
√
6)Θ + c2
)2 ,
u = e
∫
f(t)dt℘
(
x√
6
, 0, Ĉ
)
,
äå Θ =
∫
f(t)e
∫
f(t)dtdt, c1, c2, Ĉ � äîâiëüíi ñòàëi.
Àâòîðêà âäÿ÷íà ïðîôåñîðó Ð.Î. Ïîïîâè÷ó çà öiííi ïîðàäè.
[1] Ablowitz M.J., Zeppetella A., Explicit solutions of Fisher's equation for a special
wave speed, Bull. Math. Biology 41 (1979), 835�840.
[2] Barannyk T., Symmetry and exact solutions for systems of nonlinear
reaction�di�usion equations, in Proceedings of Fourth International Conference
�Symmetry in Nonlinear Mathematical Physics� (9�15 July, 2001, Kyiv),
Proceedings of Institute of Mathematics 43 (2002), Part 1, 184�193.
Òî÷íi ðîçâ'ÿçêè ðiâíÿíü Ôiøåðà çi çìiííèìè êîåôiöi¹íàìè 49
[3] Cariello F., Tabor M., Similarity reductions from extended Painlev�e expansions
for nonintegrable evolution equations, Phys. D 53 (1991), 59�70.
[4] Danilov V.G., Subochev P.Yu., Wave solutions of semilinear parabolic equa-
tions, Theoret. and Math. Phys. 89 (1991), 1029�1046.
[5] Fisher R.A., The wave of advance of advantageous genes, Ann. Eugenics 7
(1937), 353�369.
[6] Hammond J.F., Bortz D.M., Analytical solutions to Fisher's equation with time-
variable coe�cients, Appl. Math. Comput. 218 (2011), 2497�2508.
[7] Êîëìîãîðîâ À. Í., Ïåòðîâñêèé È. Ã., Ïèñêóíîâ Í. Ñ., Èññëåäîâàíèå óðàâ-
íåíèÿ äèôôóçèè, ñîåäèíåííîé ñ âîçðàñòàíèåì âåùåñòâà, è åãî ïðèìåíåíèå
ê îäíîé áèîëîãè÷åñêîé ïðîáëåìå, Áþëëåòåíü ÌÃÓ Ñåð. À ìàò. ìåõ., 1
(1937), no. 6, 1�26.
[8] Kudryashov N.A., Exact solutions of a family of Fisher equations, Theoret. and
Math. Phys. 94 (1993), 211�218.
[9] Nikitin A.G., Barannyk T.A., Solitary wave and other solutions for nonlinear
heat equations, Cent. Eur. J. Math. 2 (2005), 840�858, arXiv:math-ph/0303004.
[10] Îâñÿííèêîâ Ë.Â. Ãðóïïîâîé àíàëèç äèôôåðåíöèàëüíûõ óðàâíåíèé, Ìîñê-
âà, Íàóêà, 1978.
[11] �O�g�un A., Kart C., Exact solutions of Fisher and generalized Fisher equations
with variable coe�cients, Acta Math. Sin. (Engl. Ser.) 23 (2007), 563�568.
[12] Polyanin A.D., Zaitsev V.F., Handbook of nonlinear partial di�erential equa-
tions, 2nd ed., Chapman & Hall/CRC, Boca Raton, FL, 2012.
[13] Popovych R.O., Bihlo A., Symmetry preserving parameterization schemes,
J. Math. Phys. 53 (2012), 073102, 36 pp., arXiv:1010.3010.
[14] Popovych R.O., Kunzinger M., Eshraghi H., Admissible transformations and
normalized classes of nonlinear Schr�odinger equations, Acta Appl. Math. 109
(2010), 315�359, arXiv:math-ph/0611061.
[15] Vaneeva O.O., Group classi�cation via mapping between classes: an example of
semilinear reaction-di�usion equations with exponential nonlinearity, in Proc. of
the 5th Math. Phys. Meeting: Summer School and Conf. on Modern Mathemati-
cal Physics (Belgrade, Serbia, 2008), Belgrade, 2009, 463�471, arXiv:0811.2587.
[16] Vaneeva O.O., Popovych R.O., Sophocleous C., Enhanced group analysis and
exact solutions of variable coe�cient semilinear di�usion equations with a power
source, Acta Appl. Math. 106 (2009), 1�46, arXiv:0708.3457.
[17] Vaneeva O.O., Popovych R.O., Sophocleous C., Group classi�cation of the Fi-
sher equation with time-dependent coe�cients, in Proceedings of 6th Interna-
tional Workshop �Group Analysis of Di�erential Equations and Integrable
Systems� (Protaras, Cyprus, June 17�21, 2012), University of Cyprus, Nicosia,
2013, 225�237.
[18] Vaneeva O.O., Po�sta S., Sophocleous C., Enhanced group classi�cation of
Benjamin�Bona�Mahony�Burgers equations, Appl. Math. Lett. 65 (2017), 19�
25.
|
| id | oai:trim.imath.kiev.ua:article-366 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-08-04T01:06:10Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/ce/38538ed5541d76c5f900be186c502fce.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3662020-08-13T08:52:53Z Exact solutions of Fisher equations with time-dependent coefficients Точні розв'язки рівнянь Фішера з коефіцієнтами, що залежать від часової змінної Vaneeva, O. Ванєєва, О. The equivalence method and the method of mapping between classes of differential equations proposed in [O.Vaneeva et al. Acta Appl. Math. 106 (2009), 1-46] are used for construction of exact solutions for Fisher equations with time-dependent coefficients.   Метод еквівалентності, а також метод перетворень між класами диференціальних рівнянь, запропонований у роботі [O.Vaneeva et al. Acta Appl. Math. 106 (2009), 1-46], застосовано для побудови точних розв'язків рівнянь Фішера з коефіцієнтами, що залежать від часової змінної. Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/366 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 44-49 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 44-49 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 44-49 3083-7529 1815-2910 uk https://trim.imath.kiev.ua/index.php/trim/article/view/366/361 Авторське право (c) 2019 О. Ванєєва |
| spellingShingle | Vaneeva, O. Ванєєва, О. Exact solutions of Fisher equations with time-dependent coefficients |
| title | Exact solutions of Fisher equations with time-dependent coefficients |
| title_alt | Точні розв'язки рівнянь Фішера з коефіцієнтами, що залежать від часової змінної |
| title_full | Exact solutions of Fisher equations with time-dependent coefficients |
| title_fullStr | Exact solutions of Fisher equations with time-dependent coefficients |
| title_full_unstemmed | Exact solutions of Fisher equations with time-dependent coefficients |
| title_short | Exact solutions of Fisher equations with time-dependent coefficients |
| title_sort | exact solutions of fisher equations with time-dependent coefficients |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/366 |
| work_keys_str_mv | AT vaneevao exactsolutionsoffisherequationswithtimedependentcoefficients AT vanêêvao exactsolutionsoffisherequationswithtimedependentcoefficients AT vaneevao točnírozv039âzkirívnânʹfíšerazkoefícíêntamiŝozaležatʹvídčasovoízmínnoí AT vanêêvao točnírozv039âzkirívnânʹfíšerazkoefícíêntamiŝozaležatʹvídčasovoízmínnoí |