Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form
We use the functional-discrete method for the solution of the Strum-Liouville problem with coefficients of a special form and obtain the estimates of accuracy. The numerical experiment is performed by using the Maple-10 software package.
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| Date: | 2007 |
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| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509456430792704 |
|---|---|
| author | Klymenko, Ya. V. Makarov, V. L. Клименко, Я. В. Макаров, В. Л. |
| author_facet | Klymenko, Ya. V. Makarov, V. L. Клименко, Я. В. Макаров, В. Л. |
| author_sort | Klymenko, Ya. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:34Z |
| description | We use the functional-discrete method for the solution of the Strum-Liouville problem with coefficients of a special form and obtain the estimates of accuracy. The numerical experiment is performed by using the Maple-10 software package. |
| first_indexed | 2026-03-24T02:41:23Z |
| format | Article |
| fulltext |
UDK 517.983.27
V. L. Makarov, Q. V. Klymenko (In-t matematyky NAN Ukra]ny, Ky]v)
ZASTOSUVANNQ FD-METODU DO ROZV'QZANNQ ZADAÇI
ÍTURMA – LIUVILLQ Z KOEFICI{NTAMY
SPECIAL|NOHO VYHLQDU
The functional-discrete (FD-) method is applied to the solution of the Sturm – Liouville problem with
coefficients of special form and estimates of exactness are obtained. A numerical experiment is carried
out with the use of Maple-10.
Funkcyonal\no-dyskretn¥j (FD-) metod prymenen k reßenyg zadaçy Íturma – Lyuvyllq s ko-
πffycyentamy specyal\noho vyda y poluçen¥ ocenky toçnosty. Proveden çyslenn¥j πkspery-
ment s pomow\g paketa Maple-10.
1. Vstup. U roboti rozhlqnuto zastosuvannq FD-metodu do zadaçi Íturma –
Liuvillq z osoblyvistg (koefici[nt pry poxidnij druhoho porqdku vyrodΩu[t\-
sq na kincqx intervalu), wo [ rozpovsgdΩennqm rezul\tativ z [1, 2] na novyj
klas zadaç na vlasni znaçennq.
2. Postanovka zadaçi. Rozhlqnemo zadaçu Íturma – Liuvillq
( ) ( ) ( ) ( ( )) ( )1 2 02− ′′ − ′ + − =z u z zu z q z u zλ , z ∈(– , )1 1 ,
(1)
u( )− < ∞1 , u( )1 < ∞ ,
de q z( ) — polinom stepenq N – 1. Qkwo funkciq q z( ) ne [ polinomom, to ]]
spoçatku nablyΩa[mo z toçnistg, z qkog my xoçemo oderΩaty rozv’qzok vyxid-
no] zadaçi (1), polinomom ˜( )q z , a potim rozv’qzu[mo zadaçu
( ) ˜ ( ) ˜ ( ) ˜ ˜( ) ˜( )1 2 02− ′′ − ′ + −( ) =z u z zu z q z u zλ , z ∈(– , )1 1 ,
˜( )u − < ∞1 , ˜( )u 1 < ∞ .
Vvedemo poxybku
w z u z u z( ) ( ) ˜( )= − .
Dlq ne] ma[mo zadaçu
( ) ( ) ( ) ˜ ( ) ( )1 22− ′′ − ′ + −( )z w z zw z q z w zλ = − −( ) + −( )λ λq z u z q z u z( ) ˜( ) ˜ ˜( ) ˜( ),
w( )± < ∞1 .
Umova rozv’qznosti ostann\o] zadaçi pryvodyt\ do spivvidnoßennq
˜
( ) ˜( ) ˜( ) ( )
˜( ) ( )
λ λ− =
−( )∫
∫
0
1
0
1
q z q z u z u z dz
u z u z dz
.
Dali pokazu[mo, wo pry ˜( )q z → q z( ) rozv’qzok ˜( )u z → u z( ) i
0
1
0∫ ≥ >˜( ) ( )u z u z dz α .
Pislq c\oho oderΩu[mo ocinku
© V. L. MAKAROV, Q. V. KLYMENKO, 2007
1140 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
ZASTOSUVANNQ FD-METODU DO ROZV’QZANNQ ZADAÇI ÍTURMA – LIUVILLQ … 1141
˜ max ( ) ˜( )λ λ
α
− ≤ −q z q z
(2)
pry umovi normuvannq ˜( )u z ta u z( ).
Qkwo funkcig q z( ) nablyzyty interpolqcijnym polinomom LahranΩa
L zN −1( ) , koly interpolqcijni vuzly zbihagt\sq z nulqmy polinoma Çebyßova
perßoho rodu T xN ( ), xk N, = cos
( / )k
N
− 1 2 π
, k = 1, 2, … , N, to budemo maty
ocinku
max ( ) ( ) ln ( )q z L z C N E qN N− ≤ ⋅− −1 1 . (3)
Tut E qN −1( ) — poxybka najkrawoho nablyΩennq funkci] q z( ) polinomamy ne
vywe (N – 1)-ho stepenq, a mnoΩnyk ln N vynyka[ za raxunok stalo] Lebeha
(dyv. [3, 4]).
Ocinky (2), (3) dozvolqgt\ pobuduvaty polinom ˜( )q z = L zN −1( ) iz zadanog
toçnistg.
3. Zastosuvannq FD-metodu. Dlq pobudovy rozv’qzku zadaçi (1) „zanurg-
[mo” cg zadaçu v bil\ß zahal\nu
( ) ( , ) ( , ) ( ( ) ( )) ( , )1 2 02− ′′ − ′ + − =z u z t zu z t t t q z u z tz z λ , (4)
u t( , )±1 < ∞.
Oçevydno, wo
u z u z( , ) ( )1 = , λ λ( )1 = ,
u z CP zn n( , ) ( )0 = , λn n n( ) ( )0 1= + ,
de P zn( ) — polinomy LeΩandra [5].
Rozv’qzok zadaçi budemo ßukaty u vyhlqdi
u z t t u zn
j
j
n
j( , ) ( )( )=
=
∞
∑
0
, (5)
λ λn
j
j
n
jt t( ) ( )=
=
∞
∑
0
. (6)
Qkwo radius zbiΩnosti rqdiv (5), (6) R > 1, to rqd (5) moΩna poçlenno dyfe-
rencigvaty pry t R< (dyv., napryklad, [6, s.M361]).
Pidstavlqgçy rqdy (5), (6) u (4) i pryrivnggçy koefici[nty pry odnakovyx
stepenqx t, otrymu[mo rekurentnu poslidovnist\ rivnqn\, de liva çastyna zaly-
ßa[t\sq odni[g i ti[g Ω, a prava zming[t\sq vidpovidnym çynom:
( )
( )( )
1 2
2 1
2−
+
z
d u z
dz
n
j
– 2
1
z
du z
dz
n
j( )( )+
+ n n u zn
j( ) ( )( )+ +1 1 =
= –
p
j
n
j p
n
pu z
=
+ −∑
0
1λ( ) ( )( ) + q z u zn
j( ) ( )( ) , j = 0, 1, … , (7)
un
j( )( )+ ± < ∞1 1 ,
λn n n( ) ( )0 1= + .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1142 V. L. MAKAROV, Q. V. KLYMENKO
Tut u zn
( )( )0
— rozv’qzok bazovo] zadaçi
( ) ( ) ( ) ( ) ( )( ) ( ) ( )1 2 1 02 0 0 0− ( )′′ − ( )′ + + =z u z z u z n n u zn n n , z ∈ −( , )1 1 ,
un
( )( )0 1± < ∞ ,
qkyj ma[ vyhlqd
u z C P zn n
( )( ) ( )0
0= .
Stalu C0 vyberemo z umovy normuvannq
−
∫ ( ) =
1
1
0
2 1C P z dzn( ) ,
wo pryvodyt\ do formuly
C
n
n0
2 1= +
.
Nevidomi λn
j( )+1
, j = 0, 1, … , budemo ßukaty z umovy rozv’qznosti rivnqn\ (7),
tobto z umovy ortohonal\nosti pravo] çastyny do rozv’qzku bazovo] zadaçi
u zn
( )( )0
ta dodatkovo] umovy
−
+∫ 1
1 1 0u z u z dzn
j
n
( ) ( )( ) ( ) = 0. Todi oderΩu[mo
λn
j( )+1 =
−
∫
1
1
0q z u z u z dzn
j
n( ) ( ) ( )( ) ( ) , j = 0, 1, … .
Dlq koΩnoho j = 0, 1, … rozv’qzok u zn
j( )( )+1
rivnqnnq (7) ßuka[mo u vyhlqdi
u zn
j( )( )+1 = C P z w zj n n
j
( )
( )( ) ( )+
++1
1
,
de w zn
j( )( )+1
— çastynnyj rozv’qzok neodnoridnoho rivnqnnq (7), qkyj u svog
çerhu budemo ßukaty u vyhlqdi rozkladu za polinomamy LeΩandra P zp( ) :
w zn
j( )( )+1 =
p
n N j
p pP z
=
+ +
∑
0
1( )
( )β , (8)
P zp( ) — polinomy LeΩandra.
TakoΩ rozklademo pravu çastynu rivnqnnq (7) za polinomamy LeΩandra
F zn
j( )( )+1 = − +
=
+ −∑
p
j
n
j p
n
p
n
ju z q z u z
0
1λ( ) ( ) ( )( ) ( ) ( ) =
p
n N j
p pP z
=
+ +
∑
0
1( )
( )α , (9)
α p =
2 1
2
1
1
1p
F x P x dxn
j
p
+
−
+∫ ( )( ) ( ) .
Pidstavyvßy (8) u livu çastynu rivnqnnq (7), otryma[mo
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
1 2 12
0
1
0
1
0
1
− ′′ − ′ + +
=
+ +
=
+ +
=
+ +
∑ ∑ ∑z P z z P z n n P z
p
n N j
p p
p
n N j
p p
p
n N j
p pβ β β =
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
ZASTOSUVANNQ FD-METODU DO ROZV’QZANNQ ZADAÇI ÍTURMA – LIUVILLQ … 1143
=
p
n N j
p p p
p
n N j
p pz P z zP z n n P z
=
+ +
=
+ +
∑ ∑− ′′ − ′[ ] + +
0
1
2
0
1
1 2 1
( ) ( )
( ) ( ) ( ) ( ) ( )β β =
= − + + +
=
+ +
=
+ +
∑ ∑
p
n N j
p p
p
n N j
p pp p P z n n P z
0
1
0
1
1 1
( ) ( )
( ) ( ) ( ) ( )β β =
=
p
n N j
p pn n p p P z
=
+ +
∑ + − +( )
0
1
1 1
( )
( ) ( ) ( )β .
Vraxovugçy zobraΩennq (9) dlq funkci] F zn
j( )( )+1
, znaxodymo koefici[nty βp :
βp =
α p
n n p p( ) ( )+ − +1 1
, p ≠ n.
OtΩe,
u zn
j( )( )+1 = C P zj n+1 ( ) +
+
p
n N j
k
j
n
j k
n
k
n
j
p
p
p
u x q x u x P x dx
n n p p
P z
=
+ +
− =
+ −
∑
∫ ∑+ − +( )
+ − +
0
1
1
1
0
12 1
2
1 1
( )
( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( )
( )
λ
, p ≠ n.
Cj +1 my budemo vybyraty takym çynom, wob
−
+∫ 1
1 1 0u z u z dzn
j
n
( ) ( )( ) ( ) = 0. Zvidsy
oderΩu[mo Cj +1 = 0. Takym çynom, vyraz dlq u zn
j( )( )+1
bude takym:
u zn
j( )( )+1 =
=
p
n N j
k
j
n
j k
n
k
n
j
p
p
p
u x q x u x P x dx
n n p p
P z
=
+ +
− =
+ −
∑
∫ ∑+ − +( )
+ − +
0
1
1
1
0
12 1
2
1 1
( )
( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( )
( )
λ
, p ≠ n.
4. Ocinka toçnosti dlq vlasnyx znaçen\ i vlasnyx funkcij za FD-me-
todom. Znajdemo ocinku dlq çleniv rqdu u xn( , )1 = u xn( ):
u xn
j( )( )+1 =
p
n j N n
j
p
p
p
F P d
n n p p
P x
=
+ −
−
+
∑
∫+
+ − +
0
1
1
1 12 1
2
1 1
( )
( )( ) ( )
( ) ( )
( )
ξ ξ ξ
, n ≠ p, j = 0, 1, … ,
za L2-normog
u x u d( ) ( )
/
=
−
∫
1
1
2
1 2
ξ ξ .
Ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1144 V. L. MAKAROV, Q. V. KLYMENKO
u xn
j( )( )+1 2
=
p
n j N n
j
pF P d
n n p p=
+ −
−
+
∑
∫
+ − +
0
1
1
1 1
2
1 1
( )
( )( ) ( )
( ) ( )
ξ ξ ξ
≤
≤ 1
4 2
0
1
1
1
1
2
n
F P d
p
n j N
n
j
p
=
+ −
−
+∑ ∫
( )
( )( ) ( )ξ ξ ξ ≤ 1
4 2
1 2
n
F xn
j( )( )+
.
Ocinymo F xn
j( )( )+1 2
:
F xn
j( )( )+1 2
=
−
+
=
+ −∫ ∑− − +
1
1
1 0
1
1
2
λ ξ λ ξ ξ ξn
j
n
k
j
n
j k
n
k
n
ju u q u( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) =
=
−
+∫ ( ) ( )
1
1
1 2 0 2
λ ξ ξn
j
nu d( ) ( )( ) +
−
+
=
+ −∫ ∑− −
1
1
1 0
1
12λ ξ ξ ξ λ ξn
j
n n
j
k
j
n
j k
n
ku q u u( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) +
+
− =
+ −∫ ∑ −
1
1
1
1
2
k
j
n
j k
n
k
n
ju q u dλ ξ ξ ξ ξ( ) ( ) ( )( ) ( ) ( ) =
= λn
j( )+( )1 2
– 2 1 2
λn
j( )+( ) +
− =
+ −∫ ∑ −
1
1
1
1
2
k
j
n
j k
n
k
n
ju q u dλ ξ ξ ξ ξ( ) ( ) ( )( ) ( ) ( ) ≤
≤
− =
+ −∫ ∑ −
1
1
1
1
2
k
j
n
j k
n
k
n
ju q u dλ ξ ξ ξ ξ( ) ( ) ( )( ) ( ) ( ) .
Zvidsy znaxodymo
F xn
j( )( )+1 ≤
k
j
n
j k
n
ku x
=
+ −∑
1
1λ( ) ( )( ) + q x u xn
j( ) ( )( )
∞ . (10)
Z formuly dlq vyznaçennq λn
j( )+1
otrymu[mo ocinku
λn
j( )+1
≤ q x u xn
j( ) ( )( )
∞ . (11)
Vykorystovugçy nerivnist\ (11), zapysu[mo (10) u vyhlqdi
F xn
j( )( )+1 ≤ q x u x u x
k
j
n
k
n
j k( ) ( ) ( )( ) ( )
∞
=
−∑
0
.
Teper oderΩu[mo
u xn
j( )( )+1 ≤ M u x u xn
k
j
n
k
n
j k
=
−∑
0
( ) ( )( ) ( ) , M
q x
nn = ∞( )
2
. (12)
Vykona[mo zaminu
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
ZASTOSUVANNQ FD-METODU DO ROZV’QZANNQ ZADAÇI ÍTURMA – LIUVILLQ … 1145
un
j( ) = M Un
j j( ) ˜ ( ) ≤ M Un
j j( ) ( )
.
Todi nerivnist\ (12) nabere vyhlqdu
˜ ˜ ˜( ) ( ) ( )U U Uj
k
j
k j k+
=
−≤ ∑1
0
, j = 0, 1, … . (13)
NevaΩko perekonatys\, wo qkwo zamist\ nerivnosti (13) rozhlqnuty rekurentni
rivnqnnq
U U Uj
k
j
k j k( ) ( ) ( )+
=
−= ∑1
0
, j = 0, 1, … , (14)
to ]x rozv’qzok bude maΩoruvaty zverxu rozv’qzok nerivnostej (13), tobto bude
maty misce ocinka
˜ ( )U j ≤ U j( )
.
Rozv’qΩemo rivnqnnq (14) metodom tvirnyx funkcij. Poznaçymo
f z z U
j
j j( ) ( )=
=
∞
∑
0
,
domnoΩymo obydvi çastyny rivnqnnq (14) na z j +1
i pidsumu[mo vid 0 do ∞:
j
j jz U
=
∞
+ +∑
0
1 1( ) =
j
j
k
j
k j kz U U
=
∞
+
=
−∑ ∑
0
1
0
( ) ( )
, j = 0, 1, … .
Zvidsy otryma[mo kvadratne rivnqnnq vidnosno f z( ):
f z( ) – 1 = z f z( )( )2
,
rozv’qzok qkoho
f z( ) =
1 1 4
2
± − z
z
, 4 1z < .
Oskil\ky f ( )0 = 0, zalyßa[mo druhyj korin\
f z( ) =
1 1 4
2
− − z
z
, 4 1z < . (15)
Rozklademo 1 4− z v rqd v okoli z = 0:
1 4− z = 1 +
j
j
j
j
z
=
∞
∑
−
… − +
−
1
1
2
1
2
1 1
2
1
4
!
( ) = 1 –
j
j
jj
j
z
=
∞
∑ −
0
2 2 3( )!!
!
i pidstavymo cej rozklad v (15):
f z( ) =
j
j
jj
j
z
=
∞ −
−∑ −
1
1
12 2 3( )!!
!
=
j
j
jj
j
z
=
∞
∑ −
0
2 2 1( )!!
!
=
=
j
j
jj
j
z
=
∞
∑ −
0
4 2 1
2
( )!!
( )!!
, ( )!!− =1 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1146 V. L. MAKAROV, Q. V. KLYMENKO
Z inßoho boku,
f z( ) =
j
j jz U
=
∞
∑
0
( )
.
Tomu, pryrivnqvßy pravi çastyny, otryma[mo
U
j
j
j
j
( ) ( )!!
( )!!
= −4 2 1
2
.
OtΩe,
u
q
n
j
jn
j
j
( ) ( )!!
( )!!
+ ∞
+
≤
+
+
1
14
2
2 1
2 2
,
λn
j
j
q
q
n
j
j
( ) ( )!!
( )!!
+
∞
∞
+
≤
+
+
1
14
2
2 1
2 2
.
Teper my moΩemo sformulgvaty teoremu pro toçnist\ FD-metodu.
Teorema. Nexaj vykonu[t\sq umova
r
q
nn = ≤ <∞4
2
1β ,
todi FD-metod zbiha[t\sq ne povil\niße heometryçno] prohresi] iz znamennykom
rn i magt\ misce nastupni ocinky toçnosti:
u x u x
r
rn
j
n
n
j
n
j( ) ( )
( )− ≤
−∞
+
+
1
11
α , α j +1 =
( )!!
( )!!
2 1
2 2
j
j
+
+
,
λ λ αn
j
n
n
j
n
jq
r
r
− ≤
−∞ +
( )
1 1 ,
de
u x
j
n( ) =
p
j
n
pu x
=
∑
0
( )( ), λ λ
j
n
p
j
n
p=
=
∑
0
( )
.
Zaznaçymo, wo otrymani ocinky [ dvostoronnimy.
5. Çysel\nyj eksperyment. Nexaj q z( ) = z2
, todi v ocinkax toçnosti
q ∞ = 1. V tablyci v rqdkax v porqdku zrostannq navedeno j-ti utoçnennq dlq
λn
( )0 = n ( n + 1 ) pry n = 1 4, . V ostann\omu stovpçyku vkazano j-te nablyΩennq
toçnoho znaçennq λn , qke ßuka[t\sq za formulog λ
j
n =
p
j
n
p
=∑ 0
λ( )
.
n
j
0 1 2 3
2 6 0,5238095238 0,0101500917 – 0,0004760811
3 12 0,5111111111 0,0032941768 0,0000595989
4 20 0,5064935065 0,0017750507 0,0000053147
5 30 0,5042735043 0,0011298966 0,0000011652
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
ZASTOSUVANNQ FD-METODU DO ROZV’QZANNQ ZADAÇI ÍTURMA – LIUVILLQ … 1147
n
j
4 5 λ λn k n
j= =∑ 0
5 ( )
2 – 0,0000141089 0,0000024412 6,533471867
3 – 0,0000026542 –8,877561728 ⋅ 10 8− 12,51446215
4 5,040382526 ⋅ 10 7− –1,321932266 ⋅ 10 8− 20,50827436
5 5,770297106 ⋅ 10 8− 1,479911013 ⋅ 10 9− 30,50540463
6. Vysnovok. Otrymani apriorni ocinky toçnosti pokazugt\ dostatng efek-
tyvnist\ FD-metodu. TakoΩ vydno vaΩlyvu vlastyvist\ metodu pokrawennq
toçnosti z rostom porqdkovoho nomera vlasnoho znaçennq. V çysel\nomu ekspe-
rymenti ßvydkist\ zbiΩnosti [ nabahato krawog, niΩ ce harantugt\ apriorni
ocinky, do toho Ω [ praktyçna zbiΩnist\ navit\ pry n = 2, koly ]] ne harantu[
dovedena teorema.
1. Makarov V. L. O funkcyonal\no-raznostnom metode proyzvol\noho porqdka toçnosty re-
ßenyq zadaçy Íturma – Lyuvyllq s kusoçno-hladkymy koπffycyentamy // Dokl.
ANMSSSR. – 1991. – 320, # 1. – S. 34 – 39.
2. Bandyrskii B. I., Makarov V. L., Ukhanev O. L. Sufficient convergence conditions of nonclassical
asymptotic expansions for the Sturm – Liouville problem with periodic conditions // Differents.
Uravneniya. – 1999. – 35, # 3. – P. 1 – 12.
3. Babenko K. Y. Osnov¥ çyslennoho analyza. – M.: Nauka, 1986. – 744 s.
4. Sehe H. Ortohonal\n¥e polynom¥. – M.: Fyzmathyz, 1962.
5. Bejtmen H., ∏rdejy A. V¥sßye transcendentn¥e funkcyy. – M.: Nauka, 1974.
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OderΩano 26.09.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
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| id | umjimathkievua-article-3377 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:23Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b6/7f485d120175d3d7a0b550520dc084b6.pdf |
| spelling | umjimathkievua-article-33772020-03-18T19:52:34Z Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form Застосування FD-методу до розв'язання задачі Штурма – Ліувілля з коефіцієнтами спеціального вигляду Klymenko, Ya. V. Makarov, V. L. Клименко, Я. В. Макаров, В. Л. We use the functional-discrete method for the solution of the Strum-Liouville problem with coefficients of a special form and obtain the estimates of accuracy. The numerical experiment is performed by using the Maple-10 software package. Функционально-дискретный (FD-) метод применен к решению задачи Штурма - Лиувилля с коэффициентами специального вида и получены оценки точности. Проведен численный эксперимент с помощью пакета Maple-10. Institute of Mathematics, NAS of Ukraine 2007-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3377 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 8 (2007); 1140–1147 Український математичний журнал; Том 59 № 8 (2007); 1140–1147 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3377/3497 https://umj.imath.kiev.ua/index.php/umj/article/view/3377/3498 Copyright (c) 2007 Klymenko Ya. V.; Makarov V. L. |
| spellingShingle | Klymenko, Ya. V. Makarov, V. L. Клименко, Я. В. Макаров, В. Л. Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form |
| title | Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form |
| title_alt | Застосування FD-методу до розв'язання задачі Штурма – Ліувілля з коефіцієнтами спеціального вигляду |
| title_full | Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form |
| title_fullStr | Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form |
| title_full_unstemmed | Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form |
| title_short | Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form |
| title_sort | application of the fd-method to the solution of the sturm-liouville problem with coefficients of special form |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3377 |
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