APPLICATION OF FINITE INTEGRAL TRANSFORMATION METHOD TO THE SOLUTION OF MIXED PROBLEMS FOR PARABOLIC EQUATIONS WITH A CONTROL

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1 PROCEEDINGS OF IAM V7 N 8 pp94-5 APPLICATION OF FINITE INTEGRAL TRANSFORMATION METHOD TO THE SOLUTION OF MIXED PROBLEMS FOR PARABOLIC EQUATIONS WITH A CONTROL * Elmaga A Gaymov Baku Sa Univriy Az 4 Baku Azrbaijan gaymov-lmagha@ramblrru Abrac In h prn papr w u h mho of fini ingral ranformaion o h oluion of mi problm for parabolic quaion wih a conrol an wih irrgular bounary coniion Th analyic rprnaion of h oluion of h conir problm i obain For h accibiliy of a wi rang of rar on om parabolic mol problm h mho of fini ingral ranformaion i lf i a Kywor: Claic oluion fini ingral ranformaion mho conrol quaion irrgular boynary coniion AMS Subjc Claificaion: 35K 35K35 Inroucion Throughou h horical an appli imporanc a mi problm for parial iffrnial quaion rfr o on of h urgn problm of mahmaic an mahmaical phyic Som problm of lcroynamic problm of unrgroun hyromchanic nonaionary problm for mahmaical phyic quaion ar ruc o uch problm Symbolic calculu u by nginr-lcrician O Hvyi wa on of h convnin bu mahmaically no raonabl ool In h bginning of h XIX cnury Fourir ugg h mho of paraion of variabl for ingraion of om linar parial iffrnial quaion unr h givn bounary an iniial coniion Applicaion of h Fourir mho o h oluion of mi problm wih para variabl ruc o h problm of panion of an arbirary funcion from om cla in ign funcion corrponing o h pcral problm In 87 for olving mi problm wih conan cofficin Cauchy [] ugg h riu mho h nc of which i in rprnaion of an "arbirary" funcion in h form of ingral riu On of h mho for olving mi problm for parial iffrnial quaion i h mho of ingral ranformaion ha wa uccfully u by Laplac AL Cauchy [] ML Raulov [] an ohr In hi rarch ML Raulov u h ingral ranformaion ϕ = ϕ * Работа была представлена на семинаре Института Прикладной Математики

2 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION an for ϕ aum ha ϕ ning o zro a i an analyic funcion in h omain π R δ = : R arg + δ δ > 4 arg h coniion uniformly wih rpc o of horm on pag 5 of h papr [] No ha if for Rδ aifying h inqualiy R > ML Raulov rmin ϕ by formula hn hi ingral gnrally paking ivrg Bu if Ω = π R arg Raulov woul ak ϕ in 4 h form of an in omain R δ \ Ω rmin h funcion ϕ by analyic coninuaion hn h coniion of inc of uch a coninuaion houl b clarifi aiionally o ha hi aumpion b fulfill Furhrmor ML Raulov cp on pcial funcion ϕ = from p45 [] o no how h cla of funcion ϕ for which ϕ aifi hi aumpion In [] [] i i aum ha unr uiabl numbring θ j θ j ar h roo of h characriic quaion of pcral problm h following coniion ar fulfill R[ θ ] R[ θ ] R[ θn ] [ α β ] ω whr ω i om infini par of plan whrin w look for aympoic bhavior of h ym of funamnal paricular oluion of a homognou quaion corrponing o pcral problm In applicaion faibiliy of coniion ruc o h fac ha: A argumn θ j an argumn of hir iffrnc ar inpnn of [ α β ] for ampl : rrain 3 on pag 3 [] ha in our opinion i a mor rigi rrain No ha whn olving mi problm [4] w ar rric in coniraion of a paramric problm no on h whol infini par of - plan bu only in om π par in h cor arg + δ of -plan an bcau of ha w g ri of 4 rigi conrain A Thr i a wi rang of mi problm of horical an pracical imporanc ha ar no olv by h known Fourir GD Birkhoff [] Ya D Tamarkin [] MA Naimark [9 ] ML Raulov [] mho In h ca of 95

3 PROCEEDINGS OF IAM V7 N 8 irrgular bounary coniion panion in ign an aocia funcion ha a numbr of pcific faur Th auhor rcrch how ha whn olving h problm i i no obligaory o u Brikhoff- Tamarkin-Naimark-Raulov' panion formula In hi papr w ugg h mho of fini ingral ranformaion ha ami o fin h oluion of irrgular mi problm unr mor gnral bounary coniion an wakr conrain on h problm aa Fini ingral ranformaion L f b a compl ω a ral funcion of h ral argumn T T f ω f ω L T i om poiiv numbr an [ ] ~ Dfiniion W call h funcion f whr W hav: ~ f ~ f + = ω p ω η η f T ± rmin by h formula = ω p ω η η f [ T ] i a compl numbr h imag of h funcion f Thorm E L C T ] L [ T ] ω ω η η > < T f b boun an coninuou cp numrabl numbr of poin a which i may hav iconinuiy of fir kin wih rpc o} [ T ] Thn for all < < T f ± lim f i h funcion ± rprn by i own imag in h form f ± = p ω f 4 ± π θ L whr L i an in fini mooh lin in -plan who a rahr ian par coinci wih coninuaion of h ray π arg + a =± + θ ; a θ < θ < π / ar om conan an in 4 h ingral wih rpc o L i unroo in h n of principal valu Aum f i = f for i < < i+ fi i = f i + f i i+ = f i+ whr i = < < < m = T ar om poin 96 for 3

4 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION Corollary If ω C [ T ] ω > for [ T ] funcion f i ar aboluly coninuou wih rpc o [ i+ ] i = m θ = a wll > an h i m i a naural numbr hn 4 hol for Fini ingral ranformaion mhofor parabolic quaion Hr for a wi rang of rar on h following irrgular mol problm for parabolic quaion w a h fini ingral ranformaion mho Mol problm To fin h claic oluion u = u of h quaion u u = a + F < < < T 5 aifying h irrgular coniion u = ϕ K u = ϕ = < T 6 an h iniial coniion u = f 7 = whr F ϕ ϕ f ar h known funcion a-con L quaion b parabolic in IG Provky n i l arg a π a = a arg a θ a > 4 whr θ i om numbr aifying h inqualiy π < θ < 8 4 L h funcion F ϕ f T Sag Obaining opraion problm For > applying h fini ingral ranformaion ϕ b coninuou for φ = φ i a compl paramr o 5-6 an uing 7 an hn prforming on h imag u ~ opraion corrponing o h givn opraion on u w g h following opraion problm : a u = p u f F < < 9 97

5 PROCEEDINGS OF IAM V7 N 8 whr ~ u = v + f + ~ ϕ ; = K ~ u ~ = ϕ u ~ = u v u = ~ F = ~ ϕ = ϕ k k F Rmark On our iincion from h auhor ngag in uch problm i ha unlik h opraion problm conruc by u in h prn ca h righ han i of h opraion problm conain h ough-for funcion u a wll Sag Suying h paramric problm For olving h opraion problm - a fir w uy h paramric problm 3-4 corrponing o i a y'' y = ψ 3 U y y / = = γ U y K y = γ 4 whr ψ C[ ] γ γ ar arbirary numbr In [4] w hav hown ha whn uying mi problm for parabolic ym of orr p i uffic o conir h omain of chang of paramr in h form π R δ = : R arg + δ δ > 4 p Conqunly hr an in h qul w will aum ha π Rδ = : R arg + δ 4 p whr R i a rahr larg poiiv numbr rmark 5 δ < δ < θ 5 i an arbirarily fi numbr rmark 4 an 7 W ak h ym of funamnal paricular oluion of a homognou quaion corrponing o 3 in h form 98

6 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION y = p [ ] y = p a a 6 h funamnal oluion of quaion 3 w ak in h form P = p aπ a 7 No ha hr funamnal oluion P an h ym of funamnal paricular oluion y y ar chon o ha for δ hy ar craing funcion wih rpc o Rmark I i known ha h Grn funcion of h problm 3-4 ar inpnn on h choic of h funamnal oluion P an h ym of funamnal paricular oluion y y Hr hir uch choic R for δ fr u from h opraion prform in [ ] ovr h rminan conain in h prion of h Grn funcion Th nominaor of h Grn funcion of h problm 3-4 will b U y U y = 8 U y U y Epaning h rminan 8 w hav M M M S M S = α M + α M + + α M S + O Rδ 9 whr M i h high poibl gr wih rpc o S i om nonngaiv ingr α ν ar om numbr Incinally w no ha w can ak h numbr S conain in 9 rahr larg i for Rδ for w can g mor ac aympoic if h funcion conain in h lf han i of h paramric problm 3 4 ar rahr mooh Rmark 3 In [4] for h quaion wih variabl cofficin an "gnral" bounary coniion ufficin coniion impo on h cofficin of h paramric problm ha provi o obain aympoic o h rquir accuracy S for Rδ ar givn Dfiniion W ay ha bounary coniion of paramric problm 3-4 ar wll-impo if a la on of h numbr αm αm αm S from 9 i nonzro From hi finiion w immialy g: 99

7 PROCEEDINGS OF IAM V7 N 8 -If h bounary coniion of h paramric problm ar rgular in h n of Birkhoff -Tamarkin - Naimark-Raulov hn hy ar wll-po by our finiion Bu h invr amn i no ru 3 L irrgular coniion 4 b wll-po an in h qunc h fir nonzro numbr b α whr q q Z i om ingr q W cho h numbr R o larg ha q α q for R δ Now how h ufficincy of h coniion proviing wll-pon of irrgular coniion 4 4 n n j L K C [ ] K K = for j n whr n i om naural numbr Taking ino accoun 6 in 8 w g hav = K a a K a Uing h conrain 4 ingraing h following ingral by par w n = + a a n a a j K K K + j= n n a K a + Taking ino accoun w hav a a ε n whr in > ε = θ δ a δ j 3 R 4 Rmark 4 In 5 lcion of h numbr δ aifying h inqualiy δ < θ provi poiiviy of h numbr ε conain in 4 No ha p c p [ lim = ε whr p i any ral numbr c i om poiiv conan p p 5 Taking ino accoun in w g n+ n n = a K + O R n n+ δ 6

8 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION ha how wll -pon of irrgular coniion 4 Thu w ablih Lmma L conain 4 b fulfill Thn for Rδ for h quaion 3 h irrgular coniion 4 ar wll-po Now w choo h numbr R o larg ha n n a K a Rδ 7 n Rmark 5 Th numbr R i chon from h following wo coniion: -in h omain R δ o fin o h rquir accuracy h aympoic rprnaion of h ym of funamnal paricular oluion of homognou quaion corrponing o h quaion of h paramric problm in h prn mol ampl w i no n i -in h omain R δ for -nominaor of h Grn funcion of h paramric problm o g h lowr ima of yp in h prn ca of yp 7 From 7 i follow ha for Rδ hi inqualiy impli h valiiy of h following claic horm Thorm L h conrain' 4 b fulfill ψ C[ ] γ an γ b arbirary numbr hn for Rδ h paramric problm 34 -ha a uniqu oluion -hi oluion i rprn by h formula [4] y = δ γ γ + G ψ 8 G = P + G Th funcion δ γ γ wih rpc o γ an γ i linar i δ γ + qβγ + qβ = δ γ γ + qδ β β 9 Taking ino accoun 4 in 6 for Rδ w g h valiiy of h following inqualii y k ρ [ ] k = 3 whr ρ = p ε + p ε a a Sag 3Invrion formula A hi ag w g om invrion formula connc wih:

9 PROCEEDINGS OF IAM V7 N 8 δ γ γ i h oluion of homognou invrion formula quaion corrponing o 3 aifying inhomognou irrgular coniion 4 - G i h Grn funcion of h paramric problm 3 4 L L b an infini opn mooh lin in h omain R δ who rahr ian par coinci wih h coninuaion of h ray π arg = ± + δ In h 4 qul l R < R < R < an lim R = m m Lm b a par of L rmaining inrior o h circl of raiu R m ; an Cm b a par of a circl of raiu R m ; cnr a h origin of coorina of - plan rmaining in h omain R δ W hav Lmma L g b an analyic funcion wih rpc o Rδ S an g con for R δ whr Z i om ingr Thn for < < w hav h following formula of invrion o zro L g y = k = k whr y k i from 6 Hr an in h qul h ingral wih rpc o L i unroo in h n of principal valu Lmma 3 Unr conrain an 4 for any numbr γ γ an for any ingr Z w hav h following formula of invrion o zro δ γ γ = < < 3 L Lmma 4 Unr conrain an 4 if ψ i a pic-wi coninuou funcion in [] hn for any ingr Z invrion o zro i hol h following formula of G ψ = < < 3 L Accoring o [4] w hav h following Lmma 5 Unr conrain if ψ h gmn [] i Holr coninuou wih h ponn q < q hn for < < w hav h following invrion formula:

10 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION L a a ψ π δ ψ = = + = whr ~ ψ ψ P P = i from 78 Th following horm on invrion formula follow from lmma 4 an 5 Thorm Unr conrain an 4 if ψ on h gmn [] i Holr coninuou wih h ponn < q q hn for < < w hav h following invrion formula L a G a ψ π δ ψ = = + = 33 whr G i h Grn funcion of irrgular paramric problm 3 4 Sag 4 Th oluion of h mi problm Uing horm for δ R accoring o h formula 8 from h opraion problm w hav = ~ ~ ~ ϕ ϕ δ f v u [ ] + ] [ ~ T F f u G Thu w ablih h following Thorm 3 L conrain an 4 b fulfill Thn if irrgular mi problm 5-7 ha a claic oluion hn: -i i uniqu -hi oluion i rprn by analyic formula L u f δ π = + + ϕ ϕ δ f G > < < F G 34 3

11 PROCEEDINGS OF IAM V7 N 8 Rmark 6 From rivaion of formula 34 i follow ha unr conrain an 4 if h funcion u fin by formula 34 i no h oluion of problm 5-7 hn hi problm ha no claic oluion Rmark 7 Th poiiviy of h numbr δ δ > ami u o g h inqualiy ω CT for L T 35 whr < ω = in δ Inqualiy 35 ami o ak for > h opraion of iffrniaion wih rpc o an wih rpc o unr h ingral ign wih rpc o L Impoing fini rricion on h funcion F ϕ ϕ f [4] i i ay o ha h funcion u rmin by h formula 35 in fac i h claic oluion of irrgular mi problm 5-7 Mol problm To fin h oluion of h quaion u u = 36 > unr h bounary coniion u = ϕ > u CT p MT [ T ] a 37 = C T M T ar om conan in h iniial coniion u = Φ 38 whr ϕ Φ ar om coninuou funcion an Φ con for [ Thi mol problm i among h problm conir by ML Raulov [] Accoring o formula 35 p5 [] by Raulov h oluion of h problm i u = φ + G Φ 39 π L whr [ + = ] = ~ G ϕ ϕ 4 Now uing h fini ingral ranformaion mho a in mol problm for olving h problm w g h following prion 4

12 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION Φ u = φ + + G Φ 4 π L whr G i from 4 Thu h oluion of problm i rprn by iffrn formula 39 an 4 Comparing 39 an 4 w ha h con umman in brac 4 39 o no i furhr in 39 in h fir umman h ingral ϕ i rplac by no uccful In h problm w aum ϕ Thi rplacmn in our opinion i ϕ Φ 4 Thn from 39 w hav an from 4 w g u = π / p 43 u 44 On h ohr han by horm p 5 [] h oluion of h problm i uniqu on h ohr han for h oluion u of problm ML Raulov obain h prion 43 hough in fac h claic oluion of h problm i u from 44 Samn of h problm Fin hn claic oluion of h quaion u Z u f = v < < < < T 45 Z a + b + c aifying h ingro-iffrnial "bounary" coniion V i D D u { n= j= α i jn D j D n u i j n = + β jn D D u = + i j n + γ j n D D u = ϕi < < T i = 46 h iniial coniion = f < 47 an h fini coniion u < = 5

13 PROCEEDINGS OF IAM V7 N 8 whr u u = f < < = T u i h ough-for coninuou claic oluion; a b c f γ 48 v i h ϕ f ough-for coninuou conrol i i ar known funcion ; T T > α β ar known numbr jn jn L quaion 45 b parabolic in IG Provky n i l δ δ i om numbr R a δ [] whr i jn > m+ m+ m L a C [] b C [] c C [] i jn q γ C i = j = n = In coniion an 3 m m q 3 L [] ; ; rmark 4 L f C [] ϕ i C [ T ] i = From conrain i follow ha i i q ar om ingr f C [] [ T ] i π π + 4δ arg a 4δ [] whr π δ < δ < i om poiiv numbr 8 Dfiniion I i ai ha h funcion u u i a claic oluion of problm if a : h funcion u i coninuou for T a : h funcion u for < < T ha a coninuou rivaiv of h form u u ; a 3 : h funcion u for < < < < T ha coninuou rivaiv of h form u ; i a 4 : if α hn u C [ α] T whr α < α < i om numbr; u β hn C [ α] T ; i if 49 6

14 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION i if α hn u u C [ α ] T C [ α] [ T ; i if β hn u u C [ α ] T C [ α] [ T ; if γ i i no inically qual o zrohn u C [] T if γ i i no inically qual o zro hn u C [] T u C [] [ T ; a 5 : h funcion u aifi qualii in h orinary n Paramric problm For olving problm a fir w olv h following paramric problm Z y = ψ 5 Vi y = γi i = 5 ψ C [] γ ar om numbr -i a compl paramr whr i Hr an i h qul unl ohrwi ipula w aum π Rδ : R arg + δ 4 whr δ i from 49 R i a rahr larg poiiv numbr Dno by i = θ i h roo of h characriical quaion [ 9] covrponing o 5 i l arg a θ = θ = a p Accoring o [9] [4] unr conrain an a [] hr i h funcion [] = m g [] gi C i ha for h funcion 7

15 PROCEEDINGS OF IAM V7 N 8 yim p θi gi + gi + + g m im i hol E p im Z yim = θi [] i = m i any compl numbr C [ ] Eim i om funcion In [4] in omain R δ h funamnal oluion P of quaion 5 i conruc in h form P = P + P η h η 5 η h -i a ough -for krnl whr an uing funamnal oluion unr conrain an in [4] i i prov ha a homognou quaion corrponing o 5 ha h ym of funamnal paricular oluion y i i = ha oghr wih fir orr rivaiv ar rprn by h aympoic formula y = ym + Em = p y θ ym + Em = ; 53 whr E im ar om coninuou funcion wih rpc o [] an analyic wih rpc o Rδ an h following imaion hol: C Em p ε m C Em p ε y p m C ε y C p ε 54 Hr an in h qul by C an ε w no iffrn poiiv numbr concr valu of which ar no imporan inpnn of [] an Rδ Rmark In h claic papr [9 ] h mho u by hm compll hm o know aympoic of h ym of funamnal paricular oluion of a homognou quaion wih rpc o along h ircion of - plan an bcau of ha on h roo of characriical quaion hy impo h conrain: 8

16 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION A: h argumn θ i an argumn of hir iffrnc θ i ar inpnn of θ j For quaion 5 conrain A ruc o h aumpion B: a = αp whr P > α con for [] Unlik claic papr [9 ] in h prn papr an in [4] conrain A ar no impo on h roo of h characriical quaion an hrfor fulfilmn of conrain B i no uppo Epaning h rminan rminaor of h Grn funcion w hav = Q + E 55 whr 6 5 Q = α6 + α5 + + α6k 56 K 6 E = O K 5 K K i om ingr No ha h numbr K conain in 56 may b akn rahr larg i for on can obain ufficinly ac aympoic rprnaion if h numbr m from an q from 3 ar rahr larg Dfiniion I i ai ha bounary coniion 46or 5 wll-fin righ if a la on h numbr α6 α5 α6 K from 56 i non-zro If bounary coniion of h pcral problm ar rgular in Brikhoff - Tamarkin - Naimark - Raulov n hn hy ar wll-fin in h n of our finiion Bu h invr amn i no ru Show hi on h following ampl y" y = ψ 57 y y = y + y' + y' = 58 If w ak ym of funamnal paricular oluioun of a homognou quaion corrponing o 57 in h form y = p y = p hn h nominaion of h Grn funcion of problm will b = 6 Hnc i i n ha bounary coniion 58 for quaion 57 ar rgular in h n of Brikhoff Tamarkin Naimak Raulov I w ak h ym of funamnal paricular oluion of homognou quaion corrponing o 57 in h form 53 i y = p y = p hn h nominaor of h Grn funcion of problm will b 55 h Grn funcion ilf of h paramric problm i inpnn on h choic of h ym of funamnal paricular oluion whr Q = E = 6 9

17 PROCEEDINGS OF IAM V7 N 8 = O R S δ S i any naural numbrha inica wll-pou of bounary coniion 58 in h n of finiion L coniion 5 conain only h ingral i for ampl l Vi y Ki y = γ i i = 59 5 L K i C [] α K K K K Thn w hav Vi y Ki p = Ki Ki + ' + Ki p = Ki + O ; Vi y Ki p = Ki Ki ' Ki p = Ki + O ; hrfor from 56 w hav α Q = E = O Conqunly unr conrain 5 for paramric problm 5759 "bounary coniion" 59 ar fin by our finiion 6 L bounary coniion 46 an 5 b wll fin Furhr l among h non-zro numbr α 6 α5 α6k from 56 wih h gra in hr i α 6M Rmark 3 I i approria o ak in coniion h numbr m an in coniion 3 h numbr q o la a which for Q from 56 i hol Q = α6 6 M 6 M Uing 6 for rahr larg R from 55 w hav α 6 6M M 6 Inqualiy 6 how ha for Rδ Thrfor accoring o [9 ] an [4] i i hol following

18 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION Lmma L conrain an 6 b fulfill Thn for an ψ C[] paramric problm 5 5: i ha a uniqu oluion y ii hi oluion i an analyic funcion wih rpc o iii h oluion y i rprn by h formula y = G ψ + δ γ γ Uing 54 an 6 w g h imaion N G C [p ε + p ε ] γ + γ [p ε + N δ γ γ C + p ε ] [] Rδ Rδ 6 Rδ 63 [ whr N i om ingr numbr C i a conan inpnn of ] R δ an γ γ In [4] h following horm on invrion formula wa prov by h funamnal oluion P from 5 Thorm L conrain an b fulfill Thn for any aboluly coninuou funcion ψ on [] i hol h following invrion formula S ψ for S = < < 64 P ψ = π for S = δ + L whr L i an infini mooh lin in R δ who rahr ian par coinci wih coninuaion of h ray π arg = ± + δ morovr in 64 h ingral along h 4 lin L i unroo in h u of prinpal valu Taking ino accoun imaion 63 by h mho a in [4] w aily prov h following Thorm L conrain 3 an 6 b fulfill Thn for any funcion ψ C[] an arbirary numbr γ an γ i hol h following invrion formula S G ψ = < < 65 L

19 PROCEEDINGS OF IAM V7 N 8 S δ γ γ = < < 66 L whr S -i any ingr L -i from horm From Thorm an w obain h following Thorm 3L conrain 3 an 6 b fulfill Thn for any aboluly coninuou funcion ψ on []i hol h following invrion formula: S ψ for S = < < 67 G ψ = π for S = δ + L whr L i from horm In [4] h following horm on invrion formula i prov Thorm 4 If h funcion ϕ i coninuou for hn i hol h following invrion formula φ ϕ π = > 68 δ L φ p ϕ L i from horm 3Soluion of h mi problm L h ough-for coninuou conrol v b a priori known an problm hav h claic oluion u u Applying o h fini ingral ranformaion [4] w g Thorm 3 L conrain 3 4 an 6 b fulfill Thn if problm ha a uniqu oluion hi i a uniqu oluion an i rprn by formula u G v F = π p L + < < < T 69 ~ F = δ ψ ψ G p f + f [ ] Subiuing 69 in fini coniion 48 for rmining h unknown conrol w g Frholm con orr ingral quaion < < f p T G v F T = + 7 π L Bcau of rricion on h papr volum w on giv invigaion of h oluion of quaion 7

20 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION Solving quaion 7 w fin h unknown conrol an accoring o 67 w hav G v = < < 7 L Taking ino accoun 7 in 69 w g < < < T u p G v F = + 7 π L Improing rrain on rahr mooh n of h funcion conain in 3 an 4 an uing h inqualiy p Cp in δ L 73 u by h mho a in [4] i i aily hown ha h funcion rmin by formula 73 i h oluion of problm Rmark 3 In 7 whn ubiuing z = h obain imag of h L lin from h Laplac raigh lin an avanag of h iffr L from h Laplac raigh lin i ha accoring o inqualiy 73 in 7 h ingran facor for > L cra ponn ially an hi allow aily o juify convrgnc of h ingral in unboun lin L 4 Mol problm Problm amn Fin h oluion of h quaion u u = + v < < T wih h unknown conrol v aifying h bounary coniion u = u = < < T = = iniial coniion = f < u < = an fini coniion u = f < < C = T l l k k [] = { : ϕ C [] ϕ = ϕ = L k = l } ϕ for All conrain of horm 3 ar fulfill for problm an accoring o formula 69 w hav 4 Thorm 4 L f C [] f C [] Thn mi problm74-77 wih a conrol ha a uniqu claical oluion an hi oluion i rprn by formula 3

21 PROCEEDINGS OF IAM V7 N 8 u = k = p k π in kπ f in kπ + k = k π p k π in kπ v in kπ < < < T 78 k π v in kπ = p k π T in kπ p k π T f in kπ f 79 k = v = in kπ v in kπ whr h ough-for conrol v i foun by formula 8 i Fourir cofficin [3] by formula 79 Th fini ingral ranformaion mho u hr wa ugg by u in [4] an wa uccfuly u in [5-8] an in ohr Rfrnc 8 Brikhoff GD On h aympoic characr of h oluion of crain linar iffrnial quaion conaining a paramr Tran AM Mah Soc Cauchu AL M'moir'ur l'applicaion u calcul riu a'la oluion problm phyiqu mahmaiqu Pari Fichnhol GM Cour of iffrnial inrgral calculu vol III Mocow"Nauka" p 4 Gaymov Elmaga Fini ingral ranformaion mho Baku "Elm" p 5 Gaymov EA Mi problm on conjugaion of parabolic ym of iffrn orr wih nonlocal bounary coniion Diff Uravn Gaymov EA Applicaion of fini ingral ranformaion mho o h oluion of mi problm wih ingral iffrnial coniion for on nonclaical quaion Diff Uravn Gaymov EA Suying mi problm in conjugaion of hyprbolic ym of iffrn orr Zhournal vychililnoy mamaiki i mamaichikoy fiziki Gaymov EA Guynova AO Gaanova UN Applicaion of h gnraliz mho of paraion of variabl o h oluion of h iplac 4

22 Elmaga A Gaymov: APPLICATION OF FINITE INTEGRAL TRANSFORMATION problm wih irrgular bounary coniion Zhournal vychililnoy mamaiki i mamaichikoy fiziki Naimark MA Linar iffrnial opraor M Nauka 969 Raulov ML Applicaion of h conour ingral mho M Nauka 975 Tamarkin Ya D On om gnral problm of hory of orinary linar iffrnial quaion on panion of arbirary funcion in ri Progra 97 Применение метода конечного интегрального проеобразования к решению смещанных задач для параболических уравнений с управлением ЭАГасымов gaymov-lmagha@ramblrru РЕЗЮМЕ Одним из методов решения смещанных задач является классический метод разделения переменных метод Фурье Когда граничных условия смещанной задачи нерегульярны то вообще говоря этот метод неприменимы В настоящей работе применяется метод конечного интегрального преобразования к решению смещанных задач для параболических уравнений с управлением и с нерегульярными граничными условиями Получено аналитическое представление решения рассматриваемой смещанной задачи Ключевые слова: классическое решение метод конечного иетегрального преобразования уравнения с управлением нерегульярное граничное условие 5