The Cauchy Problem for the Heat Equation with a Random Right Part from the Space

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Appled Mhemcs, 4, 5, 38-333 Publshed Ole Augus 4 ScRes hp://wwwscrporg/jourl/m hp://dxdoorg/436/m4556 The Cuch Problem for he He Equo wh Rdom Rgh Pr from he Spce Sub ϕ ( Ω Yur Kozcheo, A Slv-Tlshch

1 Appled Mhemcs, 4, 5, Publshed Ole Augus 4 ScRes hp://wwwscrporg/jourl/m hp://dxdoorg/436/m4556 The Cuch Problem for he He Equo wh Rdom Rgh Pr from he Spce Sub ϕ ( Ω Yur Kozcheo, A Slv-Tlshch Deprme of Probbl Theor, Sscs d Acurl Mhemcs, The Fcul of Mechcs d Mhemcs, Trs Shevcheo Nol Uvers of Kv, Kv, Ure Eml: v@uvevu, slv@ure Receved 6 M 4; revsed 4 Jue 4; cceped 7 Jul 4 Coprgh 4 b uhors d Scefc Reserch Publshg Ic Ths wor s lcesed uder he Creve Commos Arbuo Ierol Lcese (CC BY hp://crevecommosorg/lceses/b/4/ Absrc The fluece of rdom fcors should ofe be e o ccou solvg problems of mhemcl phscs The he equo wh rdom fcors s clsscl problem of he prbolc pe of mhemcl phscs I hs pper, he he equo wh rdom rgh sde s exmed I prculr, we gve codos of exsece wh probbl, oe clsscl soluos he cse whe he rgh sde s rdom feld, smple couous wh probbl oe from he spce Sub ϕ ( Ω Esmo for he dsrbuo of he supremum of soluos of such equos s fouded Kewords Cuch Problem, He Equo, Sochsc Process Iroduco The subjec of hs wor s he erseco of wo brches of mhemcs: mhemcl phscs d sochsc processes The phscl formulo of problems of mhemcl phscs wh rdom fcors ws suded b Kmpe de Fere [] I he wors [] d [3], ew pproch sudg he soluos of prl dfferel equos wh rdom l codos ws proposed The uhors vesge he covergece probbl of he sequece of fuco spces of prl sums pproxmg he soluo of problem The meoed pproch ws used he wors [4]-[7] I he pper [3], he pplco of he Fourer mehod for he homogeeous How o ce hs pper: Kozcheo, Y d Slv-Tlshch, A (4 The Cuch Problem for he He Equo wh Rdom Rgh Pr from he Spce Sub φ (Ω Appled Mhemcs, 5, hp://dxdoorg/436/m4556

2 Y Kozcheo, A Slv-Tlshch hperbolc equo wh Guss l codos s jusfed The codos of he exsece of he clsscl soluo of hs equo erms of correlo fucos re lso suded Homogeeous hperbolc equo wh rdom l codos from he spce Sub ϕ ( Ω s cosdered [8]-[] The model of soluo of hperbolc pe equo wh rdom l codos ws vesged he ppers [] [3] There s sud o boudr-vlue problem of mhemcl phscs for he homogeeous hperbolc equo wh ϕ -subguss rgh pr [8] [4] The prbolc pe equos of Mhemcl Phscs wh rdom fcors of Orlcz spces hve bee suded he ppers [5] [6] Furher refereces c be foud [8] [7]-[] We cosder Cuch problem for he he equos wh rdom rgh pr We sud he homogeeous he equo o le wh rdom rgh pr We cosder he rgh pr s rdom fuco of he spce Sub ϕ ( Ω The Guss sochsc process wh zero me belogs o Sub ϕ ( Ω [] The codos of exsece wh probbl oe of he clsscl soluo of hs problem re vesged For such problem hs bee go he esmo for he dsrbuo of he supremum soluo The pper cosss of he roduco d hree prs Seco cos ecessr defos d resuls of he heor of he Sub ϕ ( Ω spce I Seco 3, we cosder he equos wh rdom rgh-hd sde For such problem codos of exsece, wh probbl oe, of clsscl soluo wh rdom rgh-hd sde from he spce Lp ( Ω re foud The esmo for dsrbuo of supremum of hs problem hs bee go Seco 4 Rdom Processes from Sub ( Ω ϕ Spce Defo [3] A eve couous covex fuco u( x, u( x u( x for x d lm =, lm = s clled - x R such h u = d u( x > N fuco x x x x Defo [] We s N- fuco u ssfes he q - codo f here exs coss z >, >, A > such h u ( x u ( Au ( x for ll x > z, > z Lemm [] Le u( x be N- fuco The u( αx αu( x for α d x R ; u αx αu( x for α > d x R ; 3 u( x + + u( for x, R ; 4 The fuco u( x x s o decresg for x > ( ( Lemm [] Le u ( x be he verse o N- fuco u( x for x > The u ( x s covex cresg fuco such h u αx αu x for α d x R ; 3 ( ( α ( u x αu x for α > d x R ; u x + u x + u for x, R ; 4 he fuco Defo 3 [3] Le u x x s ocresg for x > u x be - N fuco The fuco u ( x x u ( Fechel rsform of he fuco u( x The fuco u ( x Le { ΩI,, P} be sdrd probbl spce Defo 4 [] Le ϕ ( x be - h ϕ ( x = cx for x < x The se of rdom vrbles ered b he N- fuco ϕ ( x f { λξ} { ϕ ( λ ξ } E exp exp for ll = sup( s clled he Youg- R s N- fuco s well N fuco for whch here exs coss x > d c > such ξ ϖ, R ϖ, s clled he spce Eξ = d here exss cos ξ such h λ R The spce Sub ϕ ( Ω s Bch spce wh respec o he orm [] ( ( leexp{ λξ} ϕ τϕ = sup λ λ Sub ϕ Ω ge- 39

3 Y Kozcheo, A Slv-Tlshch Defo 5 [3] The sochsc process X = { X (, T} belogs o spce Sub ϕ ( Ω, X Sub ϕ Sub ϕ ( Ω for ll T Remr [4] The Guss sochsc process X ( wh zero me belogs o X φ ( x = x d ( X ( E( X ( τ = A Fml of Srogl Subϕ ( Ω Rdom Vrbles d Fml Srogl Subϕ ( Ω Sochsc Processes Lemm 3 [] If ξ Sub ϕ ( Ω, he here exss cos C > such h E ( ξ Cτφ ( ξ Ω f Sub ϕ Ω, where SSub ϕ Ω rdom Defo 6 [] The rdom vrble ξ Sub ϕ ( Ω s clled srogl Sub ϕ ( Ω, vrble f τφ ( Eξ ξ = Properes d pplcos of SSub ϕ ( Ω rdom vrbles d sochsc processes from SSub ϕ be foud [] Defo 7 [7] A fml of rdom vrbles ξ of he spce Sub ϕ ( Ω s clled for ll τϕ λξ = E λξ I I λ R, where I s mos couble d Ω c SSub ϕ Ω fml f ξ, I Sub ϕ Ω fml of rdom vrbles The he ler closure of Sub ϕ Ω fml X = X, T, I s clled SSub ϕ ( Ω process f he Theorem [7] Le be srogl he fml he spce L ( Ω d he me squre sese s srogl Defo 8 [] The sochsc process { } fml of rdom vrbles X = { X(, T, I} s SSub ϕ ( Ω Theorem [7] Le X = { X(, T, I} be fml of jol srogl Sub ϕ cesses The ( T, θ, µ s mesurble spce If ( T, θ, µ d he egrl ξ φ ( X ( dµ ( T rdom vrbles = {, I, =, } Theorem 3 [9] Le d ( s, mx s ξ { I,, } Ω sochsc pro- φ = s fml of mesurble fucos = s well defed he me squre sese, h he fml of Sub ϕ Ω fml ξ s R be he -dmesol spce, =, T { T,,,, } = =, he process X ( s seprble d sup τφ ( X( X( s σ ( h where ( h d ( s, h T > X { X (, T} Sub ϕ = Ω Assume h σ s moooe cresg couous fuco such h σ ( h s h We lso ssume h ψ l dε < ( + σ ( ε,where ( ψ u = ( φ ( u d σ ( ε s he verse fuco o σ ( ε If he processes X ( coverge probbl o he process T, he X ( coverge probbl he spce C( T Theorem 4 [9] Le T = { x b, =,, m} d le ξ ( X, X T h ξ ( X SSub ϕ ( Ω Pu B ( XY, = Eξ( X ξ( Y d ssume h he prl dervves B( XY, B ( XY, =, =,, m d u X for ll, be seprble rdom feld such 3

4 Y Kozcheo, A Slv-Tlshch (, 4 B XY B ( XY, =, =,, m, =,, m exs Le here exs moooe cresg couous fuco σ z ( h >, h >, such h z ( h h for z = (,,,, z = (,,,, =,, m d z= (,,,,, =,, m Assume h sup ( Bz ( X, X + Bz ( Y, Y Bz ( X, Y σ z ( h x h =,, m σ s ε u If ψ l ( du < for ll z d for suffcel smll ε > where ψ ( u = φ ( u (, he φ ( u ξ ( X ξ ( X wh probbl oe he prl dervves,,, j =,, m, exs d re couous 3 The He Equos wh Rdom Rgh Pr We cosder he Cuch problem for he he equo (, u( x, u x = + < x<, >, subjec o he l codo ξ j ( x,, u x, =, < x< ( Le he fuco ξ( x, { ξ( x,, xr, } he spce Sub ϕ ( Ω, such h Eξ ( x, =, E( ξ ( x, < Le us deoe B( xzs,,, Eξ( x, ξ( zs, B( xzs,,, be couous fuco Problem whe he fuco ( x, Lemm 4 Le ξ ( x, s rdom feld, smple cou for ech ξ ( x, couous dervve The for he fuco ( x, = > s rdom feld smple cou wh probbl oe from for exs d ξ( τ ξ( τ x R d ssf codo ( ( = Le ξ ordom hs bee see [5] > wh probbl oe, here s E ξ x, d x< (3 R ξ for ech > he egrl Fourer rsform x, = cos x x, d π ξ(, τ = cos xξ( x, τ dx π Proof Sce, b Fub s heorem, Eξ( x, d x< E ξ ( x,, we deduce h he egrl R R ξ ( x, dx< exs wh probbl oe, d herefore he egrl ξ( τ R ples from [6] h he egrl Fourer rsform exs, d he verse egrl Fourer rsform ξ(, τ = cos xξ( x, τ dx π cos x x, dx, d herefore m- 3

5 Y Kozcheo, A Slv-Tlshch ξ( x, τ = cos x ξ ( x, τ dx π exs Theorem 5 Le he codos of Lemm 4 be ssfed d d If he followg egrls exs ( G(, = e τ ξ (, τ d τ, π ξ (, τ = cos xξ( x, τ dx π u x, = cos xg, d (4 s xg (, d, s cos xg, d, s =, d for ll A > d T > here exss sequece, egrls + + for, such h he sequece of s xg, d, (5 s cos xg, d, s =, (6, he coverges probbl, uforml for x A, T u x, s he clsscl soluo o he problem ( d ( Proof Sce he egrls (5 d (6 coverges probbl uforml for x A, T, here exss subsequece b, b s, such h + b + b s b coverges wh probbl oe o uforml for x s xg, d, cos xg, d, s =,, b s A, T, Le s xg, d, cos xg, d, s =,, + b u x, = cos xg, d (7 b b B dervg (7 wh respec o x d, we esl see h u + b (, b x = + ξ ( τ b cos xg, d cos x, d, π u b ( x, b + b b = cos xg (, d, b 3

6 Y Kozcheo, A Slv-Tlshch >, x R Sce for uforml for x Ideed, A, T ub x, u x, = + cos, d, u b ( x, b b x ξ ( τ x π b coverges o (, u x wh probbl oe, we coclude h u x, = + π cos xg, d cos x, d (, u x = + x, ξ ( τ ξ ( x, ( x, ub, d u x, coverges o u x, ssfes Equo ( Lemm 5 [9] Le ξ be rdom feld, smple cou from he spce Sub ϕ ( Ω Le B( xvs,,, be he correlo fuco of he feld ξ ( x, For ll >, s > ssume h: B( xvs,,, The dervves (,,, v l m, =,, 4, l+ m= exs; B xvs dd xv B (, l, m <, =,, 4, l+ m = l m v (,,, B xvs 3, =,, 4, l+ m=, x or v l m v The Lebesgue egrls s xg (, d, s exs wh probbl oe Proof We shll prove he exsece of he egrl cos xg, d, s =, cos xg (, d For exsece of hs egrl wh probbl oe s eough o prove h here exss followg egrl There s equl Cosder EG, d ( ( (, E G = π EG, d E G, d ( τ ( s ( τ ( e e Eξ, τ ξ s, dτds = E ( ξ(, τ ξ(, s = cosxcos v ξ( x, τξ ( v, s dxdv π = π cosxcos vb x,, v, s dxd v Iegrg b prs d usg he codos of he lemm, we ob for 33

7 Y Kozcheo, A Slv-Tlshch The Therefore (,,, 4 cosxcosv B xvs E( ξ (, τ ξ ( s, = dd xv 4 s π v ( 4,, (,,, 4 B xvs E( ξ (, τ ξ ( s, = dd xv π ( E( G(, 4 s v B 4 π ( τ ( τ = B 4 ( 4,, e e dτ ds π = B(4,, e 4 8 π E G B (, d ( 4,, ( e π d for The ler egrl coverges uder R cos xg, d c be proved smlrl Lemm 6 [5] Le fuco X (, u sup X ( λ, u B; u<, λ< (, (, fuco such h The exsece of egrls λ, λ > d u > be such h: X λ u X λ v Cλ u v for ll u >, ϕ λ > for ll cos v > ϕ( λ+ v The X ( λ, u X ( λ, v mx ( C, B ϕ + v u v for ll λ d v > v > Le λ >, d he fuco Corollr Le he codos of Lemm 6 he fuco ϕ( λ ( l ( λ ( λ, ( λ, mx (, s xg, d, ϕ λ, λ > be couous cresg λϕ λ s cresg for λ > v, d for some ( l ( λ + e X u X v C B l + e u v for ll > Proof Ideed, s es o show h he fuco λϕ λ creses wh v e >, we ob he equl 8 = +, λ >, > The = Therefore Lemm 6 g fuco ϕ( λ ( l ( λ (8 = + λ >, 34

8 Y Kozcheo, A Slv-Tlshch Corollr ( ( e mx, cosx cosx ( l ( + e l ( ( l + e l x x + e + e (9 ( for some > Remr If he codos of Corollr h, x x h, he for suffcel smll h equl (9 d ( wll hve he form Le + ( u x, = cos xg, d, ( l ( + e ( l ( h + e e mx, ( l ( + e l ( h + e cosx cos x + ( u x, = s xg, d, + ( u x, = cos xg, d, Theorem 6 Le ξ ( x, be rdom feld, smple couous wh probbl oe from he he codos of Lemm 4 d Lemm 5 hold, For,, h, moreover, ( ( ( u x u ( x σ h sup τ,,, ϕ x x h, h Sub ϕ Ω d =, where σ ( h s moooe cresg couous fuco such h ( h u =, d σ where ψ ( u ( φ ( u Exmple Le ( x + ( ( ε σ s ψ l ( dε <, ( σ ( ε s he verse fuco o σ ( ε The he fuco (, ϕ be fuco such h ( x u x whch s represeed he form (4 s clsscl soluo o he problems ( d ( Proof Ths heorem follows from Theorems 5 d 3 p ϕ = x, for some p > d ll x > The ψ ( x p = x for x > d codo ( holds for ll ε > Codo ( holds f ( h + p l ( dε < ( σ ( ε C σ =, for > p, l h 35

9 Y Kozcheo, A Slv-Tlshch C >, =,, I hs cse, he codo of Theorem 6 s ssfed f for =,, here exs coss C > such h For (, (, l h E u x u x C, (3 > p ll =,,, d suffcel smll h ξ x, be rdom feld, smple couous wh probbl oe from he spce Theorem 7 Le SSub ϕ ( Ω, where ϕ ( x s fuco such h ( x dos of Lemm 4 d Lemm 5 hold d ξ( x, τ E dx< θ for some p φ = x for some p > d ll x > d he co- E ξ x, τ dx< θ θ >, θ >, ξ( x, τ E dx< θ, θ > The he fuco (, ed he form (4 s clsscl soluo o he problems ( d ( Proof I follows from Lemm 5 h here exs egrls wh probbl oe s xg (, d s, Accordg o Theorem 5 o me he fuco (, cos xg, d, s =, u x whch s represe- u x be he soluo of problems ( d ( s suffce o prove h egrls (5 d (6 coverge uforml probbl x A, T o he egrls s xg (, d s, cos xg, d, s =,, for A >, T > Accordg o Theorem 6, usg he Exmple (, o me egrl (5 d (6 coverge C T he followg codos mus hold probbl ( C E u x, u (,, x l h =,, Usg geerlzed Movsoho equl we ob ( E u ( x, u, x = E cos xg (, d cos xg (, d = E cos xg (, cos xg (, d = E ( cosx cos x G (, + ( G (, G (, cosx d ( cosx cos x E G, + EG, G, d Le x x h d for suffcel smll h, usg he equl (, we hve (4 36

10 Y Kozcheo, A Slv-Tlshch Cosder ( l ( + e l ( h + e cosx cos x ( τ ( ( = ( G, e Eξ, τ d τ π (5 I follows from Lemm 4 h Therefore Le ( ξ ( τ = ξ( τ E, E cos x x, dx π < π ( τ ( EG( θ < he EG(, G(, ( θ π ( ξ( τ E x, dx θ, e d e (6 τ π π ( τ ( e (, d e τ = E ξ τ τ ξ (, τ dτ π ( τ ( τ ( τ ( τ = E e e e ξ (, τ dτ + e ξ (, τ dτ π π ( τ ( τ ( E ( ( τ e e ξ, τ dτ + e E ξ, τ d τ Le h d for suffcel smll h, usg he equl (9, we hve Therefore ( ( l ( + e τ τ τ τ τ = ( e e e e e mx, l ( EG(, G(, ( l ( e τ ( τ ( θ τ + e mx, d e d π + θ τ l h ( l ( e θ + mx (, e τ = e d τ π + l h Thus we ob from (4, (5, (6 d (7 h ( h (7 37

11 Y Kozcheo, A Slv-Tlshch ( E u ( x, u, x ( ( ( ( ( ( ( τ ( θ l + e l + e e mx, e e dτ d π + + l h l h ( ( ( = + h h θ l + e l + e e mx, e π l l θ = + π ( ( ( I I ( ( ( τ ( + ( τ e dτ d θ l + e l + e = e mx, e e dτ d π + + l l h h θ ( l ( + e ( l ( + e ( τ + e + mx (, e + e π dτ d l h l h Cosder ( ( ( ( ( τ l + e l + e I = e + mx, e + e dτ d l h l h ( l ( e mx (, l ( e ( τ + + = e d e d e d d + τ l h + l h ( = mx, I + I + I l h l h 3 Sce e T, we hve ( l ( + e = ( ( + = I e d T l e d TC ( l ( + e = ( ( + = I e d T l e d TC Usg h So we hve ( τ e, h, he he > d for suffcel smll h, we hve ( τ I3 = e dτ d ( d h l h ( ( I TC + mx, TC + l h 38

12 ( ( Y Kozcheo, A Slv-Tlshch ( ( ( τ l + e l + e I = e mx, e e dτ + + d l l h h ( l ( + e mx (, ( l ( + e ( τ = e d e d e dτ d + l h + l h mx, = I + I + I l h l h 3 ( l ( + e ( l ( + e I = e d d = C ( l ( + e ( l ( + e = = I e d d C mx, l + e mx, τ ( τ τ = ( = ( e d d e d d C3 l h l h + mx, + The for > p, we hve l h Therefore I C ( ( C C 3 (, (, l h E u x u x C, where θ C = TC + mx (, TC + + ( C + mx (, ( C + C3, Cj, =,, j =,, 3 re some coπ ss Cosder ( E u x, u (, s (, d s (, d x E xg x G = From Lemm 4, we ob = E s xg (, s xg (, d = E ( sx s x G (, + ( G (, G (, sx d ( sx sx EG(, + EG, G, d ( = + sx s x EG, EG, G, d ( τ ( ( ( G, e Eξ, τ d τ π 39

13 Y Kozcheo, A Slv-Tlshch Smlrl ( ξ( τ = ξ( τ E, E cos x x, dx π ( C E u x, u (,, x l h ( x, τ ( x, τ ξ E sx dx π ξ E d x< θ π π θ where C = TC + mx (, TC + + ( C + mx (, ( C + C3 coss Cosder, Cj, =,, j =,, 3 re some π ( E u ( x, u, x = E cos xg (, d cos xg (, d E cos xg (, cos xg (, d = = ( cos cos (, + ( (, (, cos d E x x G G G x From Lemm 4 we ob ( cosx cosx EG(, + EG, G, d 4 4 ( = + cosx cos x EG, EG, G, d ( τ ( ( ( 4 4 G, e Eξ, τ d τ π 4 ( ξ( τ = ξ( τ E, E cos x x, dx π ξ( x, τ E cosx dx π ξ( x, τ E d x< θ π π 33

14 Y Kozcheo, A Slv-Tlshch ( The ( C E u x, u (,, x l h θ where C = TC + mx (, TC + + ( C + mx (, ( C + C3 coss π C, =,, j =,, 3 re some j 4 Esmes of he Dsrbuo of he Supremum of Soluo Theorem 8 [9] Le T > Assume h X X (, T where ( h R be he -dmesol spce, d ( s, mx s = { } s seprble d X Sub ϕ =, T { T,,,, } Ω If sup τ φ d ( s, h σ s moooe cresg couous fuco such h ( h ψ l ( dε <, where ψ + σ ( ε P{ sup X( u} Auθ (, θ T ( u = u u d σ φ >, for ll < < d Theorem 9 Le he codos of Theorem 6 hold where I u > θ ( ( ε ϕ ( θε ( θ = =, ( X ( X ( s σ ( h σ s h, d, s he verse fuco o σ ( ε The, where Au (, θ = exp ϕ u( θ I φ ( θε, ε θ ( EX( ε = sup, T T I ϕ = ψ l ( + d ε = σ ( ε u x, = cos xg, d ( G(, = e τ ξ (, τ d τ, π ξ (, τ = cos xξ( x, τ dx π ( x, D, D= [ AA, ] [, T] The for ll < θ < d I u > θ ϕ ( θε ( θ P sup u( x, > u Auθ (,, ( x, D, where Au (, θ = exp ϕ u( θ I ϕ ( θε, ε θ 33

15 Y Kozcheo, A Slv-Tlshch ε = ( x, D ( Eu( x sup,, θε T A I φ ( θε = ψ l ( + + l ( + d ε, σ ε σ ( ε where σ ( ε s moooe cresg couous fuco such h he verse fuco o σ ( ε Proof Ths heorem follows from Theorem 8 σ ε s ε, d σ ( ( ε s Refereces [] de Fere, K (96 Sscl Mechcs of Couous Med Proceedgs of Smpos Appled Mhemcs, Amerc Mhemcl Soce, Provdece, [] Besebev, E d Kozcheo, YuV (979 Uform Covergece Probbl of Rdom Seres, d Soluos of Boudr Vlue Problems wh Rdom Il Codos Theor of Probbl d Mhemcl Sscs,, 9-3 [3] Buldg, VV d Kozcheo, YuV (979 O Queso of he Applcbl of he Fourer Mehod for Solvg Problems wh Rdom Boudr Codos Rdom Processes Problems Mhemcl Phscs, Acdem of Sceces of UrSSR, Isue of Mhemcs, Kuv, 4-35 [4] de L Krus, EB d Kozcheo, YuV (995 Boudr-Vlue Problems for Equos of Mhemcl Phscs wh Srcl Orlcs Rdom Il Codos Rdom Operors d Sochsc Equos, 3, - hp://dxdoorg/55/rose99533 [5] Kozcheo, YuV d Edzhrgl (994 Jusfco of Applcbl of he Fourer Mehod o he Boudr-Vlue Problems wh Rdom Il Codos I Theor of Probbl d Mhemcl Sscs, 5, [6] Kozcheo, YuV d Edzhrgl (994 Jusfco of Applcbl of he Fourer Mehod o he Boudr-Vlue Problems wh Rdom Il Codos II Theor of Probbl d Mhemcl Sscs, 53, [7] Kozcheo, YuV d Kovlchu, YA (998 Boudr Vlue Problems wh Rdom Il Codos d Seres of Fucos of Sub φ (Ω Ur Mhemcl Jourl, 5, [8] Dovg, BV, Kozcheo, YuV d Slv-Tlshch, GI (8 The Boudr-Vlue Problems of Mhemcl Phscs wh Rdom Fcors Kv Uvers, Kv, 73 p (Ur [9] Kozcheo, YuV d Slv, GI (4 Jusfco of he Fourer Mehod for Hperbolc Equos wh Rdom Il Codos Theor of Probbl d Mhemcl Sscs, 69, hp://dxdoorg/9/s [] Slv, AI ( A Boudr-Vlue Problem of he Mhemcl Phscs wh Rdom Ils Codos Bulle of Uvers of Kv Seres: Phscs & Mhemcs, 5, 7-78 [] Slv-Tlshch, AI ( Jusfco of he Fourer Mehod for Equos of Homogeeous Srg Vbro wh Rdom Il Codos Ales Uverss Scerum Budpesess de Roldo Eövös Nome Seco Mhemc, 38, -3 [] Kozcheo, YV d Slv, GI (7 Modellg Soluo of Hperbolc Equo wh Rdom Il Codos Theor Probbl d Mhemcl Sscs, 74, [3] Tlshch, AIS ( Smulo of Vbros of Recgulr Membre wh Rdom Il Codos Ales Mhemce d Iformce, 39, [4] Dovg, BV d Kozcheo, YV (5 The Codo for Applco of Foure Mehod o he Soluo of Nogomogeeous Srg Oscllo Equo wh φ-subgussrgh Hd Sde Rdom Operors d Sochsc Equos, 3, 8-96 [5] Kozcheo, YV d Veresh, KJ ( The He Equo wh Rdom Il Codos from Orlcz Spce Theor of Probbl d Mhemcl Sscs, 8, 7-84 hp://dxdoorg/9/s [6] Kozcheo, YV d Veresh, KJ ( Boudr-Vlue Problems for Nohomogeeous Prbolc Equo wh Orlcz Rgh Sde Rdom Operors d Sochsc Equos, 8, 97-9 hp://dxdoorg/55/rose5 [7] Agulo, JM, Ruz-Med, MD, Ah, VV d Grecsch, W ( Frcol Dffuso d Frcol He Equo Advces Appled Probbl, 3, hp://dxdoorg/39/p/ [8] Kozcheo, YV d Leoeo, GM (6 Exreml Behvor of he He Rdom Feld Exremes, 8, 9-5 hp://dxdoorg/7/s

16 Y Kozcheo, A Slv-Tlshch [9] Begh, L, Kozcheo, Y, Orsgher, E d Sho, L (7 O he Soluo of Ler Odd-Order He-Tpe Equos wh Rdom Il Jourl of Sscl Phscs, 7, hp://dxdoorg/7/s x [] Rov NE, Shuhov, AG d Suhov, YM (99 Sblzo of he Sscl Soluo of he Prbolc Equo Ac Applcde Mhemce,, 3-5 [] Buldg, VV d Kozcheo, YV ( Merc Chrcerzo of Rdom Vrbles d Rdom Processes Amerc Mhemcl Soce, Rhode [] Ao, RG, Kozcheo, Y d N, T (3 Spces of φ-subguss Rdom Vrbles Memore d Memc e Applczo, Accdem Nzole delle Scze de de XL, Vol 7, 95-4 [3] Krsosels, MA d Ruc, YB (96 Covex Fucos d Orlcz Spces Noordhof, Gröge [4] Kozcheo, YV d Osrovsj, EV (986 Bch Spces of Rdom Vrbles of Sub-Guss Tpe Theor of Probbl d Mhemcl Sscs, 53, 4-53 [5] Mrovch, BM ( Equos of Mhemcl Phscs Lvv Polechc Publshg House, Lvv, 384 p (Ur [6] Budl, AM ( Fourer Seres d Iegrls S Peersburg, 37 p 333

Integral Equations and their Relationship to Differential Equations with Initial Conditions

Integral Equations and their Relationship to Differential Equations with Initial Conditions Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs GLM 6 3-3 Geerl Leers Mhemcs GLM Wese: hp://wwwscecereflecocom/geerl-leers--mhemcs/ Geerl Leers Mhemcs Scece Refleco Iegrl Equos d her Reloshp