INTEGRAL TRANSFORM METHODS FOR SOLVING FRACTIONAL PDES AND EVALUATION OF CERTAIN INTEGRALS AND SERIES

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1 ITEGRAL TRASFORM METHODS FOR SOLVIG FRACTIOAL PDES AD EVALUATIO OF CERTAI ITEGRALS AD SERIES *A. Aghl nd H. Znl *Drmn of Ald Mhmcs, Unvrsy of Guln Rsh-Irn *Auhor for Corrsondnc ABSTRACT In hs work, h uhors mlmnd wo dmnsonl Llc rnsform o vlu crn ngrls, srs nd solvng rl frconl dffrnl quons. Consrucv xmls r lso rovdd o llusr h ds. Th rsul rvls h h ngrl rnsform mhod s vry ffcv nd convnn. Mhmcs Subjc Clssfcon : 6A33; 34A8; 34K37; 35R. Ky Words: on-homognous Tm Frconl H Equons, Llc Trnsform, Klvn s Funcon nd Tm Frconl Wv Equons ITRODUCTIO Th objc of rsn work s o xnd h lcon of Llc rnsform o drv nlycl soluon for crn boundry vlu roblm of wv quon. In 985 Dhy consdrd mhod of comung Llc rnsform rs of n-dmnsons nd rov horm o obn nw n-dmnsonl Llc rnsform rs. Lr n 99 Dhy nd Vnygmoorhy sblshd svrl nw horms nd corollrs for clculng Llc rnsform rs of n-dmnsons. Thy lso consdrd wo boundry vlu roblms. Th frs ws rld o H rnsfr for coolng off vry hn sm-nfn homognous l no h surroundng mdum solvd by usng doubl Llc rnsform.th scond, ws H quon for h sm-nfn slb whr h sds of h slb r mnnd rscrbd mrur. In 99 Sbr jf nd Dhy sblshd svrl nw horms for clculng Llc horms of n-dmnsons nd n h scond r lcon of hos horms o numbr of commonly usd scl funcons ws consdrd, nd fnlly, on-dmnsonl wv quon nvolvng scl funcons ws solvd by usng wo dmnsonl Llc rnsform. Lr n 999 Dhy rovd crn horms nvolvng h clsscl Llc rnsform of -vrbls nd n h scond r non-homognous rl dffrnl quons of rbolc y wh som scl sourc funcon ws consdrd. Rcnly n 4, 6, 8 h uhors sblshd nw Thorms nd corollrs nvolvng sysms of wo-dmnsonl Llc rnsforms connng svrl quons. Th gnrlzon of h wll-known Llc rnsform o n-dmnsonl s gvn by L s { f ( ); s} f ( ) d L { f ( ); s }... x s. f ( ) P d, n n,,...,,,,...,,., whr s s s s s s nd n n n P d d. In hs r w focus on wo dmnsonl Llc rnsform of h funcon f ( x, y ), whch s dfnd s n n k k 7

2 x qy F (, q) f ( x, y ) dxdy. Dfnon..: Th nvrs of wo dmnsonl Llc rnsforms F (, q ) s dfnd by c c ' x qy f ( x, y ) F (, q) dq d, c c ' n whch R c,r q c '. Invrson of Two Dmnsonl Llc Trnsforms In hs scon, n lgorhm o nvr on nd wo dmnsonl Llc rnsforms s rsnd. Thorm.. (Efros Thorm): L L{ f ( )} F( s) nd L{ u(, )} U ( s)x( q ( s)) nd ssum U ( s), q( s ) r nlyc, hn by kng Llc rnsform w.r., on hs L{ f ( ) u (, ) d; s} U ( s ) F ( q( s )). Proof: S (Dkn 96). Lmm.. (Schoun-Vn dr Pol): Consdr funcon f() whch hs h Llc rnsform F(s) whch s nlyc n h hlf ln R(s)>s. If q(s) s lso nlyc for R(s)>s, hn h nvrs of F (q(s)) s s followng c q( s) s L { F( q( s)); s } f ( ) ds d. c Scl cs: q( s ) s ; 3 L F s s f d { ( ); } ( )x( ). 4 Proof: S (Duffy 4). In hs scon, nw clss of nvrs Llc rnsforms of xonnl funcons nvolvng nsd squr roos s drmnd. Invrs Llc rnsforms nvolvng nsd squr roos rs n mny rs of ld Mhmcs, usully s rsul of lnr voluon rl dffrnl quons of forh ordr n h sl vrbls. Exmls of such roblms bound n flud mchncs. Lmm.3: Th followng rlon holds ru. 3 x x L {x( x s s )} ( ) x( ) d 4 ( ) Proof: L us ssum h F ( s) x( x s s ) nd hn h xonn cn b r-wrn s follows, x( x s s ) x( x s ).x( x s ). On h ohr hnd, from Llc bl w hv So h, x L {x( x s )} x ( ) 3/ 4, 8

3 L { F( s)} L {x( x s s )} L x s L x s {x( )}* {x( )} 3 x ( ) x( ). x d 4 ( ) Exml.4: Suos h F (, q) b h wo dmnsonl Llc rnsform of h funcon f ( x, y ) Thn w hv f ( x, y) c x c' qy ( dq) d c c' q ( ) L us dfn h funcon F(, q),, R, ( q )( ) c c x y d. y G(, q), I mns h F (, q) G (, q), I suffcs o fnd h nvrs of G (, q ) nd hn usng Efros horm o obn h nvrs of F (, q ). On my us h followng wll known ngrl rrsnon b u Mkng chng of vrbl o g y y b, b, y ( ) u y du, ow, w cn vlu h nvrs of G (, q ) s bllow u du b. (.) (.) (.3) c x y ( ) u (, ) ( ) g x y du d c c y ( ) x u ( d) du, c 9

4 by chng of vrbl s, y c" ( s ) x y ( s ) u g ( x, y ) ( ds ) du s c" y y ( ) ( ) s c" sx y ( ) x u x u y s c" ( ds ) du ( x ( )) du. By nroducng nw chng of vrbl, y x ( ) w,w obn y 4x x y ( )( ) 3 w x 4x 4( x w ) y g ( x, y ) ( y )( x w ) ( w ) dw. x x A hs on, usng Efros horm for q( s ) f x y L G q (, ) { (, )} 4x s nd lmm. lds o h followng y 4 y 4 ( ) 4 4x 4 y y ( )( ) d d. 4 x 8x x 3 Alcons of Two Dmnsonl Llc Trnsform Th mul dmnsonl Llc rnsform s usd frqunly n ngnrng nd hyscs. Th wo dmnsonl Llc rnsform cn lso b usd o solv rl dffrnl quons nd s usd xnsvly n lcrcl ngnrng. Th wo dmnsonl Llc rnsforms rducs lnr rl dffrnl quon o n lgbrc quon, whch cn hn b solvd by h forml ruls of lgbr. Th PDE cn hn b solvd by lyng h nvrs wo dmnsonl Llc rnsform. Thr r lso som lcons of wo dmnsonl Llc rnsform o vlu crn ngrls nd srs s s dscussd n h followng rgrhs. 3.. Evluon of crn srs: Suos h F (, q) b h wo dmnsonl Llc rnsform of h funcon f ( x, y ) s blow Tkng q xqy F(, q ) f ( x, y ) dxdy, lds us o h followng rlonsh ( xy ) (, ) (, ). F f x y dxdy By mkng chng of vrbl, x y w on hs w w F(, ) f ( w y, y ) dwdy f ( w y, y ) dydw. y w 3

5 ow lng n, o g Summng ovr n o obn Th bov rlon cn b r- wrn s n whch w nw F( n, n ) ( f ( w y, y ) dy ) dw, nw F( n, n ) ( )( f ( w y, y ) dy ) dw. n n g( w) F( n, n ) dw, w n w g ( w ) f ( w y, y ) dy. Exml 3..: Evlu h followng srs 4 S. n ( n ) Soluon by usng h bov rocdur, w g x y 4 F(, q) L{ }. ( )(q) q Thn w hv consqunly w w/ w/ ( ), g w dy w w/ w w F( n, n ) dw dw. w n w snh On h ohr hnd, w know h (or by clculus of rsdus) x dx. snh x 4 Thrfor h fnl soluon s 4 w n ( n ). 3. Evluon of h ngrls: In ld mhmcs, h Klvn funcons Brν(x) nd Bν(x) r h rl nd mgnry rs, rscvly, of 3 /4 J ( x ), Whr x s rl, nd Jν(z), s h νh ordr Bssl funcon of h frs knd. Smlrly, h funcons /4 Krν(x) nd Kν(x) r h rl nd mgnry rs, rscvly, of K ( x ), whr Kν(z), s h νh ordr modfd Bssl funcon of h scond knd. Ths funcons r nmd fr Wllm 3

6 Thomson,s Bron Klvn. Th Klvn funcons wr nvsgd bcus hy r nvolvd n soluons of vrous ngnrng roblms occurrng n h hory of lcrcl currns, lscy nd n flud mchncs. On of h mn lcons of wo dmnsonl Llc rnsform s vlung h ngrls s dscussd n h followng. Lmm 3..: Show h followng ngrl rlons br( )cos d. b( )sn d ( ). Proof: L us dfn h followng funcon By usng h formul W hv I(, x, y) br( xy )cos d, q L{ br ( xy ); x, y q}, q q L { I (, x, y )} cos d. q ow, wh h d of Fourr rnsform w g h followng rlonsh q I (, x, y ). Howvr, lds o c c' c x ( ) q qy x I(, x, y) ( dq) d ( y ) d. c c' c ow, l us mk chng of vrbl y w nd consqunly d dw Lng x y on gs c" y w xy ( ) x I (, x, y ) ( w ) dw. c" br ( )cos d. L us consdr h funcon I (, x, y ) b ( xy )sn d. By kng wo dmnsonl Llc rnsform of h bov rlon wh rsc o x, y w g on h ohr hnd, w know h sn L { I(, x, y)} I ( ) d, q 3

7 consqunly cos q I (), I( ) d, ( q) q q I( ), q whch lds o c c' q x qy I(, x, y) ( ( dq) d) q By sng x y, on gs c c' c x c ( ( y ) d) ( xy H ( x)). b( )sn d ( ). Lmm 3..: Th followng rlonsh for R, m, n holds ru Proof: s [Dkn 979]. L nm n k / ( q) I ( xy ) n n m ( m ) k ( q ) n k { }. Lmm 3..3: Evlu h followng ngrl d sn sn. Proof: L us dfn h followng funcon y d f ( x, y ) sn x sn, ow by kng wo dmnsonl Llc rnsform w g d / d F (, q)., q q ( )( ) q nd consqunly Fnlly on gs q ( ) d ln. q q q 33

8 q F (, q) ( )( ln q). q q ow, rmns o us nvrs of Llc rnsform o g f ( x, y ). To do h, w us lmm 3.. nd convoluon horm s bllow ( x, y ) L { ln q} '() ln xy, q q ( x, y ) L { } { I ( xy ) J ( xy )}. q And consqunly y x f ( x, y) ( ( '() ln( x )( y ))( I( ) J( )) d) d. In scl cs x = y =, on gs d sn sn f (,) ( ( '() ln( )( ))( I( ) J( )) d) d. Alcon n Solvng Frconl PDEs 4.. Prlmnrs: Lmm.4..: Th followng rlonsh holds ru whr b, y. x y y b I ( bx ) xdx, x y b Proof: From h ngrl rrsnon of K n h form of W g h followng rlonsh K ( x y ) d, ( x y ) ( x y ) ( ) 4 / K ( x y ) d I ( bx ) xdx I ( bx ) x dx, ( x y ) ( ) 4 / ( x y ) chngng h ordr of ngron, w obn Sng y ( b ) u( ) du b ( b ) K ( y b ). ( ) u y, nd usng h subl rlon for, ordr zro I,, on gs K K n rms of modfd Bssl funcons of x y y b I ( bx ) xdx. x y b 34

9 Lmm.4.. (Bobylv-Crcgnn): L F () s n nlyc funcon hvng no sngulrs n h cu ln C \ R. Assum h F ( ) F ( ) nd h lmng vlu xs for lmos ll >. L () F( ) o() for nd rg, ; F ( ) lm F ( ), F ( ) F ( ) () Thr xss such h for vry, F ( ) o( ) for,unformly n ny scor F( r ) L ( R ), F( r ) ( r), r r Whr (r) dos no dnd on nd ( r) L ( R) for ny. Thn n h noon of h roblm, L [ F ] Im[ F ( )] d. Proof: S (Bobylv ). s Corollry: In h bov lmm, l F () s, wh,, hn F(s) wll ssfy h condons of lmm. W chck som of h condons of lmm s followng: Th lmng vlu (cos sn ) ( ) lm ( ) lm ( ), F F xss bcus >, β> nd snβπ nd cosβπ r boundd. F(s) ssfs h ohr condons s wll. Thrfor cos Im( ( F )) sn( sn ). W ly h bov lmm o g nvrs of Fs ( ) n h form Scl cs:,.5; mkng chng of vrbl cos ( ) sn{( sn ) }. f d u By usng bl of ngrls, w g f ( ) sn d, nd ngrng by rs, w obn u f ( ) cos udu, (4..) 4 f( ). 35

10 Lmm.4..3: Assum h Thn G( s) L{ g( ); s}, L G s J g d s s { ( ); } ( ) ( ). Proof: S ( Duffy 4 ) 4. Mn rsuls: In hs scon, h uhors consdr boundry vlu roblms for crn m frconl rl dffrnl quons. In hs work, only Llc rnsformon s consdrd s owrful ool o solv h bov mnond roblms. Ths gol hs bn chvd by formlly drvng xc nlycl soluon. Problm 4..: Consdr sm fn srng vbrons wh frcon dscrbd by h frconl quon n whch br, osv consns, wh nl nd boundry condons n h form Soluon: L us ssum h W hv u(, x) u (, x), u (,) V ( T ), lm u(, x). x k k k x U (, q) L { u(, x );, x q}. k (4..) u u Tk L{ } U (, q), L { } q U (, q) qh ( ) V, k x u L{ } U (, q), n whch H ( ) L{ u( x,); x }. Subsung n (4..) w g dnomnor mus ssfy h numror s wll ow subsung n (4..) w g u u u b u,.5, x Tk qh V k k ( ( ) ) U (, q) q ( b. ) Tk V k k ( ). H b (4..) On h ohr hnd h roos of 36

11 Tk V k k Tk q( ) V k b k U (, q), q ( b ) W cn rwr bov quon n h form Tk V k k (, ). U q b ( q b ) ow w should nvr h bov quon. L us nvr wh rsc o q frs by usng rsdu horm Tk V k (, ) k x( ). U x x b b ow w should nvr h bov rlonsh w.r. x b Tk u (, x ) V k L ;, k b k L F ( ) T f ( T k ), so suffcs o vlu on h ohr hnd Frs, l ; b n whch. 4 ow from lmm.4.. nd lng c x b h( ) d. c b /4 ( /) b x b U (, x ), b / 4( ( b / ) ) x b b y,, b w hv 4 x b u(, x ) L U (, x ); I ( ( )). b 4 4 ow by lmm.3 w hv x b 4 b 4 L { F( ); } I ( ( )) d. ( ) 4 37

12 L F ( ) f ( T k ), h fnl soluon wll b obnd s bllow k by h fc h T x b 4( T ) (, ) ( ( )) k. 4 3 b ( Tk ) ( ) u x I d 4 Problm 4..: L us consdr h followng m frconl four rms h quon u u u ( ) bu, x x wh h followng boundry condons u(, ) sn, u( x,) u ( x,), lm u( x, ). Soluon:. Tkng Llc rnsform of F.P.D.E nd B.Cs w.r.. o xx x ( s ) U ( s b) U, wh U(, s), lm U( x, s), s x whr U s Llc rnsform of h funcon u( x, ) wh rsc o. Consqunly s b U ( x, s ) x( x ). s s (4..3) A hs on, l us ssum h ( s ) Gs ( ), s s b F( s) x( x ). (4..4) s s Consdr h funcon F(s) whch s dfnd by lng α= n F(s) s followng x b F ( s) x( ). s s Assumb, hn l b k, x m,so w hv k F ( s) x( m ), s s by usng nvrson formul w g xx 38

13 k m s s f( x, ) ds. c s d By mkng lnr chng of vrbl s, ds h bov quon ks h form L q hn w hv c c' m k ( ) f( x, ) ( d). c' c' c' k m q q f( x, ) ( dq). q From h followng nvrson formul (s bls of nvrs of Llc rnsform) nd lmm 4..3 w obn k m m k 3 m ( ) k 4 3/ k 4 (, ) ( ( ) ). k, m ( ) m f x J d From lmm. nd h corollry of lmm 4.. w g f ( x, ) f ( x, ) x( cos )sn( sn ) d d. On h ohr hnd h nvrs of G(s) n (4..4)) wll b obnd s bllow G( s) s. s L{sn( )} sn L{sn( )}, s s Thrfor w hv g( ) sn( ) sn( ), (4..5) (4..6) In whch h frconl drvv s n Cuo sns, whch s dfnd s ( n) f ( ) D f ( ) d, n ( n ) ( ) In whch n s h smlls ngr lrgr hn α. ow, from (4..5),(4..6) nd convoluon horm, h fnl soluon s obnd s followng u( x, ) g( )* f ( x, ) f( x, ) x( cos )sn( sn ) d d sn( ( )) sn( ( )) d. ( ) 39

14 COCLUSIO Th r s dvod o sudy nd lcon of mul dmnsonl Llc rnsforms for som scl combnons of funcons of sclr vrbls. Th mul-dmnsonl Llc Trnsform rovds owrful mhod for nlyzng lnr sysms. Th ohr y of lcon s ornd for fndng nlyc soluon of h m frconl wv nd h quon usng Llc rnsform. Th m frconl s consdrd n h Cuo sns. I my b concludd h h mhod s vry owrful nd ffcn chnqu for solvng h modl. Fnlly, h rcn rnc of m - frconl h quon s modls n som flds such s h hrml dffuson n frcl md mks ncssry o nvsg h mhod. REFERECES Aghl A nd Slkhordh Moghddm B (4). Llc rnsforms rs of n- dmnsons nd Wv quon. Inrnonl Journl of Mhmcs 5(4) Aghl A nd Slkhordh Moghddm B (6). Mul-Dmnsonl Llc Trnsform nd sysms of rl dffrnl quons. Inrnonl Journl of Mhmcs (6) -4. Aghl A nd Slkhordh Moghddm B (8). Llc rnsforms rs of - dmnsons nd scond ordr lnr dffrnl quons wh consn coffcns. Annls Mhmc nformc Bobylv AV nd Crcgnn C (). Th nvrs Llc rnsform of som nlyc funcons wh n lcon o h rnl soluons of h Bolzmnn quon. Ald Mhmcs Lrs Dhy RS nd Sbr djf J (999). Thorms on -Dmnsonl Llc Trnsforms nd hr lcons. 5h Annul Confrnc of Ald Mhmcs, Unvrsy of Cnrl Oklhom. Elcronc Journl of Dffrnl Equons Confrnc Dhy RS nd Vnygmoorhy M (99). Llc Trnsform rs of n-dmnsons nd h conducon roblm. Mhmcl nd Comur Modllng 3() Dkn VA nd Prudnkov AP (979). Clcul oronnl,edon d Moscou, Duffy DG (4). Trnsform mhods for solvng rl dffrnl quons, Chmn nd Hll/CRC wyork. Klbs A nd Trujllo J (). Dffrnl quon of frconl ordr: mhods, rsuls nd roblms. II Ald Anlyss 8()