Solution of Laplace s Differential Equation and Fractional Differential Equation of That Type

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1 Applie Mheics Pulishe Olie oveer 3 (hp://wwwscirporg/jourl/) hp://oiorg/436/34a5 Soluio of Lplce s iffereil Equio Frciol iffereil Equio of Th Type Tohru Mori Ke-ichi So Tohou Uiversiy Sei Jp College of Egieerig iho Uiversiy Koriy Jp Eil: se@jcohoeejp eceive Augus 9 3; revise Sepeer 9 3; ccepe Sepeer 6 3 Copyrigh 3 Tohru Mori Ke-ichi So This is ope ccess ricle isriue uer he Creive Coos Ariuio Licese which peris uresrice use isriuio reproucio i y eiu provie he origil wor is properly cie ABSTACT I preceig pper we iscusse he soluio of Lplce s iffereil equio y usig operiol clculus i he frewor of isriuio heory We here suie he soluio of h iffereil equio wih ihoogeeous er lso frciol iffereil equio of he ype of Lplce s iffereil equio We here cosiere erivives of fucio u o whe u is loclly iegrle o he iegrl u coverges We ow iscr he ls coiio h u shoul coverge iscuss he se prole I Appeices polyoil for of priculr soluios re give for he iffereil equios suie Herie s iffereil equio wih specil ihoogeeous ers Keywors: Lplce s iffereil Equio; Kuer s iffereil Equio; Frciol iffereil Equio; isriuio Theory; Operiol Clculus; Ihoogeeous Equio; Polyoil Soluio Iroucio Yosi [] iscusse he soluio of Lplce s iffereil equio (E) which is lier E wih coefficies which re lier fucios of he vrile The E which he es up is y y y () l l for l re coss His iscussio is se o Miusińsi s operiol clculus [3] Yosi [] gve here oly oe of he soluios of he E () I he preceig pper [4] we iscusse he soluio of frciol iffereil equio (fe) of he ype of E () h is give y u u u f u () for Here for > is he ie-liouville frciol erivive (f) efie i Secio We use o eoe he se of ll rel uers : u u Whe is equl o ieger > Whe () is he ihoogeeous E for () We use o eoe he se of ll iegers : : : for sisfyig < We use for o eoe he les ieger h is o less h I [4] we op operiol clculus i he frewor of isriuio heory evelope for he soluio of he fe wih cos coefficies i [56] I [4] we give he recipe of oiig he soluio of he ihoogeeous equio s well s he hoogeeous oe we show how he se of wo soluios of he hoogeeous equio is ie I [4] we op he usul efiiio of he ie- -Liouville f which efies f oly for such loclly iegrle fucio f o > h f is fiie Prciclly we op Coiio B i [4] which is Ope Access

2 T MOITA K SATO 7 u H Coiio IB f H re epresse s lier coiio of g for > Here H is Hevisie s sep fucio whe f is efie o > f H is ssue o e equl o f whe > o whe g is efie y g (3) for > is he g fucio I [4] we e up Kuer s E s eple which is u c u u (4) c re coss If c oe of he soluios give i [78] is c F c ; ; : (5)! for > The oher soluio is c F c c ; ; (6) I [4] if c < we oi oh of he soluios Bu whe c (6) oes o sisfy Coiio IB we coul o ge i I rece review [9] we iscusse he lyic coiuios of f lyic coiuio of ie- -Liouville f f is such h he f eiss eve for such loclly iegrle fucio f o > h f iverges I he prese pper we op his lyic coiuio of f I plce of he ove Coiio IB we ow op he followig coiio Coiio A u H f H re epresse s lier coiio of g H for S S is se of > M \ < for soe M > As cosequece we c ow chieve oriry soluios for () of For (4) we oi oh soluios (5) (6) if c I is he purpose his pper o show how he preseio i [4] shoul e revise wih he chge of efiiio of f he replcee of Coiio IB wih Coiio A I Secio we prepre he efiiio of ie- Liouville f he epli how he fucio u is f i () re covere io he correspoig isriuio u is f i isriuio heory lso how u is covere c io u Afer hese preprio recipe is give o e use i solvig he fe () wih he i of operiol culculus i Secio 3 I his recipe he soluio is oie oly whe Whe is lso re- quire A eplio of his fc is give i Appeices C of [4] I Secio 4 we pply he recipe o () of which specil oe is Kuer s E This is eple which Yosi [] es up I Secio 5 we pply he recipe o he fe wih ssuig For he Herie E wih ihoogeeous er Levie Mle [] showe h here eis priculr soluios i he for of polyoil I Appeices A C we show h such soluio eiss for he E fe suie i Secios 4 5 respecively I Appei B we show how he resuls presee i [] re erive fro hose i Appei A Foruls We ow op Coiio A We he epress u H s follows; u H u g H () S u re coss Le For \ < g g () Proof By (3) for < we hve g g ie-liouville Frciol Iegrl erivive e loclly iegrle o We he efie he ie-liouville frciol iegrl f of orer > y f f (3) Le f H We he efie he ie-liouville f of orer y u u u > (4) if i eiss u u for > For > we hve g \ g (5) Ope Access

3 8 T MOITA K SATO If we ssue h es cople vlue g y efiiio (3) is lyic fucio of i he oi e < g efie y (4) is is lyic coiuio o he whole cople ple If we ssue h lso es cople vlue g efie y (4) is lyic fucio of i he oi e > The lyic coiuio s fucio of ws lso suie The rgue is urlly coclue h (5) shoul pply for he lyic coiuio eve i e ecep he pois < ; see [9] We ow op his lyic coiuio of g o represe g hece we ccep he followig le Le (5) hols for every \ < By () (5) we hve For u is loclly iegrle o S u u g efie y () we oe h M u H Frciol Iegrl erivive of isriuio (6) We cosier isriuios elogig o Whe fucio h is loclly iegrle o hs suppor oue o he lef i elogs o is clle regulr isriuio The isriuios i re clle righ-sie isriuios A copc forl efiiio of isriuio i is frciol iegrl erivive is give i Appei A of [4] Le f H e regulr isriuio The f H for is lso regulr isriu- io isriuio f H is efie y f H f H (7) Le le h e such regulr isri- > uio h h is coiuous iffereile o for every The h is efie y h h (8) Le > le f H e coiuous iffereile o The for every f H f H Whe h is regulr isriuio fie for ll h he ie lw: Le 3 For h h is vli for every irc s el fucio H fie y Le g g for e efie y Le 4 If > h (9) is e- () is he isriuio e- () g g H () Proof By puig f i (7) usig () (5) we oi H H g H By operig o his usig (9) (5) we oi () Correspoig o u epresse y () we efie u y u u g (3) The S u f re epresse s u uˆ f fˆ (4) uˆ u (5) S Becuse of () we hve g g \ g (6) Le 5 Le The g g (7) (8) The ls erivive wih respec o is e regrig s vrile A proof of (7) for > is give i Appei B of [4] Proof Whe > y Les 4 g g H g H g Ope Access

4 T MOITA K SATO 9 The firs equliy i (8) is oie fro (7) vice vers y usig () The followig le is cosequece of his le Le 6 Le u e epresse s lier coiio of g for The u uˆ uˆ (9) 3 Fro u o u Vice Vers Le 7 Le \ < > sisfy > The () g g H () g g H Proof Forul () is erive y pplyig (3) () (6) o he righh Forul () follows fro () y replcig g g y g g g g respecively y usig () (7) By usig Le 7 o (6) we oi S u g () u H S u H u g (3) Le 8 Le \ < > sisfy > The g H g (4) This follows fro () Coiio B u is epresse s lier coiio of g for S S is se of > M \ < for soe M > Whe his coiio is sisfie u is epresse s (3) wih S replce y S ~ Le 9 Le u sisfy Coiio B The he correspoig u is oie fro u y u H u (5) is epresse y () wih S replce y S Le Le u u e give y (3) () respecively The u u H re rele y u u H u S (6) S u u H if > u sisfies M Proof By (3) (6) we hve u u g S S u g (7) (8) Usig () i he firs er o he righh sie we oi (6) Muliplyig (8) y oig h he firs er o he righh sie is he equl o (3) we oi (7) 3 ecipe of Solvig Lplce s E fe of Th Type We ow epress he E/fE () o e solve s follows: l l l u f (3) l or 5 we suy his E for respecively I Secios 4 his fe for 3 efor o E/fE for isriuio Usig Le we epress (3) s l l l u f v l (3) l v l l ul (33) l l 3 Soluio Vi Operiol Clculus By usig (4) (9) we epress (3) s l l ˆ ˆ l u l u l l A uˆ Buˆ fˆ vˆ (34) Ope Access

5 3 T MOITA K SATO l l A l l l l l l B l vˆ u u l l l l l l l I orer o solve he Equio (34) for ˆ u u we solve he followig equio for fucio rel vrile : (35) (36) û of A u ˆ Bu ˆ (37) fˆ vˆ Le The copleery soluio (C-soluio) of equio (37) is give y û C ˆ C is rirry cos ˆ ep B C A (38) he iegrl is he iefiie iegrl C is y cos Le Le ˆ e he C-soluio of (37) û e he priculr soluio (P-soluio) of (37) whe he ihoogeeous er is for The uˆ ˆ C3 ˆ A ˆ (39) C 3 is y cos Sice f H sisfies Coiio A ˆv is give y (36) he P-soluio û of (37) is epresse s lier coiio of u ˆ for > of uˆ M for > respecively Fro he soluio û of (37) û is oie y susiuig y The we cofir h (34) is sisfie y h û opere o 33 eu Series Epsio Filly he oie epressio of û is epe io eu series [] Prciclly we ep i io he su of ers of egive powers of he we oi he soluio û of (34) If he oie û is lier coiio of for > M wih soe M > he uˆ is he soluio u of (3) If i sisfies Coiio B i is covere o soluio u of (3) for > wih he i of Le 9 34 ecipe of Oiig he Soluio of (3) ˆf ) We prepre he : y (4) A B ˆv y (35) (36) ) We oi ˆ y (38) The C-soluio of (3) is give y u C ˆ If v he C-soluio of (3) is oie fro his wih he i of Le 9 3) If ˆ f or vˆ we oi û give y (39) vˆ c f he so- 4) If luio of (3) is give y u C ˆ c ˆ u (3) c re coss The C-soluio of (3) is he oie fro his wih he i of Le 9 5) If ˆ f he P-soluio of (3) is give y u uˆ > M re coss The P-soluio of (3) wih ihoogeeous er c f g is oie fro his wih he i of Le 9 35 Coes o he ecipe I he ove recipe we firs oi he C-soluio of (37) h is û C ˆ I gives he C-soluio û of (34) hece he C-soluios u of (3) A C-soluio u of (3) is he oie wih he i of Le 9 We e oi he P-soluio û of (37) whe he ihoogeeous pr is for As oe ove he P-soluios û of (37) for f ˆ for ˆv re epresse s lier coiio of û for > M of uˆ for > respecively The su of he P-soluios û of (37) for ˆf for ˆv gives he P-soluio û of (34) hece he P-soluio u of (3) The C-soluio u of (3) coes fro he C-soluio of (37) he P-soluio of (37) for ˆv 36 ers Whe we oi û he e of Secio 3 we us eie wheher i is copile wih Coiio B We will fi h if l for l > he oû is o cceple Hece we hve o solve ie Ope Access

6 T MOITA K SATO 3 he prole ssuig h l for ll l > Whe we pu Whe we pu = iscussio of his prole is give i Appeices C of [4] I he prese cse he iscussio us e re ig Coiio B here o represe he prese Coiio B 4 Lplce s Kuer s E We ow cosier he cse of σ = = The (3) reuces o By (35) (36) u u u f B A B vˆ u A ˆv re (4) (4) (43) 4 Copleery Soluio of (37) (3) (4) ˆ I orer o oi he C-soluio of (37) y usig (38) we epress B A s follows: B (44) A B() is ow epresse s (45) B By usig (38) we oi ˆ! he ioil coefficies The C-soluio of (3) is give y (46) for > re ˆ ˆ u u C If > C C (47) we oi C-soluio of (4) y usig Le 9: u H C C H C! C H F ; ; H (48) er I Iroucio Kuer s E is give y (4) I is equl o (4) for c I his cse c c (49) We he cofir h he epressio (48) for c > grees wih (6) which is oe of he C-soluios of Kuer s E give i [78] 4 Priculr Soluio of (37) We ow oi he P-soluio of (37) whe he ihoogeeous er is equl o for Whe he C-soluio of (37) is ˆ he P-soluio of (37) is give y (39) By usig (4) (46) he followig resul is oie i [4]: ˆ u C ˆ 3 C (4) pp C p p p p (4) Le 3 Whe p p < C p efie y p (4) is epresse s Ope Access

7 3 T MOITA K SATO? Cp p Cp p p p C p p p p p This le is prove i [4] 43 Priculr Soluios of (3) (4) (4) Equio (4) shows h if he ihoogeeous er is for he P-soluio of (3) is give y u C (43) Theore Le \ < < f g The we hve P-soluio u of (4) give y u u H u H (44) H H Proof Applyig Le 9 o (43) we oi u H C H (45) (46) By usig (4) i (46) we oi (44) wih (45) We oe h u H is epresse s u H! H F ; ; H 44 Copleery Soluio of (4) (47) (48) By (43) (45) vˆ u Whe fˆ vˆ he P-soluio of (47) is give y uˆ u uˆ (49) By usig (44) for if < we oi C-soluio of (4): u H u u H u! H u F ; ; H (4) I Secio 4 we hve (48) h is oher C-soluio of (4) If we copre (48) wih (45) whe > i c e epresse s (4) C H C u H Proposiio Whe he copleery soluio of (4) uliplie y H is give y he su of he righh sies of (48) of (4) which C u H u u H re equl o respecively er As se i er for Kuer s E re give i (49) c (4) We he cofir h if c he se of (48) (4) grees wih he se of (45) (46) 45 ers I [] i ws show h here eis P-soluios epresse y polyoil for ihoogeeous Herie s E e l We c oi he correspoig resul for Lplce s E We iscuss his prole i Appei A he iscuss he P-soluio of ihoogeeous Herie s E i he prese forulio i Appei B 5 Soluio of fe (3) for I his secio we cosier he cse of The he Equio (3) o e solve is u u u f > ow (35) (36) re epresse s A B M vˆ u (5) (5) (53) Ope Access

8 T MOITA K SATO 33 5 Copleery Soluio of (37) By usig (5) B A is epresse s B A (54) (55) By (38) he C-soluio ˆ of (37) is give y ˆ 5 Copleery Soluio of (3) (5) The C-soluio of (3) is give y ˆ ˆ u u C If > C (56) (57) y pplyig Le 9 o his we oi he C-soluio of (5): u H C H C H (58) 53 Priculr Soluio of (3) (5) ˆ give y By usig he epressios of A (5) (56) i (39) we oi he P-soluio of (37) whe he ihoogeeous er is for : uˆ C 3 C ˆ (59) of (5) give y u H u H u H (5) H (5) I Appei C iscussio is give o show h here eis P-soluios i he for of polyoil for (5) 54 Copleery Soluio of (5) We oi he soluio u oly for < Eve hough we hve P-soluios of (3) for v vˆ whe ˆv is give y (53) wih ozero vlues of u i oes o sisfy Coiio B oes o give soluio of (5) Hece u give y (58) is he oly C-soluio of (5) If we copre (58) wih (5) we oi he followig proposiio Proposiio Le > The he C-soluio of (5) is give y (5) C H C u H EFEECES [] K Yosi The Algeric erivive Lplce s iffereil Equio Proceeigs of he Jp Acey Vol 59 Ser A 983 pp -4 [] K Yosi Operiol Clculus Spriger-Verlg ew Yor 98 Chper VII [3] J Miusińsi Operiol Clculus Pergo Press Loo 959 [4] T Mori K So ers o he Soluio of Lplce s iffereil Equio Frciol iffereil Equio of Th Type Applie Mheics Vol 4 o A 3 pp 3- [5] T Mori K So Soluio of Frciol iffereil Equio i Ters of isriuio Theory Ieris- C cipliry Iforio Scieces Vol o 6 pp is efie y (4) is give y 7-83 (4) if < [6] T Mori K So eu-series Soluio of By usig (4) i (59) we c show h if he i- Frciol iffereil Equio Ieriscipliry Iforio Scieces Vol 6 o pp 7-37 hoogeeous er is for he P-solu- io of (3) is u û By pplyig Le [7] M Arowiz I A Segu Hoo of Mhe- 9 o his we oi he followig heore icl Fucios wih Foruls Grphs Mhe- Theore Le < < icl Tles over Pul Ic ew Yor 97 Chper 3 f The we hve P-soluio u [8] M Mgus F Oerheiger Foruls Theo- res for he Fucios of Mheicl Physics Chelse Ope Access

9 34 T MOITA K SATO Pul Co ew Yor 949 Chper VI [9] T Mori K So Liouville ie-liouville Frciol erivives vi Coour Iegrls Frciol Clculus Applie Alysis Vol 6 o 3 3 pp [] L Levie Mleh Polyoil Soluios of he Clssicl Equios of Herie Legere Cheyshev Ieriol Jourl of Mheicl Eucio i Sciece Techology Vol 34 3 pp 95-3 [] F iesz B Sz-gy Fuciol Alysis over Pul Ic ew Yor 99 p 46 Ope Access

10 T MOITA K SATO 35 Appei A: Polyoil For of P-Soluio of (4) Le \ < > The (45) gives u H C H (A) C C u C H (A) C (A3) We oi he followig heores fro (A) wih he of P-soluio of (4): i of Proposiio Theore 3 Le > < < f g The we hve he polyoil for u H H Theore 4 Le > > > polyoil for of P-soluio of (4): f g (A4) for The we hve he u H H (A5) Appei B: Polyoil For of P-Soluio of Herie E We ow cosier he ihoogeeous Herie E give y y ycy (B) for > We pu u y The he equio for u is give y 4 4 u u c u (B) This is Lplce s E (4) wih preers c (B3) c c g he ihoogeeous er f Theore 5 Le c > The we hve he polyoil for of P- soluio of (B): u H H (B4) 4 3 Proof I his cse By Theore 3 we oi his resul Theore 6 Le c < > The we hve he polyoil for of P- soluio of (B): Ope Access

11 36 T MOITA K SATO 3 Proof I his cse By Theore 4 we oi his resul u H H 4 (B5) 3 4 Theore 7 Le c > The we hve he polyoil for of P- soluio of (B): u H H (B6) 3 Proof I his cse By Theore 4 we oi his resul Theore 8 Le c < > The we hve he polyoil for of P- soluio of (B): u H H (B7) 4 Proof I his cse By Theore 3 we oi his resul er 3 We cofir h Theores 7 5 res- pecively gree wih Theores i [] Appei C: Polyoil For of P-Soluio of (5) Le > The (5) gives u H C H C (C) u C H (C) C C (C3) We oi he followig heore fro (C) wih he i of Proposiio Theore 9 Le > > > f g for The we hve he polyoil for of P-soluio of (5): u H H (C4) () Ope Access