MEIJER G-FUNCTIONS SERIES AS EXACT SOLUTIONS OF A CLASS OF NON-HOMOGENEOUS FRACTIONAL DIFFERENTIAL EQUATIONS

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1 Pee cite thi tice : Młgozt Kie ie ziebowi Meie -uctio eie ect outio o c o o-hoogeeou ctio dieeti eutio Scietiic Reech o the Ititute o Mthetic d Coute Sciece 9 Voue 8 Iue ge The webite: htt:// Scietiic Reech o the Ititute o Mthetic d Coute Sciece MIJR -UNCTIONS SRIS AS XACT SOLUTIONS O A CLASS O NON-HOMONOUS RACTIONAL IRNTIAL QUATIONS Młgozt Kie ie ziebowi Ititute o Mthetic Czetochow Uiveity o Techoogy Pod ie@ii.cz. cuty o Mthetic Phyic d Coute Sciece Ooe Uiveity Pod Abtct. A ctio dieeti eutio o e ode cotiig vibe coeiciet t d o-hoogeeou te i oved. The gee outio i obtied u o Meie -uctio eie deteiig the outio o hoogeeou coutet o the coideed eutio d eie eeetig the ticu outio o o-hoogeeou eutio. The covegece o the eective eie i yzed i deti uig theoe o oetie o Meie -uctio. A ee two eutio with d / e tudied. Itoductio ctio ccuu i eteio o cic ccuu with iteg d dieeti oeto o o-itege ode. It i ued i thetic odeig o viou heoe i hyic cheity echic egieeig bioegieeig d ecooic coe oogh [-3] e [4-6] d the eeece theei. A ew c o itego-dieeti eutio eeged eut o the ictio o ctio ccuu to the cotuctio o ode i y ied. The ctio dieeti eutio theoy bece iott d iteetig e o ivetigtio. My o the eutio coideed i itetue e oved oy ueicy. Thu ocedue o ect ytic outio e ubect ti ude ivetigtio. The it oogh coceig thee obe icude ecte-uiuee eut we the ictio o iteg to d oetio ocedue [7-9]. Hee we h coide c o ie o-hoogeeou eutio o e o-itege ode deteied o the iite itev. I ou eviou e [] bic eutio with et-ided Rie-Liouvie deivtive d vibe coeiciet w oved i it hoogeeou veio. The gee outio o uch eutio i the u o cooet Meie -uctio eie. Thei covegece d oetie e thooughy dicued i oogh []. Now we ooe to coide o-hoogeeou eutio o thi tye. We eet the eut obtied o eutio with o-hoogeeou te i the o

2 7 M. Kie. ziebowi o bity ie cobitio o Meie -uctio. Thee uctio e deteied by e vecto but the ccutio c be eiy eteded to the ce o coe vecto. The e i ogized oow. I the et ectio we ec bic deiitio o ctio oeto d thei oetie. Sectio coti the i eut ecoed i Pooitio. d Cooy.3 o the ect gee outio o the coideed o-hoogeeou ctio dieeti eutio. I Sectio 3 we give two ee o the ictio o the ooed ethod to the ce o cott coeiciet d to the ce whe ictio o the eductio oetie o Meie -uctio iiie the outio. The e coe with cocudig e d edi whee eevt oetie o o d Meie uctio e eced.. ctio iteg d deivtive I the e we h tudy bic ctio dieeti eutio with et -ided Rie-Liouvie deivtive. Let u ec the deiitio o the et-ided iteg d deivtive o o-itege ode C [7 8 ]. eiitio. Re > oow: Let. The et-ided Rie-Liouvie iteg o ode i deied I t dt : > Γ t. Uig the bove deiitio o ctio iteg the Rie-Liouvie deivtive i cotucted. eiitio. Let Re. The et-ided Rie-Liouvie deivtive o ode i give by the oowig ou: d I :. d A we e ivetigtig the ctio dieeti eutio o e ode we h ue wht oow: tht. A detied eview o the oetie o the itoduced ctio oeto c be oud i oogh [7 8 ]. We uote hee oy oe o the cooitio ue which we h ue i the outio ocedue.

3 Le.3 Meie -uctio eie ect outio o c o o-hoogeeou ctio 73 Let. The the oowig ou i vid o C[ b] y oit [ b] I d t.3 Let. The the bove cooitio ue i uied o C [ b] d t y oit b]. I the ce o hoogeeou eutio coideed i e [] it eed tht thei outio e Meie -uctio eie. Thece we ec hee the deiitio o thi c o eci uctio. eiitio.4 Let d et Cbe bity coe N ube. The Meie -uctio z i give by the oowig ou [3] i b π i i z b Γ b i Γ i i... z b b... Γ i Γ b z d.4 whee i... ;... d Γ i the ue g uctio. Cotou ete the oe o Γ b d Γ i uctio i the ueto o the bove coe ee i the iteg. uctio o thi tye wee it itoduced by Meie i 936 d the geeized by o i 96. They e widey ied i obbiity theoy ttitic oou diuio theoy o ctio dieeti eutio theoy. We icuded i thi e oe eevt oetie o thi c o uctio - they e ecoed i Aedi A.. No-hoogeeou ctio dieeti eutio with et-ided Rie-Liouvie deivtive Let u coide the o-hoogeeou ctio dieeti eutio o ode :

4 74 M. Kie. ziebowi [ b]. whee R d i the Meie -uctio deied by e vecto d b. The hoogeeou veio o uch eutio w oved uig the Mei to ethod i ou eviou e []. The e ethod yied eut i the ce o ogou eutio with ight-ided deivtive [4] d i ce o eutio with yetic o ti-yetic ctio deivtive [5]. The outio o hoogeeou coutet o eutio. i decibed i the oowig ooitio. Pooitio. > Let { } >. The eutio. h i itev [b] gee outio i the o o whee cooet outio with vecto A c.3 e give by eie A b b.4 b B B deied oow A B [ ; ] e... R... R [ e ; ] d c R bity e coeiciet. Ou i ow i to deive the gee outio o the o-hoogeeou veio o eutio.. It wi coit o the bove gee outio o hoogeeou obe d the ticu outio o eutio.. Let u y the cooitio ue o Le.3 i ode to to the tudied eutio ito it iteg o geetig the ticu outio: I I.5

5 Meie -uctio eie ect outio o c o o-hoogeeou ctio 75 The outio o eutio.5 c be witte o eie I.6 whee.7 b d vecto b beog to the R d R ce eectivey. To obti ect ytic o o ticu outio.6 we houd ow ccute the ctio iteg o the ight-hd ide o ou.6 d yze the covegece o the bove eie. Let u begi with d ccute the ete deteiig the oetie o Meie -uctio [3]:.8 µ b i.9 i i b i o > κ i. o i [ ] o > ϑ. o We h ue tht the Meie -uctio i eutio. ui the coditio κ < ϑ γ µ <. Accodig to Theoe A. the it iteg whe eit ovided the coditio e vid: i [ b ] >.3 whe > o µ < d µ i b >.4

6 76 M. Kie. ziebowi whe < µ <. Both coditio.3 d.4 e ie eut o utio κ <. The eut o ctio itegtio i thi ce i o the Meie -uctio I b [ ; ] [ b; ].5 Whe we obti the oowig vue o coeiciet deteiig the eut o the eviou itegtio I : µ µ κ κ.6 i [ ] o > ϑ.7 o Let u ote tht o the utio o Theoe A. e uied ey ice we hve { } > d dd κ < ϑ we obti the oowig coditio vid: > i [ b ].8 i the ce > o µ < d µ i b >.9 i the ce < whee µ <. o the bove ieuitie d o Theoe A. it oow tht the ecod te i eie.6 i o the o: [ ; ] [ ] I I t t b;. Let u ow coide bity itege. The ete deteiig the oetie o the eective Meie -uctio eutig o the -th itegtio e oow:

7 Meie -uctio eie ect outio o c o o-hoogeeou ctio 77 κ κ µ µ > ] [ i o o ϑ ϑ. I thi gee ce we h ccute the ctio iteg o ode o uctio I. Sii to ce dicued eviouy the utio o Theoe A. o ctio itegtio e uied. The oowig ieuitie e iied by coditio ϑ ϑ κ < : ] [ i > b ovided o < > µ d > µ b i.5 ovided d < whee < µ. The eut o the itegtio i the oow- ig Meie -uctio t t I I.6 with vecto give by ou e ; ; e b We hve eicity ccuted the eeet o the eie decibig the ticu outio i te o Meie -uctio.7

8 78 M. Kie. ziebowi We h ow yze the covegece oetie o the bove eie. To thi i we y the coio tet. Whe κ < γ γ < i[ ϑ ϑ ] ϑ d > o γ µ < we obti the oowig etitio o the oduu o the eective Meie -uctio we y Theoe 3.4 o [3]: Aγ γ.8 whee coeiciet A γ e cott deedet o γ d. Let u ccute thee coeiciet. Accodig to the ccutio ecoed i the oo o Theoe 3.3 o [3] it i oduct o coeiciet A d A give beow / A π e µ γ.9 A e π πτ ih πτ.3 whee τ >. Uig ou.8-.3 i the etitio o outio.7 we obti oizig eie i the o o g γ Aγ.3 The bove oizig eie i boutey coveget o [ b] uig the d Aebet tet ovided eigevue obey the ieuity: i hππτ < b e πτ.3 Re: the bove etictio c be eed uig the ied oit theoe. We h ot dicu the deti o the ocedue i thi e evig it o the et tice. I the outio o the ooitio coceig the outio o eutio. we h oit utio.3. Let u ow chec eicity tht eie.7 ove eutio.. We deote the -th eeet o the eie oow ψ.33

9 Meie -uctio eie ect outio o c o o-hoogeeou ctio 79 d ccute the eut o the ctio o oeto o thi cooet: ψ.34 o the theoe o the dieetitio o Meie -uctio Theoe.8 i oogh [3] it oow tht ψ ; ; uctio c be ewitte uig the eductio oety give i Aedi A o to obti the ou: ψ.35 vid o y N with vecto d b. We ow y the deived ou o te ψ d obti o eie.7 the oowig eut.36 which te ubtitutio to eutio. yied b.37 o the bove coidetio the ooitio decibig the gee outio o eutio. oow.

10 8 M. Kie. ziebowi Pooitio. Let > d { } >. Let be bity Meie -uctio deteied t et o b] by e vecto d b uiig the utio o Theoe A. d coditio κ < ϑ whee κ d ϑ e give by ou..7. The dieeti eutio o ode : h i itev [ b] gee outio i the o o the oowig u c whee cooet outio e give by the eie A b b b B with vecto A B give by ou A B [ ; ] [ e ; ] [ b; e ] e... R... R [ e ; ] d bity e coeiciet c R. The bove ooitio c be eiy eteded to the ce whee the o-hoogeeou te o the ight-hd ide o eutio. i ie cobitio o the Meie -uctio obeyig the coeodig utio.

11 Meie -uctio eie ect outio o c o o-hoogeeou ctio 8 Cooy.3 Let > d { } >. Let be bity Meie -uctio deteied t et o b] by e vecto d b uiig the utio o Theoe A. d coditio κ < o... R whee d ϑ e give by ou..7. The dieeti eutio o ode : h i itev [ b] whee cooet outio ϑ R C gee outio i the o o the oowig u c R C e give eie A b b b B with vecto decibed i deti by Pooitio.. κ 3. e We h tudy i deti the two ee eutio oved i Pooitio.. We choe uch vue o ete to which the eductio oetie o Meie uctio c be ied. 3.. e: ce. Let u ove eutio. whe : [ b ] 3. Autio o Pooitio. ed > d κ < ϑ. oowig the eut o e [] we obti the gee outio o the hoogeeou t o eutio 3. i the o o ie cobitio o Mittg-Lee uctio [7]:

12 8 M. Kie. ziebowi c o Pooitio. d Poety A. we ie tht the ticu outio o the o-hoogeeou t o eutio 3. i give 3.4 whee vecto beog to the R d R ce eectivey d oo oow ; [ b; ] Cocudig the gee outio o eutio 3. i cotucted uig the - outio d outio the oowig u c e: ce /. I the et ee we tudy ce. Let u obeve tht ccodig to utio { } >. Thu it euie eutio. becoe eutio o ctio ode d we ue tht the coditio o Pooitio. egdig Meie -uctio o the ighthd ide e uied: / [ b] 3.6 The outio decibed i Pooitio. icude the gee outio o the hoogeeou t: whee the cooet eie w deived i e [] c 3.7 A b b b B

13 Meie -uctio eie ect outio o c o o-hoogeeou ctio 83 with vecto A B give by ou ; ; A B ;; The ecod t o the outio i ovided by Pooitio. the oowig eie 3.9 with vecto i the o o ;; b; ; Cocudig the outio o eutio 3.6 i the u o the bove eie 3.8 d 3.9 c i e We hve itoduced ethod o ovig bic o-hoogeeou ctio dieeti eutio. It icude ictio o the itegtio oetie o Meie -uctio. A c o eutio icudig o-hoogeeou te bity ie cobitio o the Meie uctio i oved yig the dicued ocedue. Ceu yi how tht the outio techiue c be eteded to eutio with o uctio i the o-hoogeeou te. Ou uthe go i to y the eut o the coideed eutio to c o eueti o-hoogeeou ctio dieeti eutio. Reeece [] Hie R. d. Aictio o ctio Ccu i Phyic Wod Scietiic Sigoe. [] Mgi R.L. ctio Ccuu i Bioegieeig Bege Houe Pubihe Reddig 6. [3] Wet B.J. Boog M. igoii P. Phyic o ctio Oeto Sige-Veg Bei 3. [4] Agw O.P. Teeio-Mchdo J.A. Sbtie J. d. ctio eivtive d thei Aictio: Noie yic 38 Sige-Veg Bei 4. [5] Metze R. Kte J. J. Phy. A 4 37 R6. [6] He R. J. Phy. : Nuc. Phy

14 84 M. Kie. ziebowi [7] Kib A.A. Sivtv H.M. Tuio J.J. Theoy d Aictio o ctio ieeti utio evie Ated 6. [8] Mie K.S. Ro B. A Itoductio to the ctio Ccuu d ctio ieeti utio Wiey d So New Yo 993. [9] Poduby I. ctio ieeti utio Acdeic Pe S iego 999. [] Kie M. ziebowi. Mei To o ctio ieeti utio with Vibe Poteti [I:] Noie Sciece d Coeity d: J.A. Teeio-Mchdo M.. Siv R. Bbo Sige-Veg Bei 9. [] Kie M. O Soutio o Lie ctio ieeti utio o Vitio Tye Czetochow Uiveity o Techoogy Pe Czetochow 9 I e. [] So S.. Kib A.A. Michev O.I. ctio Iteg d eivtive odo & Bech Ated 993. [3] Kib A.A. Sigo M. H-To Theoy d Aictio Ch & H/CRC Boc Rto 4. [4] Kie M. ziebowi. Sci. Re. It. Mth. Co. Sciece [5] Kie M. O ogue o eoeti uctio o tiyetic ctio deivtive. Coute d Mthetic with Aictio doi:.6/.cw Aedi A Hee we ec the bic oetie o o d Meie uctio which we ied i the ocedue o ovig the ctio dieeti eutio [3]. Poety A. I b the the oowig eductio ou hod z z A. b... b b... b ovided d >. I b the the eductio ou hod ovided d > z z A. b... b b... b Theoe A. Let C whee R e > ω C d σ >. I the coditio hod b R e σ i R e ω > A.3

15 Meie -uctio eie ect outio o c o o-hoogeeou ctio 85 ovided Re o < > µ o > ω µ σ e R R e b R e i A.4 ovided < whee < µ e R the the et-ided ctio iteg o the o H-uctio eit d i give by the ou beow: σ ω σ ω σ ω σ ω i i i i b H b t H t I A.5