MULTIPLE MELLIN AND LAPLACE TRANSFORMS OF I-FUNCTIONS OF r VARIABLES

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1 Jounal of Factional Calculus and Applications Vol.. July 2 No. pp. 8. ISSN: MULTIPLE MELLIN AND LAPLACE TRANSFORMS OF I-FUNCTIONS OF VARIABLES B.K. DUTTA L.K. ARORA Abstact. The aim of this pape is to study multiple Mellin and Laplace tansfoms involving multivaiable I-function. In this egad we have poved five theoems and a few coollaies have also been ecoded. Simila esults fo the H-function of two and vaiables obtained by othe authos follow as special case of ou findings.. Intoduction In ecent yeas many authos pointed out that deivatives and integals of factional ode ae suitable fo desciption of popeties of vaious eal mateials. It has poved to be vey useful tool fo modeling of many phenomena in physics chemisty engineeing bioscience and othe aeas. The main advantage of factional deivatives povide an excellent instument fo the desciption of memoy and heeditay popeties of vaious mateials and pocesses. Wheeas the classical intege ode models neglected such effects. Fo moe details we efe to [ ]. The Mellin tansfom may be egaded as the multiplicative vesion of the two-sided Laplace tansfom. It is closely connected to Laplace tansfom Fouie tansfom theoy of the gamma function and allied special functions. The poblem of deivation the exact solutions fo factional diffeential equations with mixed deivatives is an impotant and emeging aea in factional calculus. The Mellin Tansfom is widely used in factional calculus because of its scale invaiance popety [3]. Klimek Dziembowski [6] and Klimek [7] poposed to apply the Mellin tansfom method fo factional diffeential equations. Recently the multivaiable I-function has been intoduced and studied by Pasad [7] and Pasad and Yadav [8] which is a genealization of multivaiable H- function. Futhe Pasad and Singh [4] studied the Mellin and Laplace tansfom of the multivaiable I-function. They obseved that the Mellin tansfom of I-function of vaiables educes to the I-function of vaiables. The deivatives of multivaiable I-function has been studied by Saxena and Singh []. Chauasia and Kuma [5] investigated the factional integals of poduct of H-function [] and mutivaiable I-function. 2 Mathematics Subect Classification. 33C6 33C7 44A2 44A3. Key wods and phases. Multiple Integal tansfoms Multivaiable H-function Tansfoms of special functions. Submitted Jan Published July 2.

2 2 B.K. DUTTA L.K. ARORA JFCA-2/ In this pape we have studied vaious multiple Mellin and Laplace tansfom of I- function. Since the multivaiable I-function is of geneal natue its multiple Mellin and Laplace tansfoms educes to many simple special functions as paticula cases. Fo convenience we fist ecall some definitions and fundamental facts of integal tansfom and special functions. Definition. The Mellin tansfom of a function φt) t > is defined see [3] and [2]) as follows φ s) Mφ; s t s φt)dt ) and if the function φt) also satisfies the Diichlet conditions evey closed inteval [a b] ) then the function φt) can be estoed using invese Mellin tansfom fomula φx) M φ s); x 2πi γ+i γ i φ s)x s ds < x < ) 2) whee s is a complex such that γ < Rs) < γ 2 and γ < γ < γ 2. The Mellin tansfom ) exists if the function φt) is piecewise continuous in evey closed inteval [a b] ) and φt) t γ dt < φt) t γ 2 dt <. 3) Definition.2 The multidimensional Mellin tansfom fo the function φt) φt t ) define as follows [2] Mφ)s) : t s φt)dt R) n + + t s t s φt t )dt dt 4) whee t t >. Definition.3 The multidimensional Laplace tansfom define fo the function fx x ) as follows Lf)s) : Lfx x ); s s ) exp s i x i fx x )dx dx 5) whee Rs i ) > i. Definition.4 The multivaiable I-function epesent [7] as z I[z z ] : I ni 2 :mi) n i). A : B p i q i 2 :p i) q i) C : D z 2πω) ψζ ζ ) ϕ i ζ i )z L ζ i i dζ dζ 6) L whee ω

3 JFCA-2/ MULTIPLE MELLIN AND LAPLACE TRANSFORMS OF 3 ψζ ζ ) ϕ i ζ i ) [ nk k2 Γ a k + ] k αi) k ζ i) [ pk k2 n k + Γa k ] k αi) k ζ i) [ qk k2 Γ b k + k [ p i) i. Also [ m i) k Γbi) k n i) + Γai) βi) k ζ i) α i) ζ i) βi) ][ n i) ][ q i) ] 7) k ζ i) ] Γ ai) + α i) ζ i) km i) + n i 2 : n 2 : : n Γ bi) k + βi) k ζ i) ] 8) p i q i 2 : p 2 q 2 : : p q m i) n i) :m ) n ) ); ; m ) n ) ) p i) q i) :p ) q ) ); ; p ) q ) ) ) 2 A :: a i; α ) i αi) i :a 2 ; α ) 2 p α2) 2 ) p 2 ; ; a ; α ) α) ) p i ) B :: a i) αi) :a ) α ) ) p ); ; a ) α ) ) p ) p i) C :: b i ; β ) i βi) ) D :: b i) βi) q i) i )2 q i :b 2 ; β ) 2 β2) 2 ) q 2 ; ; b ; β ) β) ) q :b ) β ) ) q ); ; b ) β ) ) q ) such that n i p i q i m i) n i) p i) q i) ae non-negative integes and all a i b i α i β i a i) bi) αi) βi) ae complex numbes and the empty poduct denotes unity. The contou integal 6) conveges if whee n i) U i α i) p i) n i) + n + + α i) ag z i < 2 U iπ U i > i 9) m i) α i) + p β i) α i) n + q i) β i) + m i) + q 2 β i) n q α i) 2 β i) p2 α i) 2 n 2 + ) and I[z z ] O z α z α ) max z z whee α i min m i) Rb i) /βi) ) and β i max n i) Ra i) )/α i) ) i. Fo the condition of convegence and analyticity of multivaiable I-function we efe [7 8].

4 4 B.K. DUTTA L.K. ARORA JFCA-2/ 2. Main Results The main esults to be established hee as follows Theoem 2. Suppose the conditions 9) to be satisfied. The Multiple Mellin tansfom of I-function of vaiable define in 6) as follows MI)s) t s i i I[z t µ z t µ ]dt dt whee µ i min m i) R z s i µ i i µ i ϕ i s i µ i ) ψ s s ) µ µ ) i) b < Rs β i) i ) < µ i min R n i) ) i) a α i) ψt t ) and ϕt i ) i ae given in 7) and 8) espectively. Poof Fist we expess the multivaiable I-function in left side of the integation of ) as a poduct of multiple Mellin-Banes contou integal by using 6) and intechanging the ode of integation which is pemissible unde the above stated conditions. Afte a staightfowad calculation we finally aived at ). Coollay 2.. When n i p i q i i 2 the empty poduct denotes unity) the ) educes to the multiple Mellin tansfomation of H-function of vaiables [2 p. 25]. Coollay 2.2. When 2 and n 2 p 2 q 2 in ) it educes to the double Mellin tansfomation of H-function of two vaiables [2 p. 47]. Theoem 2.2 Suppose the conditions 9) to be satisfied. Then Multiple Laplace tansfom of multivaiable I-function is as follows ) LI)s) exp s i t i t ρ i i I[z t µ z t µ ]dt dt z s µ s ρ i n i i I 2 :m i) n i) + A : B p i q i 2 :p i) +q i). C : D 2) z s µ whee µ i > Rs i ) > Rρ i ) + µ i min R m i) A C a ; α ) α) )2 p i B b ; β ) β) )2 q i D ) i) b > i β i) ρ i µ i ) b i) βi) q i) a i) ) αi) p i) Poof Fist we expess the multivaiable I-function on left side of the integation of 2) as a poduct of multiple Mellin-Banes contou integal by using 6) and intechanging the ode of integation which is pemissible unde the above stated )

5 JFCA-2/ MULTIPLE MELLIN AND LAPLACE TRANSFORMS OF 5 conditions and appeal to the Eule s integal of the fist kind [3 p. ]. Afte little aangement we finally aive at 2). Coollay 2.3. When n i p i q i i 2 the empty poduct denotes unity) the 2) educes to the multiple Laplace tansfomation of multivaiable H-function of vaiables. Coollay 2.4. When 2 and n 2 p 2 q 2 in 2) it educes to the double Laplace tansfomation of H-function of two vaiables [2 p. 48]. Theoem 2.3 Suppose the conditions 9) to be satisfied. Then Multiple Laplace tansfom of multivaiable I-function as follows ) LI)s) exp s i t i t ρ i i I[z t µ z t µ ]dt dt z s µ s ρ i n i i I 2 :m i) +n i) A : B p iq i 2 :p i) q i) +. C : D 3) z s µ ) i) a whee µ i > Rs i ) > Rρ i ) µ i max R > i n i) A C a ; α ) α) )2 p i B b ; β ) β) )2 q i D a i) α i) αi) ρ i µ i ) b i) p i) βi) q i) Poof The poof of 3) is same as 2). Coollay 2.5. When n i p i q i i 2 the empty poduct denotes unity) the 3) educes to the multiple Laplace tansfomation of multivaiable H-function of vaiables. Coollay 2.6. When 2 and n 2 p 2 q 2 in 3) it educes to the double Laplace tansfomation of H-function of two vaiables [2 p. 48]. Theoem 2.4 Suppose the conditions 9) to be satisfied and µ i > > n i n i i. Then the Multiple Mellin tansfom of the poduct of two I-function of vaiable defined in 6) as follows MI I)s) η s i i t si i I[z t µ z t µ I 2 :m i) +n i) m i) +n i) ]I [η t ν η t ν ]dt dt p i +q i p i +q i :p i) +q i) p i) +q i) 2 z η s ν.. z η s ν A:B C :D 4)

6 6 B.K. DUTTA L.K. ARORA JFCA-2/ whee i) b µ i min R m i) µ i min n i) R β i) ) a i) α i) min m i) ) R ) b i) β i) + min R n i) < Rs i ) < ) a i) α i) i and A a i ; α ) i αi) i B a i) αi) p i) C b i ; β ) i βi) i D b i) βi) q i) ; ) 2 ; p i b i b i) s i ) 2 ; q i a i i k s k ν k β k) i ; µ β i) µ i β i) i k β ) i ν q i) s k ν k α k) i ; µ a i) s i α i) µ i α i) µ i β i) i α ) i ν p i) µ i α i) i Poof To pove the integal fomula 4) we expess the fist I-function on its lefthand-side as a multiple Mellin-Banes contou integal by using 6) and intechange the ode of integation which is pemissible unde the above stated conditions. Evaluate the inne integal with the help of ) afte staight calculation we finally aive at 4). Coollay 2.7. When p i q i p i q i i 2 the empty poduct denotes unity) the 4) educes to the multiple Mellin tansfomation of H-function of vaiables. Coollay 2.8. When 2 and ν ν 2 p 2 q 2 p 2 q 2 in 4) it educes to the double Mellin tansfomation of H-function of two vaiables [2 p. 48]. Theoem 2.5 Suppose the conditions 9) to be satisfied and µ i > > n i n i i. Then the Multiple Mellin tansfom of the poduct of two I-function of vaiable define in 6) as follows MI I)s) η s i i t si i I[z t µ z t µ I 2 :m i) +m i) n i) +n i) ) 2 q i ) 2 p i ]I [η t ν η t ν ]dt dt p i +p i q i+q i :p i) +p i) q i) +q i) 2 s ν z η. z η s ν A:B C :D 5)

7 JFCA-2/ MULTIPLE MELLIN AND LAPLACE TRANSFORMS OF 7 whee µ i min n i) R i and A a i ; α ) i αi) i B a i) αi) ) i) a α i) µ i min m i) R p i) ; C b i ; β ) i βi) i D b i) βi) q i) ; ) 2 ; p i b i) β i) a i + a i) + s i ) 2 ; q i min m i) ) i k α i) i b i + k R ) b i) β i) + min R n i) s k ν k α k) i ; µ µ i α i) α ) i ν p i) s k ν k β k) i ; µ b i) + s i β i) µ i β i) < Rs i ) < ) a i) α i) β ) i ν q i) Poof To pove the integal fomula 5) same as 4). µ i α i) i ) 2 µ i β i) i Coollay 2.9. When n i p i q i n i p i q i i 2 and n n the empty poduct denotes unity) the 5) educes to the multiple Mellin tansfomation of H-function of vaiables. Coollay 2.. When n i p i q i n i p i q i i 2 n n and ν k k then the 5) educes to the esult of []. Coollay 2.. When 2 and ν ν 2 p 2 q 2 p 2 q 2 in 5) it educes to the double Mellin tansfomation of H-function of two vaiables [2 p. 49]. 3. Conclusion The I-function of vaiables defined by Pasad [7] and Pasad and Yadav [8] in tems of the Mellin-Banes type of basic integals is most geneal in chaacte which involves a numbe of special functions. The esults deduced in the pesent pape may povide bette multiple Mellin and Laplace tansfoms of some simple multivaiable special functions. Refeences p i ) 2 [] A.A. Inayat-Hussain New popeties of hypegeometic seies deivable fom Feynman integals A genealization of the H-function J. Phy. A : Math. Gen ) [2] H.M. Sivastava K.C. Gupta S.P. Goyal The H-functions of one and two vaiables with applications South Asian Publishes New Delhi 982. [3] I. Podlubny Factional Diffeential Equations Academic Pess San Diego 999. [4] J. Sabatie O.P. Agawal J.A.T. Machado Advances in factional calculus: theoetical developments and applications in physics and engineeing Spinge 27. [5] K. Diethelm The Analysis of Factional Diffeential Equations. Spinge Belin 2. q i

8 8 B.K. DUTTA L.K. ARORA JFCA-2/ [6] M. Klimek D. Dziembowski Mellin tansfom fo factional diffeential equation with vaiable potential. Poceedings of the 2nd Confeence on Nonlinea Science and Complexity Poto 28. Eds. J.A. Teneio-Machado M.F. Silva R.S. Babosa L.B. Figueiedo CD-ROM 28. [7] M. Klimek G-Meie functions seies as solutions fo some Eule-Lagange equations of factional mechanics. Poceedings of the 6 th EUROMECH Nonlinea Dynamics Confeence Saint Petesbug 28. Eds. B.R. Andievsky A.L. Fadkov CD-ROM 28. [8] R. Hilfe Applications of factional calculus in physics Wold Scientific 2. [9] R. Magin Factional Calculus in Bioengineeing Begell House Redding 26. [] R.K. Saxena Y. Singh On the deivatives of the multivaible I-function Vianana Paishad Anusandhan Patika 362) 993) [] R.S. Gag On multidimensional Mellin convolutions and H-function tansfomations Indian J. Pue Appl. Math. 3I) 982) [2] S.G. Samko A.A Kilbas O.L. Maichev Factional integals and deivatives: theoy and applications Godon and Beach Science Publishes 993. [3] S. Das Functional factional calculus fo system identification and contols Spinge 28. [4] T. Pasad N.P. Singh Mellin and Laplace tansfom of the multivaiable I-function Ganita Sandesh 42) 99) [5] V.B.L. Chauasia D. Kuma A family of factional integals petaining to multivaiable I-function Appl. Math. Sci. 43) 2) [6] V.E. Taasov Factional dynamics: applications of factional calculus to dynamics of paticles felds and media Spinge 2 [7] Y.N. Pasad Multivaiable I-function Vianana Paishad Anusandhan Patika 294) 986) [8] Y.N. Pasad G.S. Yadav Poc. Math. Soc. B.H.U. I 985) B.K. Dutta Depatment of Mathematics NERIST Niuli Aunachal Padesh India addess: dutta.bk@gmail.com L.K. Aoa Depatment of Mathematics NERIST Niuli Aunachal Padesh India addess: lkaoa neist@yahoo.com

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