THE Mittag-Leffler (M-L) function, titled as Queen -

Transcript

1 nternational Journal of Matheatical and Coputational Sciences Vol:4, No:8, 2 Rieann-Liouville Fractional Calculus and Multiinde zrbashjan-gelfond-leontiev ifferentiation and ntegration with Multiinde Mittag-Leffler Function U.K. Saha, L.K. Arora nternational Science nde, Matheatical and Coputational Sciences Vol:4, No:8, 2 waset.org/publication/547 Abstract The ultiinde Mittag-Leffler M-L function and the ultiinde zrbashjan-gelfond-leontiev -G-L differentiation and integration play a very pivotal role in the theory and applications of generalized fractional calculus. The object of this paper is to investigate the relations that eist between the Rieann-Liouville fractional calculus and ultiinde zrbashjan-gelfond-leontiev differentiation and integration with ultiinde Mittag-Leffler function. Keywords Multiinde Mittag-Leffler function, Multiinde zrbashjan-gelfond-leontiev differentiation and integration, Rieann-Liouville fractional integrals and derivatives.. NTROUCTON THE Mittag-Leffler M-L function, titled as Queen - function of fractional calculus FC due to ainly its applications in the solutions of fractional-order differential and integral equations arising probles of atheatical, physical, biological and engineering areas. The Mittag-Leffler M-L functions E Mittag-Leffler, and E,β Wian 95, Agarwal 953 are defined by the power series E z Γk+,> } E,β z Γk+β,>,β >. An iportant identity of M-L function 4 is E,β z Γβ + ze,+βz, 2 which will be required later on. n the section 2 of the book by Sako, Kilbas and Marichev 9, the left and right sided operators of Rieann-Liouville fractional calculus are defined as follows: f ft dt, > 3 Γ t f ft dt, > 4 Γ t f d + {} f + d ft,> 5 Γ {} t {} U.K. Saha e-ail:uksahanerist@gail.co, L.K. Arora eail: lkarora nerist@yahoo.co are with the epartent of Matheatics, North Eastern Regional nstitute of Science and Technology, Nirjuli-79 9, tanagar, Arunachal Pradesh, NA. f d + {} f d + ft,>, 6 Γ {} t {} where eans the aial integer not eceeding and {} is the fractional part of.. MULTNEX M-L FUNCTON AN MULTNEX -G-L FFERENTATON AN NTEGRATON The ultiinde ultiple, -tuple M-L function is introduced by Kiryakova 6, 7 and the ultiinde -G-L differentiation and integration, generated by the ultiinde M-L function are introduced and studied by Kiryakova 8. efinition. 6, 7 Let >be an integer,,,ρ > and μ,,μ be arbitrary real nubers, then the ultiinde M-L function are defined by eans of the power series E,μ iz φ k j Γμ j + k. 7 For, this is the classical M-L function E ρ,μ z, and for ρ >,μ β>, it is the M-L function E,βz considered by Wian 95 and Agarwal 953, which is given in. For 2, 7 reduces to the M-L function considered first by zrbashjan 3. He denoted it by φ ρ,ρ 2 z; μ,μ 2 and defined in the following for, see also 8, Appendi: E, ρ,μ 2,μ 2z φ ρ,ρ 2 z; μ,μ 2 Γμ + k Γμ 2 + k ρ 2 8 efinition.2 8, 6, 5 Let fz be analytic function in a disk Δ R { z <R} and >,μ i Ri,, be arbitrary paraeters, then the correspondences: fz a k fz ρi,μ ifz, Lfz L ρi,μ ifz, defined by fz k a k j Lfz a k j Γμj+ k ρj Γμ j+ k ρ j Γμj+ k ρj Γμ j+ k+ + 9 nternational Scholarly and Scientific Research & nnovation scholar.waset.org/999.7/547

2 nternational Journal of Matheatical and Coputational Sciences Vol:4, No:8, 2 nternational Science nde, Matheatical and Coputational Sciences Vol:4, No:8, 2 waset.org/publication/547 are called ultiinde -G-L differentiations and integrations, respectively. For, the operators 9 are -G-L operators of differentiation and integration, studied by iovski and Kiryakova, 2, Kiryakova 8: ρ,μ fz L ρ,μ fz k a Γμ+ k ρ k Γμ+ k ρ zk a Γμ+ k ρ k Γμ+ k+ ρ zk+ Lea.3 6, 7 The ultiple M-L function 7 satisfy the following relations λ : ρi,μ ie,μ iλz λe,μ iλz L ρi,μ ie,μ iλz λ E,μ iλz λ j Γμ j 2 One can verify the above lea easily by applying the definition 7 and 9.. RELATONS WTH REMANN-LOUVLLE FRACTONAL CALCULUS n this section we derive certain relations that eist between the ultiinde -G-L operators of differentiation and integration connecting with ultiinde M-L function and the left and right sided operators of Rieann-Liouville fractional integrals and derivatives. Theore. Let >, >,μ i Ri,,,λ, and let be the left sided operator of Rieann-Liouville fractional integral 3, then there holds the forula t μ E ρ,μ i iλt +μ E ρ,μ i + ρ,μ 2 ρ,,μ 2 ρ λ Γμ + j2 Γμ j Proof: By virtue of 3 and, we have Γ t μ E ρ,μ i iλt t λ λ k t k +μ j Γμ j + k dt 3 nterchanging the order of integration and suation and evaluating the inner integral by eans of beta function forula, it gives +μ λ k+ k Γμ + + k j2 Γμ j + k +μ λ k k Γμ k + + k j2 Γμ j + k +μ ρ λ k k Γμ + + k j2 Γμ j + k Γμ + j2 Γμ j +μ ρ E ρ,μ i + ρ,μ 2 ρ,,μ 2 ρ λ Γμ + j2 Γμ j Corollary.2 Let >, >,μ i Ri,,,λ and, t μ E, ρ,, 2 ρ,μi λt E μ, ρ,, 2 ρ,μ,μ2 ρ,,μ 2 ρ λ Γμ j2 Γμ j 4 For, and μ μ, 3 reduces to Corollary.3 For >,ρ >,μ R and λ, t μ ρ,μ E,μ λt μ E,μ λ Γμ 5 Theore.4 Let >, >,μ i Ri,,,λ, and let be the left sided operator of Rieann-Liouville fractional integral 3, then there holds the forula t μ LE ρ,μ i iλt λ +μ E,μ +,μ 2,,μ λ Proof: By virtue of 3 and 2, we have Γ t μ LE ρ,μ i iλt t λ Γμ + j2 Γμ j 6 tμ j Γμ dt j λ k t k +μ j Γμ j + k Now splitting the integral into two integrals and changing the order of integration and suation in the first integral and then evaluating the two integrals with the help of beta function forula, we obtain λ k k λ +μ Γμ + j2 Γμ j Γμ ++ k j2 Γμj+ k λ E +μ ρ,μ i +,μ 2,,μ λ Γμ + j2 Γμ j This copletes the proof of the theore.4 nternational Scholarly and Scientific Research & nnovation scholar.waset.org/999.7/547

3 nternational Journal of Matheatical and Coputational Sciences Vol:4, No:8, 2 nternational Science nde, Matheatical and Coputational Sciences Vol:4, No:8, 2 waset.org/publication/547 For, β and μ μ, the above theore reduces to Corollary.5 For >,ρ >,μ R and λ, there holds the forula t μ L ρ,μ E β,μ λt β λ +μ E β,μ+ λ β 7 Γμ + On using the identity 2 on the right hand side of 7, it reduces to the result t μ L ρ,μ E β,μ λt β +β+μ E β,+β+μ λ β 8 Theore.6 Let >, >,μ i Ri,,,λ, and let be the right sided operator of Rieann-Liouville fractional integral 4, then t μ E ρ,μ i iλt μ+ E ρ,μ i + ρ,μ 2 ρ,,μ 2 ρ λ Γμ + j2 Γμ j 9 Proof: By virtue of 4 and, we have t μ E ρ,μ i iλt t λ k t k t μ λ Γ j Γμ j + k dt f we interchange the order of integration and suation and substitute t u to evaluate the inner integral, we obtain Γ μ μ k μ+ λ k+ μ k j Γμ j + k λ k+ k Γμ ++ k j2 Γμj+ k Γμ ++ k λ k k k Γμj+ j2 ρ j u k +μ u du λ k k Γμ + + k j2 Γμ j + k Γμ + j2 Γμ j μ+ E ρ,μ i + ρ,μ 2 ρ,,μ 2 ρ λ Γμ + j2 Γμ j Corollary.7 Let >, >,μ i Ri,,,λ and, t μ E, ρ,, 2 ρ,μi λt E μ+, ρ,, 2 ρ,μ,μ2 ρ,,μ 2 ρ λ Γμ j2 Γμ j For, and μ μ, 9 reduces to 2 Corollary.8 For >,ρ >,μ R and λ, t μ ρ,μ E,μ λt μ+ E,μ λ Γμ 2 Theore.9 Let >, >,μ i Ri,,,λ, and let be the right sided operator of Rieann-Liouville fractional integral 4, then t μ LE ρ,μ i iλt λ E μ ρ,μ i +,μ 2,,μ λ Γμ + j2 Γμ j 22 Proof: By virtue of 4 and 2, we have t μ LE ρ,μ i iλt t t μ λ k t k Γ λ j Γμ j + k j Γμ dt j Now splitting the integral into two integrals and interchanging the order of integration and suation in the first integral and substitute t u to evaluate the integrals, we obtain λ k k λ μ Γμ + j2 Γμ j Γμ ++ k j2 Γμj+ k λ E μ ρ,μ i +,μ 2,,μ λ Γμ + j2 Γμ j For, β and μ μ, 22 reduces to Corollary. For >,ρ >,μ R and λ, t μ L ρ,μ E β,μ λt β λ μ E β,μ+ λ β Γμ + 23 nternational Scholarly and Scientific Research & nnovation scholar.waset.org/999.7/547

4 nternational Journal of Matheatical and Coputational Sciences Vol:4, No:8, 2 nternational Science nde, Matheatical and Coputational Sciences Vol:4, No:8, 2 waset.org/publication/547 Making use of the identity 2, 23 reduces to the result t μ L ρ,μ E β,μ λt β μβ E β,+β+μ λ β 24 We now proceed to derive certain other theores of E ρ,μ i iλz and LE ρ,μ i iλz associated with the fractional derivative operators and defined by 5 and 6, respectively. Theore. Let >, >,μ i Ri,,,λ, and let be the left sided operator of Rieann-Liouville fractional derivative 5, then there holds the forula t μ E ρ,μ i iλt μ E,μ,μ 2 ρ 2,,μ ρ λ Γμ j2 Γμ j Proof: By virtue of 5 and, we have d t μ E ρ,μ i iλt + {} λ k+ j Γμ j + k 25 t μ E ρ,μ i iλt + d Γ {} t {} t k ρ +μ dt λ k+ Γ k + μ +{} j2 Γμ j + k + d k ρ +μ {} λ k+ k ρ +μ Γ k + μ j2 Γμ j + k μ λ k+ k Γμ + k j2 Γμ j + k μ λ k k Γμ k + k j2 Γμ j + k μ ρ λ k k Γμ + k j2 Γμ j + k Γμ j2 Γμ j μ ρ E ρ,μ i ρ,μ 2 ρ,,μ 2 ρ λ Γμ j2 Γμ j For, β and μ μ, 25 reduces to Corollary.2 For >,ρ >,μ R and λ, t μ ρ,μ E β,μ λt β μβ E β,μβ λ β Γμ β 26 Theore.3 Let >, >,μ i Ri,,,λ, and let be the left sided operator of Rieann-Liouville fractional derivative 5, then there holds the forula t μ LE,μ iλt λ E μ ρ,μ i,μ 2,,μ λ Γμ j2 Γμ j 27 The proof of the above theore can be developed by the siilar lines to that of the theore. For, β and μ μ, 27 reduces to Corollary.4 For >,ρ >,μ R and λ, t μ L ρ,μ E β,μ λt β λ μ E β,μ λ β Γμ On using the identity 2, 28 reduces to the result t μ L ρ,μ E β,μ λt β 28 μ+β E β,μ+β λ β 29 Theore.5 Let >, >,μ i Ri,,,μ >,λ, and let be the right sided operator of Rieann-Liouville fractional derivative 6, then there holds the forula t μ E,μ iλt μ+ E ρ,μ i ρ,μ 2 ρ,,μ 2 ρ λ Γμ j2 Γμ j Proof: By virtue of 6 and, we have t μ E ρ,μ i iλt 3 d + {} t μ E ρ,μ i iλt nternational Scholarly and Scientific Research & nnovation scholar.waset.org/999.7/547

5 nternational Journal of Matheatical and Coputational Sciences Vol:4, No:8, 2 nternational Science nde, Matheatical and Coputational Sciences Vol:4, No:8, 2 waset.org/publication/547 λ k+ j Γμ j + k d t {} t k +μ dt λ k+ + Γ {} Γμ + k j2 Γμ j + k d + {} k ρ +μ + λ k+ k ρ μ Γμ + k j2 Γμ j + k λ k k μ Γμ k + k j2 Γμ j + k μ+ λ k k Γμ + k j2 Γμ j + k Γμ j2 Γμ j μ+ E ρ,μ i ρ,μ 2 ρ,,μ 2 ρ λ Γμ j2 Γμ j f we set, β,μ μ, 3 reduces to Corollary.6 f >,ρ >,μ R,μ > and λ, then t μ ρ,μ E β,μ λt β μ+β E β,μβ λ β 3 Γμ β Theore.7 Let >, >,μ i Ri,,,μ >,λ, and let be the right sided operator of Rieann-Liouville fractional derivative 6, then there holds the forula t μ LE,μ iλt λ E μ ρ,μ i,μ 2,,μ λ Γμ j2 Γμ j 32 The above theore can be proved by the siilar lines to that of the theore.5 and λ, t μ L ρ,μ E β,μ λt β λ μ E β,μ λ β Γμ 33 On using the identity 2, 33 reduces to the result t μ L ρ,μ E β,μ λt β μβ E β,β+μ λ β 34 V. CONCLUSON t is epected that soe of the results derived in this survey ay find applications in the solution of certain fractional order differential and integral equations arising probles of physical sciences and engineering areas, where the -G-L differentiation and integration as well as ultiinde M-L functions leading a pivotal role. REFERENCES. iovski, V. Kiryakova, Convolution and Coutant of Gelfond- Leontiev Operator of ntegration, Proceedings of the Constructive Function Theory, Varna 98, Publ. House BAS, Sofia, 983, iovski, V. Kiryakova, Convolution and ifferential Property of the Borel-zrbashjan Transfor, in: Proceedings of the Cople Analysis and Applications, Varna 98, Publ. House BAS, Sofia, 984, M.M. zrbashjan, On the ntegral Transforation Generated by the Generalized Mittag-Leffler function in Russian, zv. AN Ar. SSR 33 96, A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricoi, Higher Transcendental Functions,Vol., NewYork, Toronto, London: McGraw-Hill Book Copany, V. Kiryakova, The Multi-inde Mittag-Leffler Functions as an portant Class of Special Functions of Fractional Calculus, Coputers and Math. With Appl , V. Kiryakova, Multiinde Mittag-Leffler Functions, Related Gelfond- Leontiev Operators and Laplace Type ntegral Transfors, Fract. Calc. Appl. Anal , V. Kiryakova, Multiple ultiinde Mittag-Leffler Functions and Relations to Generalized Fractional Calculus, J. Coput. Appl. Math. 8 2, V. Kiryakova, Generalized Fractional Calculus and Applications, Research Notes in Math. Series, Vol. 3, NewYork: Prita Longan, Harlow and Wiley, S.G. Sako, A.A. Kilbas and O.. Marichev, Fractional ntegrals and erivatives: Theory and Applications, Gordon and Breach Science Publishers, Reading, PA, 993., β,μ μ, 32 reduces to Corollary.8 f >,ρ >,μ R,μ > nternational Scholarly and Scientific Research & nnovation scholar.waset.org/999.7/547