Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 S317 GENERAL FRACTIONAL CALCULUS OPERATORS CONTAINING THE GENERALIZED MITTAG-LEFFLER FUNCTIONS APPLIED TO ANOMALOUS RELAXATION by Xiaojun YANG a b * a Sae Key Laboraory for Geomechanic an Deep Unergroun Engineering China Univeriy of Mining an Technology Xuzhou China b School of Mechanic an Civil Engineering China Univeriy of Mining an Technology Xuzhou China Original cienific paper hp://oi.org/1.2298/tsci1751196y In hi paper we are a family of he general fracional calculu operaor of Wiman an Prabhakar ype for he fir ime. The general Miag-Leffler funcion o rucure he kernel funcion of he fracional orer erivaive operaor an heir Laplace inegral ranform are coniere in eail. The formulaion are a he mahemaical ool propoe o inveigae he anomalou relaxaion. Key wor: general fracional calculu operaor Miag-Leffler funcion general Miag-Leffler funcion anomalou relaxaion Inroucion The Miag-Leffler funcion (ML an i generalizaion have been playe imporan role in he fiel of he mahemaical phyical an pracical cience [1-8]. The general fracional calculu (GFC operaor of Riemann-Liouville an Liouville-Capuo ype involving he family of he ML funcion an i generalizaion wa evelope by he ifferen auhor. The general fracional erivaive (GFD an general fracional inegral (GFI operaor were ue o moel he phyical phenomena conaining he power-law an ML funcion wih power-law. The rheological [9 1] hea ranfer [1 11] an anomalou iffuion [12-14] moel involving he GFC operaor. The evelopmen of he heory of he GFC operaor ue o he ifferen kernel i ill open for cieni an engineer o ecribe he complex moel in he fiel of he phyical an pracical cience. The aim of he manucrip i o propoe he GFC operaor of Wiman an Prabhakar ype o conier he anomalou relaxaion moel. A family of he ML funcion an i generalizaion Le an be he e of complex number real number non-negaive real number poiive ineger an = {} repecively. The ML funcion inrouce by Sweih mahemaician Goa Miag-Leffler in 193 i efine [4]: * Auhorʼ e-mail: yangxiaojun@163.com
S318 Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 E ( = κ ( κ κ = Γ 1 where R( κ an Γ( i he familiar Gamma funcion [1]. A fir exenion of he ML funcion rucure by Wiman in 195 i efine [5]: E υ ( = κ κ = Γ κ υ where η υ R R( υ an κ. A hir exenion of he ML funcion inrouce by Prabhakar in 1971 i efine [6]: E υ ( κ ( κ ( κ υ ( κ = Γ Γ κ = 1 where η υ R R( υ R( κ an he familiar Pochhammer ymbol i preene [1]: 1 κ = Γ κ ( = ( κ κ Γ ( If he Laplace ranform of he funcion g ( i efine by [1]: Table 1. The Laplace ranform of he generalize ML funcion wih he power-law funcion Generalize ML funcion E ( Eυ E µ υ 1 υ E Laplace ranform 1 1 (1 (1 υ 1 (1 ( µ υ 1 υ (1 υ = L g : e g (5 (1 (2 (3 (4 hen he Laplace ranform of he generalize ML funcion wih he power-law funcion (ee [1 6-1] an he cie reference herein are lie in ab. 1. The GFC operaor In hi ecion we preen he GFD an GFI operaor involving he ML funcion an generalize ML funcion wih he power-law funcion. Now le a < b< an Ω Lab (. E µ υ 1 υ E υ E ( µ υ 1 µ υ ( (1 µ υ (1 υ ( ( µ υ ( (1 The GFC operaor of he ML funcion-kernel ype The GFD of Riemann-Liouville ype in he negaive ML funcion in he kernel i efine by [9 13 14]: RL ( ( Ω = E (6 where Ω Lab ( an he GFD of Liouville-Capuo ype by [9 13 14].
Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 S319 LC ( (1 ( Ω = E (7 for Ω AC 1 ( a b repecively. The relaionhip beween eq. (6 an (7 i [9 13]: LC RL ( E Ω = Ω Ω (8 The correponing GFI operaor i efine [13]: ( 1 Ω ( Ω =Ω 1 Γ( (9 Similarly he GFD of Riemann-Liouville ype in he poiive ML funcion in he kernel i efine a [9 13 14]: RL ( ( Ω = E ( ( Ω (1 an he GFD of Liouville-Capuo ype by [9 13 14]: LC ( (1 ( Ω Ω = E (11 repecively. The relaionhip beween eq. (1 an (11 i [13]: LC RL ( E Ω = Ω Ω (12 The correponing GFI operaor i efine [13]: ( 1 Ω( ( Ω =Ω Γ 1 ( (13 ( The GFD conaining he ML funcion in he kernel wih he ai of he normalizaion funcion were evelope in [12-14]. The GFC operaor of Wiman ype The GFD of Riemann-Liouville ype wih he negaive general Wiman funcion in he kernel i efine by: RL ( ( = ( E ( ( Ω υ (14 where Ω Lab ( an for Ω AC 1 ( a b he GFD of Liouville-Capuo ype by: ( LC Ω ( (1 υ = E (15 repecively. The relaionhip beween eq. (14 an (15 i: ( LC RL υ 1 Eυ Ω = Ω Ω (16
S32 Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 I GFI operaor i efine: ( υ 1 ( Ω ( 1 υ Ω = E (17 In a imilar way he GFD of Riemann-Liouville ype wih he poiive Wiman funcion in he kernel i efine: an he GFD of Liouville-Capuo ype by: repecively where: RL ( Ω υ = E (18 ( LC Ω ( (1 υ = E (19 ( LC RL υ 1 Eυ Ω = Ω Ω (2 The correponing GFI operaor i efine: ( υ 1 ( Ω ( 1 υ Ω = E (21 A he exene verion of he GFC operaor of Wiman ype we have he following. The GFD of Riemann-Liouville ype wih he negaive general Wiman funcion in he kernel i efine by: RL ( Ω Ω he GFD of Liouville-Capuo ype by: repecively where: = E (22 ( LC Ω ( (1 Ω = E (23 ( LC RL µ υ 1 E Ω = Ω Ω (24 The correponing GFI operaor i efine: ( ( µ υ 1 Ω ( 1 µ υ = E (25 The GFD of Riemann-Liouville ype wih he poiive general Wiman funcion in he kernel i efine: he GFD of Liouville-Capuo ype by: RL ( Ω Ω = E (26 ( LC Ω = ( (1 Ω E (27
Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 S321 repecively where: ( LC RL µ υ 1 E Ω = Ω Ω (28 The correponing GFI operaor i efine: ( ( µ υ 1 Ω ( 1 µ υ = E (29 The previou formulaion of he GFC operaor of Wiman ype are exene from he reul in [1]. For he more reul of he GFC operaor of Wiman ype reaer refer o he reference [1]. The GFC operaor of Prabhakar ype The GFD of Riemann-Liouville ype wih he negaive Prabhakar funcion in he kernel i efine by: RL ( ( = ( E ( ( Ω υ (3 where Ω Lab ( an for Ω AC 1 ( a b he GFD of Liouville-Capuo ype by: ( LC Ω ( (1 υ = E (31 repecively. The relaionhip beween eq. (3 an (31 i: ( LC RL υ 1 Eυ Ω = Ω Ω (32 The correponing GFI operaor i efine: ( υ ( Ω ( 1υ ( Ω = E (33 The GFD of Riemann-Liouville ype wih he poiive Prabhakar funcion in he kernel i efine: an he GFD of Liouville-Capuo ype by: repecively where: RL ( Ω υ = E (34 ( LC Ω = ( (1 υ E (35 ( LC RL υ 1 Eυ Ω = Ω Ω (36 The correponing GFI operaor i efine: ( υ ( Ω = ( 1υ ( Ω E (37
S322 Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 Accoring o he rule [13] he previou formulaion are exene from he reul in he pecial cae ee [1]. A he exene verion of he GFC operaor of Prabhakar ype we have he following: The GFD of Riemann-Liouville ype wih he negaive Prabhakar funcion in he kernel i efine by: RL ( µ υ 1 ( = ( E ( ( Ω Ω (38 where Ω Lab ( an for Ω AC 1 ( a b he GFD of Liouville-Capuo ype by: ( LC µ υ 1 Ω ( (1 Ω = E (39 repecively. The relaionhip beween eq. (38 an (39 i: ( LC RL µ υ 1 E Ω = Ω Ω (4 The correponing GFI operaor i efine: ( µ υ ( 1 ( µ υ Ω = E Ω (41 The Laplace ranform of eq. (38 (39 an (41 are preene: RL ( 1 µ υ ( Ω = 1 Ω Ω (42 LC ( 1 ( µ υ ( Ω = 1 Ω( (43 1 ( µ υ Ω = 1 Ω( (44 The GFD of Riemann-Liouville ype wih he poiive Prabhakar funcion in he kernel i efine: RL ( µ υ 1 Ω Ω an he GFD of Liouville-Capuo ype by: repecively where = E (45 ( LC µ υ 1 Ω = ( (1 Ω E (46 ( LC RL µ υ 1 E Ω = Ω Ω (47 I GFI operaor i efine: ( µ υ ( 1 ( µ υ Ω = E Ω (48
Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 S323 The Laplace ranform of eq. (45 (46 an (48 are preene: RL ( 1 µ υ ( Ω = 1 Ω Ω (49 LC ( 1 µ υ ( 1 Ω = Ω (5 ( 1 ( µ υ 1 Ω = Ω (51 A he irec reul we propoe he following GFC operaor conaining he negaive general Prabhakar funcion. In paricular if µ = we have he following. The GFD of Riemann-Liouville ype wih he negaive general Prabhakar funcion in he kernel i efine by: ( RL ( υ 1 Ω υ an he GFD of Liouville-Capuo ype by: repecively where: = E (52 ( LC υ 1 Ω = ( (1 υ E (53 ( LC RL υ 1 E υ Ω = Ω Ω (54 The correponing GFI operaor i efine: ( LC υ 1 Ω = ( (1 υ E (55 The GFD of Riemann-Liouville ype wih he poiive general Prabhakar funcion in he kernel i efine: RL ( υ 1 Ω υ an he GFD of Liouville-Capuo ype by: repecively where: = E (56 ( LC υ 1 Ω = ( (1 υ E (57 LC RL υ 1 ( Ω = Ω E υ Ω (58 The correponing GFI operaor i efine: ( LC υ 1 Ω ( (1 υ = E (59 In paricular if = we have he following.
S324 Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 The GFD of Riemann-Liouville ype wih he negaive general Prabhakar funcion in he kernel i efine by: RL ( Ω Ω an he GFD of Liouville-Capuo ype by: repecively where: = E (6 ( LC Ω ( (1 Ω = E (61 ( LC RL µ υ 1 E Ω = Ω Ω (62 The correponing GFI operaor i efine a: ( ( µ υ Ω = 1 ( µ υ ( E (63 The GFD of Riemann-Liouville ype wih he poiive general Prabhakar funcion in he kernel i efine: RL ( Ω Ω an he GFD of Liouville-Capuo ype by: repecively where: = E (64 ( LC Ω = ( (1 Ω E (65 ( LC RL µ υ 1 E I GFI operaor i efine: Ω = Ω Ω (66 ( ( µ υ Ω 1 ( µ υ ( = E (67 Remark 1. The GFC operaor of Prabhakar ype eq. (38 (39 (45 an (46 can be exene o oher an heir Laplace ranform can be eaily reuce. Moelling anomalou relaxaion behavior In hi ecion we icu four anomalou relaxaion moel bae on he formulaion of he GFC operaor of Prabhakar ype. Moel 1. A anomalou relaxaion moel uing he GFD of Prabhakar ype wih he negaive Prabhakar funcion in he kernel i given: where κ i he relaxaion conan an: RL ( ( Ω κω = (68
Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 S325 RL ( µ υ 1 = ( E ( ( ( Ω Ω > (69 Moel 2. A anomalou relaxaion moel uing he GFD of Prabhakar ype wih he negaive Prabhakar funcion in he kernel i preene: where κ i he relaxaion conan an: LC ( ( Ω κω = (7 ( LC µ υ 1 (1 ( Ω = E Ω > (71 Moel 3. A anomalou relaxaion moel uing he GFD of Prabhakar ype wih he poiive Prabhakar funcion in he kernel i given: where κ i he relaxaion conan an: RL ( ( Ω κω = (72 RL ( µ υ 1 = ( E ( ( ( Ω Ω > (73 Moel 4. A anomalou relaxaion moel uing he GFD of Prabhakar ype wih he poiive Prabhakar funcion in he kernel i preene: where κ i he relaxaion conan an: LC ( ( Ω κω = (74 ( LC µ υ 1 (1 ( Ω = E Ω > (75 Concluion In hi work we propoe a family of he GFC operaor of Wiman an Prabhakar ype. The anomalou relaxaion moel uing he GFD of Prabhakar ype wih he negaive an poiive Prabhakar funcion in he kernel were icue in eail. The formulaion are a new perpecive efficien for eveloping he mahemaical moel in mahemaical phyic. Acknowlegmen Thi work i uppore by he Sae Key Reearch Developmen Program of he People Republic of China (Gran No. 216YFC675 he Naural Science Founaion of China (Gran No. 513234 an he Prioriy Acaemic Program Developmen of Jiangu Higher Eucaion Iniuion (PAPD214. Nomenclaure κ relaxaion conan [ 1 ] fracional orer [ ] ime co-orinae [] Ω( Laplace ranform of Ω( Ω( relaxaion conan [K 1 ]
S326 Yang X. e al.: General Fracional Calculu Operaor Conaining he Generalize... THERMAL SCIENCE: Year 217 Vol. 21 Suppl. 1 pp. S317-S326 Reference [1] Gorenflo R. e al. Miag-Leffler Funcion Relae Topic an Applicaion Springer Berlin 214 [2] Yang X. J. e al. Local Fracional Inegral Tranform an Their Applicaion Acaemic Pre New York USA 25 [3] Samko S. G. e al. Fracional Inegral an Derivaive Theory an Applicaion Goron an Breach Yveron Swizerlan 1993 [4] Miag-Leffler G. M. Sur La Nouvelle Foncion Eα ( x Compe Renu e l Aca emie e Science 137 (193 pp. 554-558 [5] Wiman A. Ueber en Funamenal Saz in er Theorie er Funkionen Eα ( x Aca Mahemaica 29 (195 pp. 217-234 [6] Prabhakar T. R. A Singular Inegral Equaion wih a Generalize Miag Leffler Funcion in he Kernel Yokohama Mahemaical Journal 19 (1971 1 pp. 7-15 [7] Shukla A. K. Prajapai J. C. On a Generalizaion of Miag-Leffler Funcion an I Properie Journal of Mahemaical Analyi an Applicaion 336 (27 2 pp. 797-811 [8] Kilba A. A. e al. Generalize Miag-Leffler Funcion an Generalize Fracional Calculu Operaor Inegral Tranform an Special Funcion 15 (24 1 pp. 31-49 [9] Yang X. J. New General Fracional-Orer Rheological Moel wihin Kernel of Miag-Leffler Funcion Romanian Repor in Phyic 69 (217 4 118 [1] Giui A. e al. Prabhakar-Like Fracional Vicoelaiciy Communicaion in Nonlinear Science an Numerical Simulaion 58 (218 1 pp. 138-143 [11] Yang X. J. Fracional Derivaive of Conan an Variable Orer Applie o Anomalou Relaxaion Moel in Hea-Tranfer Problem Thermal Science 21 (217 3 pp.1161-1171 [12] Aangana A. e al. New Fracional Derivaive wih Nonlocal an Non-Singular Kernel: Theory an Applicaion o Hea Tranfer Moel Thermal Science 2 (216 2 pp. 763-769. [13] Yang X. J. e al. Anomalou Diffuion Moel wih General Fracional Derivaive wihin he Kernel of he Exene Miag-Leffler Type Funcion Romanian Repor in Phyic 69 (217 4 115 [14] Yang X. J. General Fracional Derivaive Proceeing A Tuorial Commen Sympoium on Avance Compuaional Meho for Linear an Nonlinear Hea an Flui Flow Avance Compuaional Meho in Applie Science an Fracional (Fracal Calculu an Applie Analyi Songjiang Shanghai China 217 Paper ubmie: May 1 217 Paper revie: June 25 217 Paper accepe: July 1 217 217 Sociey of Thermal Engineer of Serbia Publihe by he Vinča Iniue of Nuclear Science Belgrae Serbia. Thi i an open acce aricle iribue uner he CC BY-NC-ND 4. erm an coniion