General Fractional Derivatives with Applications in Viscoelasticity

447 Pages·2020·2.509 MB·English

by  Xiao-Jun Yang, Feng Gao, Yang Ju

General Fractional Derivatives With
Applications in Viscoelasticity

General Fractional
Derivatives With
Applications in
Viscoelasticity
Xiao-Jun Yang
Feng Gao
Yang Ju
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Contents
Preface ix
1 Specialfunctions 1
1.1 Eulergammaandbetafunctions 3
1.2 Laplacetransformandproperties 5
1.3 Mittag-Lefflerfunction 6
1.4 Miller–Rossfunction 12
1.5 Rabotnovfunction 16
1.6 One-parameterLorenzo–Hartleyfunction 19
1.7 Prabhakarfunction 23
1.8 Wimanfunction 47
1.9 Thetwo-parameterLorenzo–Hartleyfunction 53
1.10 Two-parameterGorenflo–Mainardifunction 56
1.11 Euler-typegammaandbetafunctionswithrespecttoanother
function 59
1.12 Mittag-Leffler-typefunctionwithrespecttoanotherfunction 61
1.13 Miller–Ross-typefunctionwithrespecttofunction 64
1.14 Rabotnov-typefunctionwithrespecttoanotherfunction 66
1.15 Lorenzo–Hartley-typefunctionwithrespecttoanotherfunction 68
1.16 Prabhakar-typefunctionwithrespecttoanotherfunction 70
1.17 Wiman-typefunctionwithrespecttoanotherfunction 85
1.18 Two-parameterLorenzo–Hartleyfunctionwithrespecttoanother
function 89
1.19 Gorenflo–Mainardi-typefunctionwithrespecttoanotherfunction 91
2 Fractionalderivativeswithsingularkernels 95
2.1 Thespaceofthefunctions 98
2.2 Riemann–Liouvillefractionalcalculus 100
2.3 Oslerfractionalcalculus 107
2.4 Liouville–Weylfractionalcalculus 111
2.5 Samko–Kilbas–Marichevfractionalcalculus 113
2.6 Liouville–Sonine–Caputofractionalderivatives 115
2.7 Liouvillefractionalderivatives 120
2.8 Almeidafractionalderivativeswithrespecttoanotherfunction 120
2.9 Liouville-typefractionalderivativewithrespecttoanotherfunction 122
2.10 Liouville–Grünwald–Letnikovfractionalderivatives 123
vi Contents
2.11 Kilbas–Srivastava–Trujillofractionaldifferencederivatives 127
2.12 Rieszfractionalcalculus 128
2.13 Fellerfractionalcalculus 132
2.14 Herrmannfractionalcalculus 137
2.15 Samko–Kilbas–Marichevsymmetricfractionaldifference
derivative 143
2.16 Grünwald–Letnikov–Herrmann-typesymmetricfractional
differencederivative 144
2.17 Grünwald–Letnikov–Feller-typesymmetricfractionaldifference
derivative 144
2.18 Samko–Kilbas–Marichevsymmetricfractionaldifference
derivativeonaboundeddomain 147
2.19 Grünwald–Letnikov–Herrmann-typesymmetricfractional
differencederivativeonaboundeddomain 147
2.20 Grünwald–Letnikov–Feller-typesymmetricfractionaldifference
derivativeonaboundeddomain 148
2.21 Erdelyi–Kober-typecalculus 150
2.22 Hadamardfractionalcalculus 160
2.23 Marchaudfractionalderivatives 171
2.24 Riemann–Liouville-typetemperedfractionalcalculus 173
2.25 Liouville–Weyl-typetemperedfractionalcalculus 175
2.26 Riemann–Liouville-typetemperedfractionalcalculuswithrespect
toanotherfunction 191
2.27 Hilferderivatives 195
2.28 Mixedfractionalderivatives 198
3 Fractionalderivativeswithnonsingularkernels 209
3.1 Historyoffractionalderivativeswithnonsingularkernels 211
3.2 Soninegeneralfractionalcalculuswithnonsingularkernels 224
3.3 GeneralfractionalderivativeswithMittag-Lefflernonsingular
kernel 237
3.4 GeneralfractionalderivativeswithWimannonsingularkernel 252
3.5 GeneralfractionalderivativeswithPrabhakarnonsingularkernel 263
3.6 GeneralfractionalderivativeswithGorenflo–Mainardinonsingular
kernel 285
3.7 GeneralfractionalderivativeswithMiller–Rossnonsingularkernel 293
3.8 Generalfractionalderivativeswithone-parameterLorenzo–Hartley
nonsingularkernel 298
3.9 Generalfractionalderivativeswithtwo-parameterLorenzo–Hartley
nonsingularkernel 303
4 Variable-orderfractionalderivativeswithsingularkernels 311
4.1 Riemann–Liouville-typevariable-orderfractionalcalculuswith
singularkernel 311
Contents vii
4.2 Variable-orderHilfer-typefractionalderivativeswithsingular
kernel 318
4.3 Liouville–Weyl-typevariable-orderfractionalcalculus 323
4.4 Riesz-,Feller-,andHerrmann-typevariable-orderfractional
derivativeswithsingularkernel 330
4.5 Variable-ordertemperedfractionalderivativeswithweaklysingular
kernel 340
5 Variable-ordergeneralfractionalderivativeswithnonsingular
kernels 349
5.1 Riemann–Liouville-typevariable-ordergeneralfractional
derivativeswithMittag-Leffler–Gauss-likenonsingularkernel 350
5.2 Hilfer-typevariable-orderfractionalderivativeswithMittag-Leffler
nonsingularkernel 357
5.3 Variable-ordergeneralfractionalderivativeswith
Gorenflo–Mainardinonsingularkernel 359
5.4 Variable-orderHilfer-typefractionalderivativeswith
Gorenflo–Mainardinonsingularkernel 364
5.5 Variable-ordergeneralfractionalderivativeswithone-parameter
Lorenzo–Hartleynonsingularkernel 365
5.6 Variable-orderHilfer-typefractionalderivativeswith
Gorenflo–Mainardinonsingularkernel 370
5.7 Variable-ordergeneralfractionalderivativewithMiller–Ross
nonsingularkernel 371
5.8 Variable-orderHilfer-typefractionalderivativeswithMiller–Ross
nonsingularkernel 376
5.9 Variable-ordergeneralfractionalderivativewithPrabhakar
nonsingularkernel 377
5.10 Variable-orderHilfer-typefractionalderivativeswithPrabhakar
nonsingularkernel 382
6 Generalderivatives 385
6.1 Classicalderivatives 385
6.2 Derivativeswithrespecttoanotherfunction 386
6.3 Generalderivativeswithrespecttopower-lawfunction 390
6.4 Generalderivativeswithrespecttoexponentialfunction 392
6.5 Generalderivativeswithrespecttologarithmicfunction 394
6.6 Othergeneralderivatives 395
7 Applicationsoffractional-orderviscoelasticmodels 399
7.1 Mathematicalmodelswithclassicalderivatives 399
7.2 Mathematicalmodelswithgeneralderivatives 400
7.3 Mathematicalmodelswithfractionalderivatives 409
7.4 Mathematicalmodelswithfractionalderivativeswithnonsingular
kernels 411
viii Contents
7.5 Mathematicalmodelswithfractionalderivativeswithrespectto
anotherfunction 417
References 429
Index 439
Preface
The main purpose of this monograph is to provide an introduction to the newly-
established fractional-order calculus operators involving singular and nonsingular
kernels with applications to fractional-order viscoelastic models from the general
fractional-ordercalculusoperators’view-point.Anotheraimistopresentanomalous
relaxation and rheological models in the light of nature complexity. The topics are
important and interesting for scientists and engineers in the fields of mathematics,
physics,chemistry,andelasticity.
Duetotheabove-mentionedavenuesoftheirpotentialapplicationsinwide-spread
real-world phenomena in the fields of physical and engineering sciences, we sys-
tematically illustrate the different calculi and the viscoelastic models with different
derivatives. More specifically, we have clearly illustrated the special functions, frac-
tional derivatives with singular, weakly-singular, and nonsingular kernels, variable-
order fractional derivatives with singular, weakly-singular, and nonsingular kernels,
andgeneralderivatives.Moreover,wehaveinvestigatedtheviscoelasticmodels,e.g.,
dashpot,Maxwell-like,Kelvin–Voigt-like,Burgers-like,andZener-likeelements.
Themonographisdividedintosevenchapters.
Chapter 1 introduces the Euler gamma and beta functions, and the families
of Mittag-Leffler, Miller–Ross, Rabotnov, Lorenzo–Hartley, Kilbas–Saigo–Saxena,
Gorenflo–Mainardi,Wiman,anPrabhakarfunctions,includingthesubsine,subcosine,
hyperbolicsubsine,andhyperbolicsubcosinefunctions.
Chapter 2 investigates the functional spaces as well as the fractional deriva-
tive and integral operators with singular kernels, for example, power function, and
the function related to the power-law. More specifically, we have introduced the
Riemann–Liouville, Osler, Liouville–Weyl, Samko–Kilbas–Marichev fractional cal-
culi, Liouville–Sonine–Caputo and Liouville fractional derivatives, Almeida and
Liouville-type fractional derivatives with respect to another function, Liouville–
Grünwald–Letnikov fractional derivatives, Kilbas–Srivastava–Trujillo fractional dif-
ference derivatives, Riesz fractional calculus, Liouville–Sonine–Caputo–Riesz-type
fractional derivatives, Feller fractional calculus, Liouville–Sonine–Caputo–Feller
type-fractionalderivatives,Herrmannfractionalcalculus,Liouville–Sonine–Caputo–
Herrmann-typefractionalderivatives,Samko–Kilbas–Marichevsymmetricfractional
difference derivative, Grünwald–Letnikov–Herrmann-type symmetric fractional dif-
ference derivative, Grünwald–Letnikov–Feller-type symmetric fractional difference
derivative, Erdelyi–Kober-type calculus, Hadamard fractional calculus, Hadamard-
type fractional calculus involving the exponential function, Marchaud fractional
derivatives Marchaud-type fractional derivatives with respect to another function,
Riemann–Liouville-typetemperedfractionalcalculus,Liouville–Weyl-typetempered
fractional calculus, Liouville–Sonine–Caputo-type tempered fractional derivatives,