Miller K.S., Ross B. — An Introduction to the Fractional Calculus and Fractional Differential Equations :: Электронная библиотека попечительского совета мехмата МГУ

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Miller K.S., Ross B. — An Introduction to the Fractional Calculus and Fractional Differential Equations

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Название: An Introduction to the Fractional Calculus and Fractional Differential Equations

Авторы: Miller K.S., Ross B.

Аннотация:

Commences with the historical development of fractional calculus, its mathematical theory—particularly the Riemann-Liouville version. Numerous examples and theoretical applications of the theory are presented. Features topics associated with fractional differential equations. Discusses Weyl fractional calculus and some of its uses. Includes selected physical problems which lead to fractional differential or integral equations.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1993

Количество страниц: 366

Добавлена в каталог: 06.12.2009

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Предметный указатель

$A^{-1}_{k}$       276
$C_{t}(v, a)$       49 300
$e_{i}(t)$       142
$E_{t}(v, a)$       48 300
$E_{w}(t)$       132
$S_{t}(v, a)$       49 300
$\mathbf{c}$       45
$\mathbf{S}$       238
$\mathscr{C}$       88
$\mathscr{C}(v, x)$       337
$\mathscr{E}(v, x)$       330
$\mathscr{L}$       67
$\mathscr{L}^{-1}$       68
$\mathscr{S}(v, x)$       337
$\mathscr{S}^{r}_{j}$       41
$\psi(z)$       299
(n, q)      127
*      143
Abel's integral equation      3 255
Abel's tautochrone problem      3 255
Abel, N.H.      3 4 6 255
Asymptotic expansions, gamma function      40 298 299 331
Bagley, R.      16
Bernoulli's theorem      270
Bernoulli, J.      1 256
Bessel equation      126 207 301 302
Bessel function      301
Bessel function, fractional      308
Bessel operator, fractional      207
beta function      54 299 300
Brachistochrone      256
Bromwich, T.J.      14 261
Brychkov, Y.A.      16
Camko      see "Samko"
Campos, L.M.B.C.      16
Carson, J.R.      261
Cauchy integral formula      22
Cayley, A.      8
Center, W.      6 7 12
Complementary polynomials      290
Complete elliptic integrals      306
Complex variables      28
Convolution      143
Convolution of fractional Green's functions      165
D      35
Davis, H.T.      8 15 22
De Morgan, A.      7
Derivatives of fractional integrals      59 62 65
Design of a weir notch      269
Differential equations      see also "Fractional differential equations"
Differential equations, adjoint operator      33
Differential equations, Bessel      126 207 302
Differential equations, indicial polynomial      127
Differential equations, Kummer      251 305
Differential equations, Laguerre      307
Differential equations, Legendre      307
Differential equations, linear      126
Differential equations, one-sided Green's function      26 154
Differential equations, Wronskian      26 154
Digamma function      51 299
Dirichlet's formula      57
Double loop      32
e      240
Elliptic integrals      306
Erdelyi, A.      15
Error function      49 301
Euler's constant      51 297
Euler, L.      2 3
Exponential order, function of      67
Fluid flow, weir notch      269
Fourier, J.B.J.      3
Fractional Bessel function      208
Fractional Bessel operator      207
Fractional calculus      see also "Fractional derivative" "Fractional "Fractional
Fractional calculus, notation      22
Fractional calculus, origin      1 6 9
Fractional Cauchy equation      200
Fractional derivative      35
Fractional derivative of a constant      7
Fractional derivative, applications      99 100 101
Fractional derivative, definition      82
Fractional derivative, examples      83 84 85 97
Fractional derivative, Gruenwald's definition      38
Fractional derivative, Marchaud's definition      42
Fractional derivative, Riemann — Liouville      3
Fractional differential equation      6 127
Fractional differential equations      127
Fractional differential equations of order (n, q)      127
Fractional differential equations with nonconstant coefficients      194
Fractional differential equations with nonconstant coefficients, examples      196 198
Fractional differential equations, Bessel operator      207
Fractional differential equations, Cauchy type      200
Fractional differential equations, comparison with ordinary differential equations      229
Fractional differential equations, equations of order (2, q)      182
Fractional differential equations, examples      130 134 137 149 160 162
Fractional differential equations, explicit representation of solutions      145
Fractional differential equations, fractional Green's function      159
Fractional differential equations, Frobenius approach      204
Fractional differential equations, homogeneous      139
Fractional differential equations, indicial polynomial      127
Fractional differential equations, initial value theorem      140
Fractional differential equations, linearly independent solutions      141
Fractional differential equations, nonhomogeneous      160
Fractional differential equations, reduction to ordinary differential equations      171
Fractional differential equations, semidifferential equations      174
Fractional differential equations, semidifferential equations, example      177
Fractional differential equations, sequential      209
Fractional differential equations, solution of homogeneous equation      139
Fractional differential equations, solution of nonhomogeneous equation      157
Fractional differential equations, vector      217
Fractional Green's function      159
Fractional Green's function, convolution      165
Fractional integral in terms of ordinary derivatives      55
Fractional integral of derivatives      59 62 66
Fractional integral, applications      99 100 101
Fractional integral, derivative of      99
Fractional integral, equations      186
Fractional integral, law of exponents      57
Fractional integral, Weyl      13 22 33 236
Fresnel integrals      49 300
Frobenius-type series      204
Fuchsian class      254
Functions of exponential order      67
Functions, $C_{t}(v, a)$       49 300
Functions, $E_{t}(v, a)$       48 300
Functions, $S_{t}(v, a)$       49 300
Functions, $\mathscr{C}(v, x)$       337
Functions, $\mathscr{E}(v, x)$       330
Functions, $\mathscr{S}(v, x)$       337
Functions, Bessel      301
Functions, beta      299
Functions, complete elliptic integrals      306
Functions, digamma      51 299
Functions, error      49 301
Functions, fractional Bessel      208
Functions, Fresnel      49 300
Functions, gamma      297
Functions, good      238
Functions, hypergeometric      303
Functions, incomplete, beta      54 300
Functions, incomplete, gamma      48 300
Functions, Kummer      305
Functions, Laguerre      307
Functions, Legendre      307
Functions, Mittag-Leffler      132 143 144 206
Functions, psi      51 299
Galileo, G.      256
Gamma function      48 51 297 299 300
Good functions      238
Gorenflo, R.      16

Greatheed, S.S.      6
Green's function      153
Greer, H.R.      9
Gregory, D.F.      14
Hadamard, J.      13
Hardy, G.H.      15
Hargreave, C.J.      9
Heaviside operational calculus      13 261
Heaviside solution of differential equations      262 264
Heaviside, O.      8 13 15 261
Heywood, P.      15
Hill, G.W.      261
Holmgren, H.      9
Homogeneous fractional differential equation      139
Huygens, C.      260
Hypergeometric function      77 303
Incomplete beta function      55 300
Incomplete gamma function      48 300
Incomplete gamma function, functions related to      314
Incomplete gamma function, functions related to, Laplace transforms of      321
Incomplete gamma function, functions related to, properties of      315 316 318 319 320
Incomplete gamma function, properties of      309
Indicial polynomial      127
Initial value theorem      140
Integral equation, Abel's      3 255
Integral of the fractional integral      58
Integral relations      118
Integral representations      111
Interchange of sign of the order      87 91 248
Iterated integrals      23
j      45
J'      45
Jordan normal form      219
Joshi, J.M.C.      16
Kalla, S.      15 16
Kelland, P.      4 6 9 14
Kernel      23
Kilbas, A.      16
Kober, H.      15
Krug, A.      13
Kummer differential equation      251 305
Kummer function      305
L'Hopital, G.F.A.      xi 1
Lacroix, S.F.      2 3 6 7 8 12
Lagrange, J.L.      2
Laguerre differential equation      307
Laguerre function      307
Lamb, W.      15
Laplace transform      28 321
Laplace transform of fractional derivative      121
Laplace transform of fractional integral      69
Laplace transform of fractional integral, examples      70
Laplace transform, initial value theorem      140
Laplace, P.S.      2 3
Laurent's formula      77
Laurent, H.      10
Law of exponents for fractional integrals      57
Law of exponents for fractional operators      104 108 110
Law of exponents for Weyl transform      239 244
Legendre differential equation      307
Legendre function      307
Legendre, A.M.      2
Leibniz's rule      9
Leibniz's rule for fractional derivatives      95 97
Leibniz's rule for fractional integrals      75
Leibniz's rule for fractional integrals, applications      76 77
Leibniz's rule for Weyl fractional integral      247
Leibniz, G.W.      xi 1 3
Letnikov, A.V.      10
Linear differential equations      126
Linear operators      22
Liouville class      11
Liouville first definition      5
Liouville fractional differential equations      6
Liouville fractional integral      21
Liouville second definition      5
Liouville, J.      4 6 7 8 9
Littlewood, J.E.      15
Loop integral      29
Love, E.R.      15 16 97
Lowndes, J.S.      15
Marichev, O.      16
Matrix, canonical form      xiii 219
McBride, A.C.      15 16
Mikolas, M.      9 15 16
Mittag-Leffler function      132 143 144 206
Nekrassov, P.A.      13
Nishimoto, K      15 16
Nonhomogeneous fractional differential equations      160
Numerical tables      332—335 340—347
O'Shaughnessy, L.      195
Oldham, K.B.      4 15 97
One-sided Green's function      26 154
Operational calculus      13 261
Ordinary differential equations      126
Ordinary differential equations, Green's function      154
Ordinary differential equations, Wronskian      154
Osier, T.J.      15 97
Owa, S.      15 16
Peacock, G.      6 8 9
Phillips, P.C.      16
Pochhammer contour      32
Poincare, H.      261
Poisson's formula      302
Polynomial, complementary      290
Potential theory, Liouville's problem      265
Prudnikov, A P.      16
Psi function      51 299
Raina, R.K.      16
Representations of functions      116
Riemann class      11
Riemann fractional integral      21
Riemann — Liouville definition      9 45
Riemann — Liouville fractional calculus      80
Riemann — Liouville fractional derivative      3
Riemann — Liouville fractional integral      9 21
Riemann — Liouville fractional integral, examples      46 47 48 50 52
Riemann, G.F.B.      7 8 10
Riesz, M.      15
Roach, G.F.      15
Rodrigues' formula      115 307
Rooney, P.G.      15
Ross, B.      4 9 15 16 190
Sachdeva, B.K.      190
Saigo, M.      16
Samko, S.      15 16 39 43
Saxena, R.K      16
Semidifferential equations      174
Sequential fractional differential equations      209
Sneddon, I.N.      15
Sonin's formula      302
Sonin, N.Ya.      9 10
Spanier, J.      4 15 97
Spitzer, S.      9
Srivastava, H.M.      15 16
Stirling numbers      41
Stirling's formula      298 299
Table of formulas      352
Tables, numerical      332—335 340—347
Tautochrone problem      3 255
Torvik, P.J.      16
V      127
Vandermonde convolution formula      96
Vandermonde determinant      156
Vandermonde matrix      225
Vector fractional differential equations      217
Wallis, J.      1
Weir notch      269
Weyl fractional derivative      241
Weyl fractional integral      13 22 33 236

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