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Kilbas A., Srivastava H.M. Ч Theory and Applications of Fractional Differential Equations
Kilbas A., Srivastava H.M. Ч Theory and Applications of Fractional Differential Equations

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Ќазвание: Theory and Applications of Fractional Differential Equations

јвторы: Kilbas A., Srivastava H.M.

јннотаци€:

This monograph provides the most recent and up-to-date developments on fractional differential and fractional integro-differential equations involving many different potentially useful operators of fractional calculus.

The subject of fractional calculus and its applications (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.

Some of the areas of present-day applications of fractional models include Fluid Flow, Solute Transport or Dynamical Processes in Self-Similar and Porous Structures, Diffusive Transport akin to Diffusion, Material Viscoelastic Theory, Electromagnetic Theory, Dynamics of Earthquakes, Control Theory of Dynamical Systems, Optics and Signal Processing, Bio-Sciences, Economics, Geology, Astrophysics, Probability and Statistics, Chemical Physics, and so on.

In the above-mentioned areas, there are phenomena with estrange kinetics which have a microscopic complex behaviour, and their macroscopic dynamics can not be characterized by classical derivative models.

The fractional modelling is an emergent tool which use fractional differential equations including derivatives of fractional order, that is, we can speak about a derivative of order 1/3, or square root of 2, and so on. Some of such fractional models can have solutions which are non-differentiable but continuous functions, such as Weierstrass type functions. Such kinds of properties are, obviously, impossible for the ordinary models.

Whatare the useful properties of these fractional operators which help in the modelling of so many anomalous processes? From the point of view of the authors and from known experimental results, most of the processes associated with complex systems have non-local dynamics involving long-memory in time, and the fractional integral and fractional derivative operators do have some of those characteristics.

This book is written primarily for the graduate students and researchers in many different disciplines in the mathematical, physical, engineering and so many others sciences, who are interested not only in learning about the various mathematical tools and techniques used in the theory and widespread applications of fractional differential equations, but also in further investigations which emerge naturally from (or which are motivated substantially by) the physical situations modelled mathematically in the book.

This monograph consists of a total of eight chapters and a very extensive bibliography. The main objective of it is to complement the contents of the other books dedicated to the study and the applications of fractional differential equations. The aim of the book is to present, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy type problems involving nonlinear ordinary fractional differential equations, explicit solutions of linear differential equations and of the corresponding initial-value problems through different methods, closed-form solutions of ordinary and partial differential equations, and a theory of the so-called sequential linear fractional differential equations including a generalization of the classical Frobenius method, and also to include an interesting set of applications of the developed theory.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2006

 оличество страниц: 540

ƒобавлена в каталог: 22.05.2008

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$\alpha$-Exponential functions      50Ч51 53
$\alpha$-Wronskian of n functions      395
Absolutely continuous function      1Ч3 92
Amorphous polymer      440
Analytic continuation      27 86 120
Asymptotic behavior      28Ч29 31 33Ч35 39Ч41 43 54Ч56 61Ч62 349
Banach space      4 5 360Ч361
Basset's problem      434Ч435
Bessel function of the first kind      17 32Ч35 55 65Ч66 342
beta function      26
Binomial coefficients      26
Burger's model      440Ч441
Caputo fractional derivatives      91Ч93 97 99 230 312 322Ч323 418
Caputo sequential fractional derivative      421 445
Cauchy type problem      135Ч140 144Ч145 147Ч148 150Ч160 162Ч177 179Ч186 191Ч192 195Ч198 211Ч219 222Ч228 234Ч238 251Ч256 266Ч270 275 277 282 309Ч311 327 349 353Ч354 356 358Ч359 362 364Ч366 368 370Ч371 373 380 388 407 415 450 462 468
Commutative property      11 13 19Ч20 22Ч23
Compact support      7Ч9 83 89 131 355
Complex conjugate solutions      402 431
Composition relations      74
Continuous embedding      5
Continuous global solution      411 414
Delayed elasticity      441
Differential operator      380
Dirac function      7 9 16 23 44 133 350 451 459 462Ч463 466
Elastic region      439
Elastic response      440
Erdelyi Ч Kober type fractional operators      105Ч106 108Ч109 143Ч144 270 352 457
Euler gamma function      24Ч27 36 38 69 289 368 372 388Ч389 456
Euler integral      24 26Ч27
Euler psi function      16 23 26
Euler transformation formula      28
Exponential weight      88
Fourier convolution operator      11 13 127 341
Fourier convolution theorem      12 13 128
Fourier integrals      10 13
Fourier transform      10Ч18 20 44 90 121 127Ч128 131Ч133 341Ч342 353Ч356 358Ч359 361Ч364 366Ч368 374 376Ч381 383Ч384 388 437 442Ч443 453 458Ч459
Fractal      349 350 357 439 451 453Ч454
Fractional damping      433 436Ч437
Fractional derivative      69Ч80 83Ч105 108Ч112 115Ч127 130Ч131 135 138 140Ч145 162Ч163 172 176 182 199 205 208 212 221Ч222 230 234 238Ч242 245Ч247 251 256Ч257 261 264Ч279 283 295 311Ч312 322Ч323 329Ч331 336 341 344Ч355 358Ч362 373 379 380 384 393 409 415Ч418 426 431Ч434 442Ч452 455Ч462
Fractional Green function      359 373 379 393 403 407 409 412
Fractional integral      69 74Ч80 83Ч84 86Ч87 90 99Ч100 103Ч106 108Ч114 116Ч117 119 123Ч127 154 157 163 168 189 238 261Ч263 272 276 352 359 449Ч452 456Ч457 464
Fractional integration by part      76 83 107
Fractional relaxation      435 455
Fractional sequential derivative      394Ч395 397
Frobenius method      394 415 417 429
Fundamental system of solutions      283Ч285 288Ч291 293Ч294 312Ч316 319Ч321 396 399Ч400 402Ч403 410
Gauss hypergeometric function      27Ч28 30 132 143 350
Gauss Ч Legendre multiplication theorem      25
Gauss Ч Weierstrass transforms      130
Generalized functions      6Ч10 14Ч16 23 133 143 351
Generalized hypergeometric functions      27 30Ч32 45 58 65 353
Generalized Stirling numbers      115
Gruenwald-Letnikov fractional derivatives      121Ч122 443 458
H-function      58Ч65 352Ч354 368Ч369 371Ч372 379 388
Hadamard type fractional derivatives      111 113 115Ч116 119Ч120 122 123
Hadamard type fractional integral      110Ч114 116 119
Hardy Ч Littlewood theorem      72 82 88 128
Holderian functions      129
Homogeneous linear FDE      132 142 144 197 224Ч225 231Ч232 235Ч236 239 242 252 256Ч257 276 280Ч281 283 295 302Ч303 309 311Ч312 322Ч323 326 359 393Ч394 396Ч397 399Ч400 403 406Ч410 415 424 427 435 466Ч467
Hypergeometric series      27 29Ч30
Hypersingular integral      130Ч132 458
Incomplete gamma functions      27 289
Incompressible viscous fluid      433Ч434
Instantaneous deformation      440
Integral representations      29 34Ч35
Kummer confluent hypergeometric function      29 45
Kummer hypergeometric functions      29Ч30 65
Laplace convolution theorem      20
Laplace transform      18Ч19 23 31 36 42 44 47Ч48 50 52 55 58 84 98 140 279Ч284 287 291 295 303Ч304 306 311Ч312 315 322Ч323 329 336 340 350 352Ч353 356Ч357 362Ч364 366Ч370 373Ч377 380Ч381 384 393 400 402Ч405 435Ч436 442 451 465
Laws of thermodynamics      443
Lebesgue measurable functions      1Ч3 79 151
Left-sided and right-sided Caputo fractional derivatives      91
Left-sided difference      121
Left-sided Gruenwald Ч Letnikov fractional derivative      122
Left-sided Liouville fractional derivative      88 336 338
Legendre duplication formula      25 390
Linearly independent solutions      28 34 36 244 281 283 286 288Ч291 293 294Ч295 315Ч316 318 320 401 408 429 432Ч433
Liouville fractional derivatives      80 83 87 89Ч90 101Ч103 105 127 239 245 257 266 270 279 283 295 322Ч323 329
Liouville fractional integrals      78Ч79 83Ч84 86Ч87 90 100 106 127 163 189 261 263 359
Liouville left- and right-sided fractional operators      80 87 338
Lizorkin space      9 128 131
Logarithmic derivative      26
Marchaud fractional derivatives      122 458
Mellin convolution operator      22Ч23 330Ч331
Mellin convolution theorem      23
Mellin transform      18 20Ч24 31Ч32 34 36Ч39 41 44 46 48 54 57 84 99 104 109 119Ч120 283 329Ч330 332 337 346 352Ч353
Mellin-Barnes contour      27 30 33Ч35 38Ч39 41 43 46Ч48 54Ч55 57Ч58
Minkowski inequality      113
Mittag Ч Leffler Functions      40Ч42 44Ч46 48Ч50 55 66Ч67 78 86 98 137 141Ч142 144 223 226 237Ч239 250 265 270 272 280 284 295 308Ч309 312 355 361 381 383 394 416 439 451
N-dimensional Fourier transform      12 17
n-dimensional Laplace transform      13 19
n-dimensional Mellin transform      22
Non-Constant or Varible Coefficients linear FDE      394 409 415 417
Non-Homogeneous linear FDE      132 245 251 256 279Ч281 295 302Ч303 310Ч311 322Ч323 327 329 341 344 346 392 394 400 403Ч410 412 435
Non-Sequential Linear FDE      407 433
Operational decomposition method      142 405
Operational methods      141 260 270 402Ч404 413
Partial Liouville fractional derivatives      348
Partial Riemann Ч Liouville fractional operators      78 123Ч124 273 348 350Ч351 355 359 362 464
Pochhammer symbol      24 27 45
Positive monotone function      99 456
Purely imaginary order      71 80 87
Riemann Ч Liouville fractional integration      71 95 136 187 323
Riemann-Liouville fractional derivatives      70Ч71 90Ч95 93Ч95 124Ч126 138 143 222 242 251 348 351
Riesz      127Ч131 279 341 344 359 458Ч459
Right-sided mixed Riemann Ч Liouville fractional integrals      124Ч126
Right-sided partial Riemann Ч Liouville fractional derivatives      124
Schwartz space      8 10 354
Semigroup properties      51 53 73 75 83 96 101 107 114 121 128 261
SEQUENTIAL      138 140 270 282 393Ч397 407Ч408 410 412 415 421 427 431 433Ч435 437 445
Sequential FDE      282 393Ч397 427 433Ч435 437
Singular Points for linear FDE      416Ч417 424Ч427 429Ч431
Sobolev theorem      128
Stirling      25 115 475 521
Substitution operator      100 456
Systems of linear FDE      294 393 474
Tempered distributions      8
Transition region      439
Truncated hypersingular integral      132
Viscoelastic material      442 519
Viscoelasticity      347 439
Vitreous region      439
Weighted $L_p$-space      1 129
Wright function      42 44 47 54Ч58 67 286 291 297 299 303 305 314 319 331 336 356 364 366 374
Young theorem      11 13
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