Index

• additivity 2.2.1.9
• adjoint operators 2.2.1.7
• analysis of singularities 2.2.2.8
• anomalous subdiffusion 2.3.1
• Banach space Appendix B, 2.2.2.6, 2.2.2.9, 2.3.3.2
• Bernstein function 2.3.3.6
• Bessel function 2.3.4.2
• binomial formula 2.1.1, 2.2.2.6
• Bochner-Levy diffusion 2.2.2.9, 2.3.4.1
• Borel measure 2.3.3.6
• Cauchy problem 2.3.3.1
• causality 2.3.3.5
• classification of phase transitions 2.2.2.8
• conjugate Riesz potential Definition 2.3
• continuity in time 2.3.3.3
• continuous time random walk 2.3.4.3
• convolution
  • distributions 2.2.1.5
  • functions 2.2.1.3
  • integral 2.2.1.1
  • kernel Appendix C, 2.2.1.3, 2.2.1.5, 2.2.2.5
  • operators 2.2.1.3, 2.2.1.5, 2.2.2.10, 2.3.3.6
  • periodic functions 2.2.1.3
  • product 2.2.1.3, 2.2.1.5
  • semigroup 2.3.3.6
• critical phenomena 2.2.2.8
• CTRW 2.3.4.3
• difference quotient 2.1.4, 2.1.4, 2.1.6, 2.2.2.6, 2.2.2.6
• diffusion 2.2.2.9, 2.3.1, 2.3.2, 2.3.2, 2.3.4, 2.3.4.1, 2.3.4.2, 2.3.4.2, 2.3.4.2, 2.3.4.2, 2.3.4.3
  • anomalous 2.3.1
  • fractional 2.2.2.9, 2.3.2, 2.3.2, 2.3.4, 2.3.4.1, 2.3.4.2, 2.3.4.2, 2.3.4.2, 2.3.4.2, 2.3.4.3
• Dirac δ function Appendix C, Appendix C, Appendix C, 2.2.1.3, 2.2.1.5
• Dirac-measure 2.3.3.6
• distribution(s) Appendix C
  • convolution Appendix C, 2.2.1.5
  • differentiation Appendix C
  • Dirac δ Appendix C, 2.2.1.5
  • direct product Appendix C
  • homogeneous Appendix C
  • regular Appendix C, 2.2.1.5
  • singular Appendix C, 2.2.1.5
  • tempered Appendix C
  • with compact support Appendix C
• Ehrenfest classification 2.2.2.8
• eigenfunction(s)
  • fractional derivatives 2.2.3
• ergodicity breaking 2.3.3.1
• essential range Appendix B
• Euler 2.1.2, 2.1.3
• Euler Beta function 2.2.1.9
• exponential series 2.1.3, 2.1.4
• Fokker-Planck equation 2.3.4.1
• Fourier 2.1.5
• Fourier series 2.2.1.3
• Fourier transformation 2.2.1.6, 2.2.2.5
• fractional derivative
  • at lower limit 2.2.2.8
  • definition
    • generalized Riemann-Liouville 2.2.2.2
    • Grünwald-Letnikov 2.2.2.6
    • lower limit 2.2.2.1
    • Marchaud-Hadamard 2.2.2.3
    • of distributions 2.2.2.7
    • Riemann-Liouville 2.2.2.1
    • Riesz 2.2.2.5
    • upper limit 2.2.2.1
    • Weyl-Liouville 2.2.2.4
    • Weyl-Marchaud 2.2.2.4
  • eigenfunction 2.2.3
  • generalized Riemann-Liouville 2.2.2.2
  • Grünwald-Letnikov 2.2.2.6
  • history 2.1.1
    • Euler 2.1.2
    • Fourier 2.1.5
    • Grünwald 2.1.6
    • Leibniz 2.1.3
    • Liouville 2.1.3, 2.1.4
    • paradoxa 2.1.1, 2.1.3
    • Riemann 2.1.7
  • interpolation 2.2.2.2
  • Liouville(-Caputo) 2.2.2.2
  • local 2.2.2.8
  • Marchaud-Hadamard 2.2.2.3, 2.2.2.7, 2.3.3.7
  • notation 2.2.2.1
  • of distributions 2.2.2.7
  • order 2.2.2.1, 2.2.2.2, 2.2.2.3, 2.2.2.4, 2.2.2.5, 2.2.2.6, 2.2.2.7
  • Riemann-Liouville 2.2.2.1, 2.2.2.2
  • Riesz 2.2.2.5
  • symbols 2.2.2.1
  • type 2.2.2.2
  • Weyl-Liouville 2.2.2.4
  • Weyl-Marchaud 2.2.2.4
• fractional difference quotient 2.2.2.6
• fractional differential equation 2.2.3, 2.2.3, 2.3.1
• fractional diffusion 2.3.4, 2.3.4.1, 2.3.4.2, 2.3.4.3
  • Bochner-Levy 2.2.2.9, 2.3.4.1
  • Montroll-Weiss 2.3.4.2
• fractional integral 2.2.1
  • by parts 2.2.1.7
  • common symbols 2.2.1.2
  • composition law 2.2.1.9
  • definition
    • lower limit Definition 2.1
    • of distributions 2.2.1.5
    • Riemann-Liouville Definition 2.1
    • Riesz 2.2.1.4
    • Riesz-Feller 2.2.1.4
    • upper limit Definition 2.1
    • Weyl Definition 2.2
  • domain of definition 2.2.1.2
  • interpolation 2.2.1.4
  • inverse 2.2.2.1
  • notation 2.2.1.2
  • of distributions 2.2.1.5
  • Riemann 2.1.7
  • Riemann-Liouville 2.2.1.2, Definition 2.1
  • Riesz 2.2.1.4
  • Riesz-Feller 2.2.1.4
  • Weyl 2.2.1.3, Definition 2.2
• fractional integration
  • by parts 2.2.1.7
• fractional master equation 2.3.4.3, 2.3.4.3
• fractional powers 2.2.2.9
• fractional stationarity 2.3.3.8
• fractional time 2.3.3
• fractional time evolution 2.3.3.6
• function(s)
  • Bernstein 2.3.3.6
  • Bessel 2.3.4.2
  • Beta 2.2.1.9
  • continuous differentiable Appendix B
  • essential range Appendix B
  • Gamma 2.1.2, 2.2.1.2
  • generalized Appendix C
  • Hölder continuous Appendix B
  • locally integrable Appendix B, 2.2.1.1
  • Mittag-Leffler 2.2.3
  • periodic 2.2.1.3, 2.2.1.3, 2.2.2.4
  • p-integrable Appendix B
  • regularly varying 2.2.2.8
  • slowly varying 2.2.2.8
  • smooth Appendix B
• Gamma function 2.1.2, 2.2.1.2
• Grünwald 2.1.6
• Grünwald-Letnikov derivative 2.2.2.6
• Hardy-Littlewood theorem 2.2.1.8
• Heaviside step function 2.2.1.3
• H-function 2.3.3.6, 2.3.4.2
• Hilbert space Appendix B, 2.2.2.9
• Hölder space Appendix B
• homogeneity of time 2.3.3.4
• homogenous divisibilty 2.3.3.4, 2.3.3.6, 2.3.3.8
• Hörmander symbol class Definition 2.19
• identity operator 2.2.2.3, 2.2.2.6
• infinitesimal generator 2.2.2.9, 2.3.3.7
• integral transforms 2.2.1.6
• integration
  • by parts 2.2.1.7
  • fractional 2.2.1.2, 2.2.1.3, 2.2.1.4, 2.2.1.5
    • Riemann-Liouville 2.2.1.2
    • Riesz 2.2.1.4
    • Weyl 2.2.1.3
  • iterated 2.2.1.1
• integroderivatives 2.1.6
• irreversibility problem
  • normal 2.3.3.1
  • reversed 2.3.3.1
• iterated integrals 2.2.1.1
• Kohn-Nirenberg pseudodifferential operator Definition 2.19
• Laplace transformation 2.2.1.6
• Laplacian 2.2.2.9, 2.3.4.1
• Lebesgue space Appendix B
• Leibniz 2.1.1, 2.1.3
• Leibniz’ paradox 2.1.1, 2.1.2, 2.1.3
• Leibniz’ product rule 2.1.1
• Liouville 2.1.4
• Lizorkin space 2.2.1.6
• local fractional derivative 2.2.2.8
• locality 2.3.2, 2.3.4.2
• locally integrable function Appendix B
• Marchaud fractional derivative 2.2.2.3
• master equation 2.3.4.3
  • fractional 2.3.4.3, 2.3.4.3
• Mellin transformation 2.3.3.6
• Mittag-Leffler function 2.2.3
• Montroll-Weiss diffusion 2.3.4.2
• nonlocality 2.3.2, 2.3.2, 2.3.3.1, 2.3.3.1
  • space 2.3.2
  • time 2.3.3.1
• operator(s)
  • adjoint 2.2.1.7
  • closed 2.2.2.9
  • convolution 2.2.1.3, 2.2.1.5, 2.2.2.10, 2.3.3.6
  • fractional powers 2.2.2.9
  • identity 2.2.2.3, 2.2.2.6
  • Kohn-Nirenberg 2.2.2.10
  • Kohn-Nirenberg symbol Definition 2.19
  • Laplace 2.2.2.9, 2.3.4.1
  • pseudodifferential 2.2.2.10
  • semigroup of 2.3.3.2
  • spectral decomposition 2.2.2.9
  • symbol Definition 2.19
  • translation 2.2.2.3, 2.2.2.6
• p-integrable function Appendix B
• principal value Appendix C
• pseudodifferential operators 2.2.2.10
• random walk
  • continuous time 2.3.4.3
• rapidly decreasing function Appendix B
• regularly varying function 2.2.2.8
• Riemann 2.1.7
• Riemann-Liouville derivatives 2.2.2.1
• Riemann-Liouville fractional integrals 2.2.1.2
• Riesz-Feller kernel 2.2.1.4
• Riesz fractional derivatives 2.2.2.5
• Riesz fractional integrals 2.2.1.4
• Riesz kernel 2.2.1.4
• Riesz potential 2.2.1.4
  • conjugate Definition 2.3
• Schwartz space Appendix B, Appendix C
• semigroup 2.2.1.9, 2.2.2.9, 2.2.2.9, 2.3.3.1, 2.3.3.2, 2.3.3.6, 2.3.3.6, 2.3.3.7
  • generator 2.3.3.2
  • of convolutions 2.3.3.6
• singularities 2.2.2.8
• slowly varying function 2.2.2.8
• Sobolev space Appendix B
• space
  • Banach Appendix B, 2.2.2.6, 2.2.2.9, 2.3.3.2
  • continuous functions Appendix B
  • Hilbert Appendix B, 2.2.2.9
  • Hölder Appendix B
  • Lebesgue Appendix B
  • Lizorkin 2.2.1.6
  • Schwartz Appendix B
  • Sobolev Appendix B
  • test functions Appendix C
• stationarity 2.3.3.1, 2.3.3.8
• subdiffusion 2.3.1
• test functions Appendix C
• time
  • continuity 2.3.3.3
  • evolution 2.3.3.2
  • fractional 2.3.3
  • homogeneity 2.3.3.4
  • homogenous divisibilty 2.3.3.4, 2.3.3.6, 2.3.3.8
  • infinitesimal generator 2.3.3.7
  • translation 2.3.3.1, 2.3.3.4, 2.3.3.6
• time evolution 2.3.3.2
• translation invariance 2.3.3.1, 2.3.3.4, 2.3.3.6
• translation operator 2.2.2.3, 2.2.2.6
• types of fractional derivative 2.2.2.2
• unit step function 2.2.1.3
• Weyl fractional integrals 2.2.1.3
• Youngs inequality 2.2.1.8