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Table Of Integrals Series And Products

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

We develop the finite element method for the numerical resolution of the space and time fractional Fokker-Planck equation, which is an effective tool for describing a process with both traps and flights; the time fractional derivative of the equation is used to characterize the traps, and the flights are depicted by the space fractional derivative. The stability and error estimates are rigorously established, and we prove that the convergent order is $O(k^{2-\alpha}+h^\mu)$, where $k$ is the time step size and $h$ the space step size. Numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims.

Finite Element Method for the Space and Time Fractional Fokker-Planck Equation

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10 years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures. DOI: 10.1115/1.4000563

Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results

We investigate the motion of a particle governed by a generalized Langevin equation with fractional derivative, nonlocal dissipative force, and long-time correlation function. We derive general expressions for the variances with a linear external force. We also analyze their asymptotic behaviors for the power-law correlation function without external force.

Generalized Langevin equation with fractional derivative and long-time correlation function.

In this present work we consider a fractional Langevin equation with Riemann-Liouville fractional time derivative which modifies the classical Newtonian force, nonlocal dissipative force, and long-time correlation. We investigate the first two moments, variances and position and velocity correlation functions of this system. We also compare them with the results obtained from the same fractional Langevin equation which uses the Caputo fractional derivative.

Fractional Langevin equation and Riemann-Liouville fractional derivative.

We investigate one-dimensional equations for the diffusion with a nonconstant diffusion coefficient inside the second derivative and between the derivatives. In particular, we employ the diffusion coefficient D(x) proportional to /x/(-theta)(theta in R) and a quartic potential. These diffusion equations present a rich variety of behaviors associated with different regimes. Results of two approaches are analyzed and compared. We also investigate the mean first passage time of these systems. We show that the system with the coefficient D(x) between the derivatives can produce different behaviors for the mean first passage time in comparison with those obtained by the system with the coefficient inside the derivatives.

Power law diffusion coefficient and anomalous diffusion: analysis of solutions and first passage time.

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the usual diffusion equation. We consider a space- and time-dependent diffusion coefficient and the presence of absorbing boundaries. We obtain analytical results for the probability distribution and the first passage time distribution for finite and semi-infinite intervals. In addition, we compare our results for the first passage time distribution with the one obtained by the usual diffusion equation with constant diffusion coefficient.

Anomalous diffusion, solutions, and first passage time: Influence of diffusion coefficient.

We analyze a linear fractional Fokker–Planck equation for the case of an external force F(x)∝x|x|α−1 and diffusion coefficient D(x)∝|x|−θ (α,θ∈R). We also discuss the connection of the solutions found here with the Fox functions and the nonextensive statistics based on the Tsallis entropy.

Kwok Sau Fa

Papers

Generalized Langevin equation with fractional derivative and long-time correlation function.

Fractional Langevin equation and Riemann-Liouville fractional derivative.

Power law diffusion coefficient and anomalous diffusion: analysis of solutions and first passage time.

Anomalous diffusion, solutions, and first passage time: Influence of diffusion coefficient.

Anomalous diffusion: Fractional Fokker–Planck equation and its solutions