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By T.D. Frank

Providing an advent to the idea of nonlinear Fokker-Planck equations, this e-book discusses basic homes of brief and desk bound strategies, emphasizing the steadiness research of desk bound suggestions. additionally taken care of are Langevin equations and correlation services.

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Extra resources for Nonlinear Fokker Planck Equations

Example text

Here, the probability density is expressed in terms of time-dependent first and higher order moments. For these moments coupled nonlinear differential equations are derived and truncated such that they can be solved numerically [77]. 36) by means of a particular finite-difference scheme [141]. 37) 30 2 Fundamentals is satisfied. 38) (see also Sect. 2). 37) guarantees the normalization of the time-dependent probability density: x xr P (x, t; u) dx = xlr u(x) dx = 1. We would like to mention that there xl are also alternative finite-difference schemes that have been applied to study the time-dependent behavior of solutions of nonlinear Fokker–Planck equations [377].

73) for all n, then we deal with a family of Markov processes [188]. 73) depend only on two time points. For example, let us consider a family of stochastic processes characterized by the initial distributions u1 , u2 , u3 , . .. Then, we deal with a set of Markov transition probability densities given by P i (xn , tn |xn−1 , tn−1 ; . . ; x1 , t1 ) = P i (xn , tn |xn−1 , tn−1 ). 74) that describes the evolution of all members of the family, we refer to the family as a linear family of Markov processes.

3. In a similar vein, one can treat nonlinear Fokker–Planck equations subjected to natural boundary conditions. Here, the probability density is expressed in terms of time-dependent first and higher order moments. For these moments coupled nonlinear differential equations are derived and truncated such that they can be solved numerically [77]. 36) by means of a particular finite-difference scheme [141]. 37) 30 2 Fundamentals is satisfied. 38) (see also Sect. 2). 37) guarantees the normalization of the time-dependent probability density: x xr P (x, t; u) dx = xlr u(x) dx = 1.

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