Fractional Fokker–Planck equation

By using the definition of the characteristic function and Kramers–Moyal Forward expansion, one can obtain the Fractional Fokker–Planck Equation (FFPE) in the domain of fractal time evolution with a critical exponent α (0<α⩽1). Two different classes of fractional differential operators, Liouville–Riemann (L–R) and Nishimoto (N) are used to represent the fractal differential operators in time. By applying the technique of eigenfunction expansion to get the solution of FFPE, one finds that the time part of eigenfunction expansion in terms of L–R represents the waiting time density Ψ(t), which gives the relation between fractal time evolution and the theory of continuous time random walk (CTRW). From the principle of maximum entropy, the structure of the distribution function can be known.     

Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

On the fundamental solution of the Fokker–Planck–Kolmogorov equation

The Fokker–Planck–Kolmogorov parabolic second-order differential operator is considered, for which its fundamental solution is derived in explicit form. Such operators arise in numerous applications, including signal filtering, portfolio control in financial mathematics, plasma physics, and problems involving linear-quadratic regulators.     

New exact solutions to the Fokker–Planck–Kolmogorov equation

A generalized Liouville theorem has been proven for Itô systems. This allows us to show that the conserved quantities of the deterministic part of the Itô systems lead to the solution of the Fokker–Planck–Kolmogorov equation. The results have been applied to a stochastic 3-species Lotka Volterra system and the semi-classical Jaynes–Cummings system.     

Decay and stability of solutions to the Fokker–Planck–Boltzmann equation in

In this paper, we use the method of constructing the compensating function introduced by Kawashima and the standard energy method to study the global existence of solutions to the Fokker–Planck–Boltzmann equation in the whole space. The time decay and uniform stability of solutions to the global Maxwellian are also obtained.     

Analytical solution of stochastic differential equation by multilayer perceptron neural network approximation of Fokker–Planck equation

The Fokker–Planck equation is a useful tool to analyze the transient probability density function of the states of a stochastic differential equation. In this paper, a multilayer perceptron neural network is utilized to approximate the solution of the Fokker–Planck equation. To use unconstrained optimization in neural network training, a special form of the trial solution is considered to satisfy the initial and boundary conditions. The weights of the neural network are calculated by Levenberg–Marquardt training algorithm with Bayesian regularization. Three practical examples demonstrate the efficiency of the proposed method.     

Finite difference approximations for the fractional Fokker–Planck equation

The fractional Fokker–Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine some practical numerical methods to solve a class of initial-boundary value problems for the fractional Fokker–Planck equation on a finite domain. The solvability, stability, consistency, and convergence of these methods are discussed. Their stability is proved by the energy method. Two numerical examples are also presented to evaluate these finite difference methods against the exact analytical solutions.     

Cauchy problem for the Fokker–Planck–Kolmogorov equation of a multidimensional normal Markovian process

We present the results of studying the fundamental solution and correct solvability of the Cauchy problem as well as the integral representation of solutions for the Fokker–Planck–Kolmogorov equation of a class of normal Markovian processes.     

A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems

A general formulation of the Fokker–Planck–Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHSs). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example.     

Stationary Fokker–Planck equation on noncompact manifolds and in unbounded domains

We investigate the Fokker–Planck equation on an infinite cylindrical surface and in an infinite strip with reflecting boundary conditions, prove the existence of a positive (not necessarily integrable) solution, and derive various conditions on the vector field f that are sufficient for the existence of a solution that is the probability density. In particular, these conditions are satisfied for some vector fields f with integral trajectories going to infinity.     

Computation of the solutions of the Fokker–Planck equation for one and two DOF systems

Uncertainty in structures may come from unknowns in the modelisation and in the properties of the materials, from variability with time, external noise, etc. This leads to uncertainty in the dynamic response. Moreover, the consequences are issues in safety, reliability, efficiency, etc. of the structure. So an issue is the gain of information on the response of the system taking into account the uncertainties [Mace BR, Worden K, Manson G. Uncertainty in structural dynamics. J Sound Vib 2005;288(3):423–9].If the forcing or the uncertainty can be modelled through a white noise, the Fokker–Planck (or Kolmogorov forward) equation exists. It is a partial differential linear equation with unknown p(X, t), where p(X, t) is the probability density function of the state X at time t.In this article, we solve this equation using the finite differences method, for one and two DOF systems. The numerical solutions obtained are proved to be nearly correct.     

Stabilization of Solutions of the Nonlinear Fokker–Planck Equation

One considers the equation $$ \mathrm{div}\left( {{u^{\sigma }}Du} \right)+b(x)Du-{u_t}=f(x)g(u),\quad x\in {{\mathbb{R}}^n},\quad t\in \left( {0,\infty } \right), $$ where $ b:{{\mathbb{R}}^n}\to {{\mathbb{R}}^n} $ and $ f:{{\mathbb{R}}^n}\to [0,\infty ) $ are locally bounded measurable functions, g: (0,∞)?→?(0,∞) is continuous and nondecreasing, One obtains the conditions ensuring that its positive solutions stabilize to zero as t?→?∞.     

On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ _t μ = L ^* μ for bounded Borel measures on ℝ ^d  × (0, T), where L is a second order elliptic operator, for example, Lu = D_xu + ( b,?_xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity

An extension of a data assimilation method based on the application of the Fokker–Planck equation

A data assimilation method based on the Kalman filter theory and on the Fokker–Planck equation is extended to assimilate Atlantic Ocean data into a new version of the well-known Modular Ocean Model (MOM_3) from NOAA/GFDL. This extension enables assimilation of non-uniformly distributed data in space and time. Numerical experiments with Levitus atlas data are carried out with the ocean model configured at a low resolution. Some results of these experiments as well as other possible expansions are discussed.