Fully Implicit Iterative Solving Method for the Fokker-Planck Equation in Tokamak Plasmas

A three dimensional bounce-averaged Fokker-Planck （FP） numerical code has been newly developed based on fully implicit iterative solving method, and relativistic effect is also included in the code. The code has been tested against various benchmark cases： Ohmic con ductivity in the presence of weak Ohmic electric field, runaway losses of electrons in the presence of strong Ohmic electric field, lower hybrid current drive and electron cyclotron current drive via two- or three-dimensional simulation. All the test cases run fast and correctly during calculations. As a result, the code provides a set of powerful tools for studying radio frequency wave heating and current drive in tokamak plasmas.

On New Invariant Solutions of Generalized Fokker-Planck Equation

The generalized one-dimensional Fokker-Planck equation is analyzed via potential symmetry method and the invariant solutions under potential symmetries are obtained. Among those solutions, some are new and first reported.

Immersed Interface Method for Fokker-Planck Equation with Discontinuous Drift

The Immersed Interface Method (IIM) is derived to solve the corresponding Fokker-Planck equation of Brownian motion with pure dry friction, which is one of the simplest models of piecewise-smooth stochastic systems. The IIM is capable of treating a discontinuity in the drift of Fokker-Planck equation and it is readily extended to the dry and viscous friction model. Analytic results of the considered model are used to confirm the effectiveness and design accuracy of the scheme.

Dissipaton Equation of Motion Theory versus Fokker-Planck Quantum Master Equation

The quest of exact and nonperturbative methods on quantum dissipation with nonlinear coupling environments remains in general a great challenge.In this review we present a comprehensive account on two approaches to the entangled system-and-environment dynamics,in the presence of linear-plus-quadratic coupling bath.One is the dissipaton-equation-ofmotion(DEOM)theory that has been extended recently to treat the nonlinear coupling environment.Another is the extended Fokker-Planck quantum master equation(FP-QME)approach that will be constructed in this work,based on its DEOM correspondence.We closely compare these two approaches,with the focus on the underlying quasi-particle picture,physical implications,and implementations.

From Chemical Langevin Equations to Fokker-Planck Equation: Application of Hodge Decomposition and Klein-Kramers Equation

The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful However, what are the su^cient and necessary conditions of the existence of the potential function is still an open problem. Use Hodge decomposition theorem in differential form theory, we focus on the general chemical Langevin equations, which reitect complex chemical reaction systems. We analysis the conditions for the existence of potential landscape of the systems. By mapping the stochastic differential equations to a Hamiltonian mechanical system, we obtain the Fokker-Planck equation of the chemical reaction systems. The obtained Fokker-Planck equation can be used in further studies of other steady properties of complex chemical reaction systems, such as their steady state entropies.

Nanofluids Transport Model Based on Fokker-Planck Equation and the Convection Heat Transfer Calculation

In current research about nanofluid convection heat transfer, random motion of nanoparticles in the liquid distribution problem mostly was not considered. In order to study on the distribution of nanoparticles in liquid, nanofluid transport model in pipe is established by using the continuity equation, momentum equation and Fokker-Planck equation. The velocity distribution and the nanoparticles distribution in liquid are obtained by numerical calculation, and the effect of particle size and particle volume fraction on convection heat transfer coefficient of nanofluids is analyzed. The result shows that in high volume fraction （ 0 _-- 0.8% ）, the velocity distribution of nanofluids characterizes as a ＂cork-shaped＂ structure, which is significantly different from viscous fluid with a parabolic distribution. The convection heat transfer coefficient increases while the particle size of nanoparticle in nanofluids decreases. And the convection heat transfer coefficient of nanofluids is in good agreement with the experimental result both in low （0 ~〈 0.1% ） and high （ q = 0.6% ） volume fractions. In presented model, Brown motion, the effect of interactions between nanoparticles and fluid coupling, is also considered, but any phenomenological parameter is not introduced. Nanoparticles in liquid transport distribution can be quantitatively calculated by this model.

GLOBAL CLASSICAL SOLUTIONS FOR QUANTUM KINETIC FOKKER-PLANCK EQUATIONS

We consider a class of nonlinear kinetic Fokker-Planck equations modeling quantum particles which obey the Bose-Einstein and Fermi-Dirac statistics, respectively. We establish the existence and convergence rate to the steady state of global classical solution to such kind of equations around the steady state.

Path prediction of flexible needles based on Fokker-Planck equation and disjunctive Kriging model

Path prediction of flexible needles based on the Fokker-Planck equation and disjunctive Kriging model is proposed to improve accuracy and consider the nonlinearity and anisotropy of soft tissues.The stochastic differential equation is developed into the Fokker-Planck equation with Gaussian noise,and the position and orientation probability density function of flexible needles are then optimized by the stochastic differential equation.The probability density function obtains the mean and covariance of flexible needle movement and helps plan puncture paths by combining with the probabilistic path algorithm.The weight coefficients of the ordinary Kriging are extended to nonlinear functions to optimize the planned puncture path,and the Hermite expansion is used to calculate nonlinear parameter values of the disjunctive Kriging optimization model.Finally,simulation experiments are performed.Detailed comparison results under different path planning maps show that the kinematics model can plan optimal puncture paths under clinical requirements with an error far less than 2 mm.It can effectively optimize the path prediction model and help improve the target rate of soft tissue puncture with flexible needles through data analysis and processing of the mean value and covariance parameters derived by the probability density and disjunctive Kriging algorithms.

The solution of the time-dependent Fokker-Planck equation of non-degenerate optical parametric amplification and its application to the optimum realization of EPR paradox

In this paper, the solution of the time-dependent Fokker-Planck equation of non-degenerate optical parametric amplification is used to deduce the condition demonstrating the Einstein-Podolsky-Rosen （EPR） paradox. The analytics and numerical calculation show the influence of pump depletion on the error in the measurement of continuous variables. The optimum realization of EPR paradox can be achieved by adjusting the parameter of squeezing. This result is of practical importance when the realistic experimental conditions are taken into consideration .

Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker-Planck Equations

Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application,a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efficiency of the scheme and conform well to theoretical analysis.

The Hydrodynamic Limit of Nonlinear Fokker-Planck Equation

The non-linear Fokker-Planck equation arises in describing the evolution of stochastic system, which is a variant of the Boltzmann equation modeling the evolution of the random system with Brownian motion, where the collision term is replaced by a drift-diffusion operator. This model conserves mass, momentum and energy;the dissipation is much weaker than that in a simplified model we considered before which conserved only mass, thus more difficult to analyze. The macro-micro decomposition of the solution around the local Maxwellian introduced by T.-P. Liu, T. Yang and S.-H. Yu for Boltzmann equation is used, to reformulate the model into a fluid-type system incorporate viscosity and heat diffusion terms, coupled with an equation of the microscopic part. The viscosity and heat diffusion terms can give dissipative mechanism for the analysis of the model.

A New Approach to Time-Dependent Solutions to the Non-Linear Fokker-Planck Equations Related to Arbitrary Functions of Tsallis Entropy: A Mathematical Study and Investigation

The following article has been retracted due to the investigation of complaints received against it. Mr. Mohammadali Ghorbani (corresponding author and also the last author) cheated the authors’ name: Alireza Heidari and Seyedali Vedad. The scientific community takes a very strong view on this matter and we treat all unethical behavior such as plagiarism seriously. This paper published in Vol.3 No.5 420-429, 2012, has been removed from this site.

On the Application of Fokker-Planck Equation to Psychological Future Time

This paper tries to make a comparison and connection between Fokker-Planck or forward Kolmogorov equation and psychological future time which is based on quantum mechanics. It will be showed that in quantum finance forward interest rate model can be further improved by noting that the predicted correlation structure for field theory models depends only on variable where t is present time and x is future time. On the other side, forward Kolmogorov equation is a parabolic partial differential equation, requiring international conditions at time t and to be solved for . The aforementioned equation is to be used if there are some special states now and it is necessary to know what can happen later. It will be tried to establish the connection between these two equations. It is proved that the psychological future time if applied and implemented in Fokker-Planck equation is unstable and is changeable so it is not easily predictable. Some kinds of nonlinear functions can be applied in order to establish the notion of psychological future time, however it is unstable and it should be continuously changed.

A Novel Approach for Numerical Computation of Fokker-Planck Equation

This paper present an implementation of＂modified cubic B-spline differential quadrature method （MCB-DQM）＂ proposed by Arora ＆ Singh （Applied Mathematics and Computation Vol. 224（1） （2013） 161-177） for numerical computation of Fokker-Planck equations. The modified cubic B-splines are used as set of basis functions in the differential quadrature to compute the weighting coefficients for the spatial derivatives, which reduces Fokker-Planck equation into system of first-order ordinary differential equations （ODEs）, in time. The well known SSP-RK43 scheme is then applied to solve the resulting system of ODEs. The efficiency of proposed method has been confirmed by three examples having their exact solutions. This shows that MCB-DQM results are capable of achieving high accuracy. Advantage of the scheme is that it can be applied very smoothly to solve the linear or nonlinear physical problems, and a very less storage space is required which causes less accumulation of numerical errors.

A Method to Solve the Fokker-Planck Equation With Coordinate-dependent Mass,Friction and Temperature

An analytical expression of the propagator is obtained by using Lie algebramethod.The fission rates at the saddle point in both constant and coordinate-dependentmass,friction and temperature cases are calculated based on the expression of thepropagator and local approximation.The numerical calculation for 240pu shows thatthe fission rates from our method are reasonable.

Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.

Dynamical distribution of continuous service time model involving non-Maxwellian collision kernel and value functions

The distribution of continuous service time in call centers is investigated.A non-Maxwellian collision kernel combining two different value functions in the interaction rule are used to describe the evolution of continuous service time,respectively.Using the statistical mechanical and asymptotic limit methods,Fokker–Planck equations are derived from the corresponding Boltzmann-type equations with non-Maxwellian collision kernels.The steady-state solutions of the Fokker–Planck equation are obtained in exact form.Numerical experiments are provided to support our results under different parameters.

Application of Elzaki Transform Method to Market Volatility Using the Black-Scholes Model

Black-Scholes Model (B-SM) simulates the dynamics of financial market and contains instruments such as options and puts which are major indices requiring solution. B-SM is known to estimate the correct prices of European Stock options and establish the theoretical foundation for Option pricing. Therefore, this paper evaluates the Black-Schole model in simulating the European call in a cash flow in the dependent drift and focuses on obtaining analytic and then approximate solution for the model. The work also examines Fokker Planck Equation (FPE) and extracts the link between FPE and B-SM for non equilibrium systems. The B-SM is then solved via the Elzaki transform method (ETM). The computational procedures were obtained using MAPLE 18 with the solution provided in the form of convergent series.

On the Connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck Control Frameworks

In the framework of stochastic processes, the connection between the dynamic programming scheme given by the Hamilton-Jacobi-Bellman equation and a recently proposed control approach based on the Fokker-Planck equation is discussed. Under appropriate assumptions it is shown that the two strategies are equivalent in the case of expected cost functionals, while the Fokker-Planck formalism allows considering a larger classof objectives. To illustratethe connection between the two control strategies, the cases of an Itō stochastic process and of a piecewise-deterministic process are considered.

A monotone finite volume method for time fractional Fokker-Planck equations

We develop a monotone finite volume method for the time fractional Fokker-Planck equations and theoretically prove its unconditional stability. We show that the convergence rate of this method is of order 1 in the space and if the space grid becomes sufficiently fine, the convergence rate can be improved to order 2.Numerical results are given to support our theoretical findings. One characteristic of our method is that it has monotone property such that it keeps the nonnegativity of some physical variables such as density, concentration,etc.