The Exact Solution of the Cauchy Problem for Two Generalized Fokker-Planck Equations — Algebraic Approach

1997 ◽  
Vol 12 (01) ◽  
pp. 165-170 ◽  
Author(s):  
A. A. Donkov ◽  
A. D. Donkov ◽  
E. I. Grancharova
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Algebraic Approach ◽  
Planck Equation ◽  
Operational Method ◽  
Suzuki Method ◽  
Fokker Planck Equation ◽  
The Cauchy Problem ◽  
Fokker Planck Equations ◽  
Fokker Planck

By employing algebraic techniques we find the exact solutions of the Cauchy problem for two equations, which may be considered as n-dimensional generalization of the famous Fokker–Planck equation. Our approach is a combination of the disentangling techniques of R. Feynman with operational method developed in modern functional analysis in particular in the theory of partial differential equations. Our method may be considered as a generalization of the M. Suzuki method of solving the Fokker–Planck equation.

scholarly journals Fokker–Planck approach to neural networks and to decision problems

2021 ◽  
Author(s):  
Sebastian Vellmer ◽  
Benjamin Lindner
Keyword(s):  
Cognitive Neuroscience ◽  
Planck Equation ◽  
Recurrent Network ◽  
Spike Trains ◽  
Sequential Decision ◽  
Fokker Planck Equation ◽  
Gaussian Colored Noise ◽  
Correlation Statistics ◽  
Fokker Planck Equations ◽  
Fokker Planck

AbstractWe review applications of the Fokker–Planck equation for the description of systems with event trains in computational and cognitive neuroscience. The most prominent example is the spike trains generated by integrate-and-fire neurons when driven by correlated (colored) fluctuations, by adaptation currents and/or by other neurons in a recurrent network. We discuss how for a general Gaussian colored noise and an adaptation current can be incorporated into a multidimensional Fokker–Planck equation by Markovian embedding for systems with a fire-and-reset condition and how in particular the spike-train power spectrum can be determined by this equation. We then review how this framework can be used to determine the self-consistent correlation statistics in a recurrent network in which the colored fluctuations arise from the spike trains of statistically similar neurons. We then turn to the popular drift-diffusion models for binary decisions in cognitive neuroscience and demonstrate that very similar Fokker–Planck equations (with two instead of only one threshold) can be used to study the statistics of sequences of decisions. Specifically, we present a novel two-dimensional model that includes an evidence variable and an expectancy variable that can reproduce salient features of key experiments in sequential decision making.

scholarly journals Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Fokker-Planck Equation

2019 ◽  
pp. 29-37 ◽  
Author(s):  
Hijaz Ahmad
Keyword(s):  
Large Class ◽  
Planck Equation ◽  
Suitable Method ◽  
Iteration Algorithm ◽  
Auxiliary Parameter ◽  
Variational Iteration ◽  
Fokker Planck Equation ◽  
Linear Problems ◽  
Fokker Planck Equations ◽  
Fokker Planck

In this paper, variational iteration algorithm-I with an auxiliary parameter is implemented to investigate Fokker-Planck equations. To show the accuracy and reliability of the technique comparisons are made between the variational iteration algorithm-I with an auxiliary parameter and classic variational iteration algorithm-I. The comparison shows that variational iteration algorithm-I with an auxiliary parameter is more powerful and suitable method for solving Fokker-Planck equations. Furthermore, the proposed algorithm can successfully be applied to a large class of nonlinear and linear problems.

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