Solutions of fractional nonlinear diffusion equation and first passage time: Influence of initial condition and diffusion coefficient

2008 ◽  
Vol 387 (18) ◽  
pp. 4547-4552
Author(s):  
Jun Wang ◽  
Wen-Jun Zhang ◽  
Jin-Rong Liang ◽  
Pan Zhang ◽  
Fu-Yao Ren
Keyword(s):  
Diffusion Coefficient ◽  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
First Passage Time ◽  
Passage Time ◽  
Nonlinear Diffusion Equation ◽  
Initial Condition ◽  
First Passage ◽  
And Diffusion

Fractional nonlinear diffusion equation and first passage time

2008 ◽  
Vol 387 (4) ◽  
pp. 764-772 ◽  
Author(s):  
Jun Wang ◽  
Wen-Jun Zhang ◽  
Jin-Rong Liang ◽  
Jian-Bin Xiao ◽  
Fu-Yao Ren
Keyword(s):  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
First Passage Time ◽  
Passage Time ◽  
Nonlinear Diffusion Equation ◽  
First Passage

scholarly journals A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process

2011 ◽  
Vol 48 (02) ◽  
pp. 420-434 ◽  
Author(s):  
Peter J. Thomas
Keyword(s):  
Lower Bound ◽  
First Passage Time ◽  
Passage Time ◽  
Initial Condition ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Fire Model ◽  
Integrate And Fire ◽  
Ornstein Uhlenbeck Process ◽  
Time Density

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck processX(t) obeying dX= -βXdt+ σdWto reach a fixed threshold θ from a suprathreshold initial conditionx0> θ > 0 has a lower bound of the form ρ(t) >kexp[-pe6βt] for positive constantskandpfor timestexceeding some positive valueu. We obtain explicit expressions fork,p, anduin terms of β, σ,x0, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.

scholarly journals Equations for nonlinear diffusion

2010 ◽  
Vol 51 ◽  
Author(s):  
Arvydas Juozapas Janavičius
Keyword(s):  
Diffusion Coefficient ◽  
Diffusion Equation ◽  
Analytical Solutions ◽  
Nonlinear Diffusion ◽  
Diffusion Time ◽  
Nonlinear Diffusion Equation ◽  
Square Root ◽  
Brownian Movement ◽  
Finite Velocity

The diffusion is the result of Brownian movement and occurs with a finite velocity. We presented the nonlinear diffusion equation, with diffusion coefficient directly proportional to the impurities concentration. Analytical solutions, showing that the maximum displacements of diffusing particles are proportional to the square root of diffusion time like for Brownian movement, was obtained. For small concentrations of impurities, nonlinear diffusion equation transforms to linear.

scholarly journals First passage dynamics of stochastic motion in heterogeneous media driven by correlated white Gaussian and coloured non-Gaussian noises

2021 ◽  
Author(s):  
Nicholas Mwilu Mutothya ◽  
Yong Xu ◽  
Yongge Li ◽  
Ralf Metzler ◽  
Nicholas Muthama Mutua
Keyword(s):  
Diffusion Coefficient ◽  
Gaussian Noise ◽  
First Passage Time ◽  
Passage Time ◽  
Mean First Passage Time ◽  
Stochastic Motion ◽  
First Passage ◽  
The Mean ◽  
Non Gaussian ◽  
Markovian Systems

Abstract We study the first passage dynamics for a diffusing particle experiencing a spatially varying diffusion coefficient while driven by correlated additive Gaussian white noise and multiplicative coloured non-Gaussian noise. We consider three functional forms for position dependence of the diffusion coefficient: power-law, exponential, and logarithmic. The coloured non-Gaussian noise is distributed according to Tsallis' $q$-distribution. Tracks of the non-Markovian systems are numerically simulated by using the fourth-order Runge-Kutta algorithm and the first passage times are recorded. The first passage time density is determined along with the mean first passage time. Effects of the noise intensity and self-correlation of the multiplicative noise, the intensity of the additive noise, the cross-correlation strength, and the non-extensivity parameter on the mean first passage time are discussed.

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