Fokker-Plank Equation Research Articles

A scaling-based model to describe temporal variability in soil hydraulic properties

Accurate knowledge of temporal variability in soil hydraulic properties (SHPs) can improve the prediction capability of flow and transport models. Lab and field measurements for determining SHPs at multiple timescales are expensive and time-consuming. Further, the existing Fokker-Plank equation (FPE) based numerical and analytical models for describing temporal variation in SHPs require parameterization. In this study, a scaling model is proposed to describe temporal variation in SHPs of four different soils. As field studies related to temporal variation in SHPs are limited, first synthetic temporally varying SHPs, soil water characteristic curves and hydraulic conductivity curves are generated using the analytical solution of FPE. Functional normalization approach is then used to develop reference curves and to obtain scale factors at different time-steps. The time-varying scale factors and the initial SHPs are related by two-step regression equations. The reference curves and the regression equations together can be used to estimate SHPs at any time given the initial SHPs. The developed model is validated using a 5-fold cross validation method and with experimental data at two study sites. The average percentage error in predicted van Genuchten parameters (θs,n and Ksat) except for α are mostly less than 10 %. The performance of the proposed model is comparable to that of FPE analytical model for experimental data. The proposed method holds promise for describing temporal variation in SHPs using only the initial SHPs.

A dynamic trap well model of hydrothermal shape-memory effect in amorphous polymers undergoing tailorable shape recovery behaviour

A dynamic trap well model is developed to describe the complex relaxations of functional segments, and explore the working principles behind the hydrothermal coupling effect in shape memory polymers (SMPs). A constitutive relationship among shape fixity strain, shape recovery strain and relaxation time has been formulated to characterize the hydrothermal coupling effect using geometrical parameters (i.e. width and height) of trap wells. Moreover, effects of temperature and solvent absorption on dynamic relaxation behaviours of SMPs have been formulated based on the Flory-Huggins theory and Fokker-Plank probability equation. The trap well model effectively analyzes the shape fixity ratio and shape recovery ratio within ranges of 50–100% and 0–100%, respectively. Finally, an extended Maxwell model is proposed to formulate the dynamic mechanical behaviours of SMPs with hydrothermal shape-memory effect (SME), and the analytical results have been verified using the experimental results reported in literature. A good agreement between the analytical results obtained from the proposed model and the experimental data is present, where the correlation coefficient ( R 2 ) is 95%. The present study firstly introduces the dynamic trap well model for shape memory behaviours and intricate relaxations, and then accurately predicts the dynamic shape recovery of SMP in response to hydrothermal stimulus.

Estimation of Optimal Lock-Down and Vaccination Rate of a Stochastic SIR Model: A Mathematical Approach

This paper utilizes a stochastic Susceptible-Infected-recovered (SIR) model with a nonlinear incidence rate to estimate the optimal lock-down intensity and vaccination rate under the COVID-19 pandemic environment. We use a Feynman-type path integral control approach to determine a Fokker-Plank type equation of this system. Since we assume the availability of information on the COVID-19 pandemic is complete and perfect, we show the existence of a unique fixed point. A non-linear incidence rate is used because, it can be raised from saturation effects that if the proportion of infected agents is very high so that exposure to the pandemic is inevitable, then the transmission rate responds slower than linearity to the increase in the number of infections. The simulation study shows that with higher diffusion coefficients susceptible and recovery curves keep the downward trends while the infection curve becomes ergodic. Finally, we perform a data analysis using UK data at the beginning of 2021 and compare it with our theoretical results.

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

Stochastinių logistinių dėsnių ir tankio mišinių taikymas medynų augimo analizėje

The processes of growth play an important role in different fields of science, such as ecology, economics, finance etc. In applications the averaged trend kinetics is represented by means of one of the types of logistic forms (Verhulst, Gompertz, Mitscherlich, Richards et al.). The article deals with the numerical solutions of the Fokker–Plank equation.

Electrical Response Estimation of Vibratory Energy Harvesters via Hilbert Transform Based Stochastic Averaging

Abstract Converting mechanical vibrations into electrical power with vibratory energy harvesters can ensure the portability, efficiency, and sustainability of electronic devices and batteries. Vibratory energy harvesters are typically modeled as nonlinear oscillators subject to random excitation, and their design requires a complete characterization of their probabilistic responses. However, simulation techniques such as Monte Carlo are computationally prohibitive when the accurate estimation of the response probability distribution is needed. Alternatively, approximate methods such as stochastic averaging can estimate the probabilistic response of such systems at a reduced computational cost. In this paper, the Hilbert transform based stochastic averaging is used to model the output voltage amplitude as a Markovian stochastic process with dynamics governed by a stochastic differential equation with nonlinear drift and diffusion terms. Moreover, the voltage amplitude dependent damping and stiffness terms are determined via an appropriate equivalent linearization, and the stationary probability distribution of the output voltage amplitude is obtained analytically by solving the corresponding Fokker–Plank equation. Two examples are used to demonstrate the accuracy of the obtained analytical probability distributions via comparisons with Monte Carlo simulation data.

Entropy- A Tale of Ice and Fire

Abstract In this review paper, we recall, in a unifying manner, our recent results concerning the Lie symmetries of nonlinear Fokker-Plank equations, associated to the (weighted) Tsallis and Kaniadakis entropies. The special values of the Tsallis parameters, highlighted by the classification of these symmetries, clearly indicate algebraic and geometric invariants which differentiate the Lie algebras involved. We compare these values with the ones previously obtained by several authors, and we try to establish connections between our theoretical families of entropies and specific entropies arising in several applications found in the literature. We focus on the discovered correlations, but we do not neglect dissimilarities, which might provide -in the future-deeper details for an improved extended panorama of the Tsallis entropies.

Analysis of growth in population through nonlinear stochastic process

Probability density evolution of an insect population is discussed by taking non linear stochastic differential equations for the growth and also using the related Fokker-Plank equations for the probability densities which are time dependent. It can be seen that the variance behavior depends on initial conditions whereas the effect of initial conditions disappears rapidly in mean case. We can observe the difference in the response of mean and variance as the growth of insects proceeds. It is found that variance of the process may increase monotonically to a level above the steady state variance before getting back to steady state variance.

Trial of identification of Non-linear 1-dof system based on the maximum likelihood estimation using the analytical solution of Fokker-Plank equation

In this research, the system identification based on the maximum likelihood estimation method using the analytical solution of the Fokker-Plank equation is discussed in case of a nonlinear one-degree-of-freedom system. In previous work, an author was already reported about the verification result of the fundamental operation. However, there is a problem which the previous study didn’t consider the estimation accuracy loss by non-stationary response. Therefore, it is necessary understanding of the relationship between estimation accuracy and non-stationary response. In this paper, the moment equation analysis was conducted in order to understanding of the estimation accuracy mechanism. The non-stationarity and steady-state values are obtained from the moment equations. As the result, relaxation time of transient behavior was revealed in a case. Furthermore, the squared error surface is calculated based on each theoretical values about the moment.

Convergence analysis of structure‐preserving numerical methods for nonlinear Fokker–Planck equations with nonlocal interactions

A class of nonlinear Fokker–Planck equations with nonlocal interactions may include many important cases, such as porous medium equations with external potentials and aggregation–diffusion models. The trajectory equation of the Fokker–Plank equation can be derived based on an energetic variational approach. A structure‐preserving numerical scheme that is mass conservative, energy stable, uniquely solvable, and positivity preserving at a theoretical level has also been designed in the previous work. Moreover, the numerical scheme is shown to satisfy the discrete energetic dissipation law and preserve steady states and has been observed to be second order accurate in space and first‐order accurate time in various numerical experiments. In this work, we give the rigorous convergence analysis for the highly nonlinear numerical scheme. A careful higher order asymptotic expansion is needed to handle the highly nonlinear nature of the trajectory equation. In addition, two step error estimates (a rough estimate and a refined estimate) are necessary in the convergence proof. Different from a standard error estimate, the rough estimate is performed to control the nonlinear term. A few numerical results are also presented to verify the optimal convergence order and the preservation of equilibria.

Two Isospectral-Nonisospectral Super-Integrable Hierarchies and Related Invariant Solutions

In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.

Multivariate Pade approximation for solving Fokker-Plank equations of fractional order

Multivariate Pade approximation for solving Fokker-Plank equations of fractional order

On Systems of Active Particles Perturbed by Symmetric Bounded Noises: A Multiscale Kinetic Approach

We consider an ensemble of active particles, i.e., of agents endowed by internal variables u(t). Namely, we assume that the nonlinear dynamics of u is perturbed by realistic bounded symmetric stochastic perturbations acting nonlinearly or linearly. In the absence of birth, death and interactions of the agents (BDIA) the system evolution is ruled by a multidimensional Hypo-Elliptical Fokker–Plank Equation (HEFPE). In presence of nonlocal BDIA, the resulting family of models is thus a Partial Integro-differential Equation with hypo-elliptical terms. In the numerical simulations we focus on a simple case where the unperturbed dynamics of the agents is of logistic type and the bounded perturbations are of the Doering–Cai–Lin noise or the Arctan bounded noise. We then find the evolution and the steady state of the HEFPE. The steady state density is, in some cases, multimodal due to noise-induced transitions. Then we assume the steady state density as the initial condition for the full system evolution. Namely we modeled the vital dynamics of the agents as logistic nonlocal, as it depends on the whole size of the population. Our simulations suggest that both the steady states density and the total population size strongly depends on the type of bounded noise. Phenomena as transitions to bimodality and to asymmetry also occur.

Mathematical aspects of application of fractional multidimensional Fokker-Plank equation to the theory of rate of chemical reactions based on spectroscopy experiments with examples.

Different types of spectroscopy experiments carried by multiple teams suggest some discrepancies between one-dimensional (1D) Kramers’ theory for the rate of chemical reactions with hydrodynamic Stokes-Einstein friction model. The disagreement continued despite multidimensional (multi-D), or non-Markovian, or not-exactly hydrodynamic modifications by Grote-Hynes, Hubbard, and other authors. The proposed fractional dimensional modification to multi-D model is based on theoretical and experimental findings of dimensional disagreement and deceptive simplicity between curves for barrier crossing at equilibrium and the boundary of the curves. Comparison of less anomalous fractional derivative exponents of 1.9 and 1.7 with more anomalous of 1.5 and 1.3 suggest Sierpinski carpet, Cesaro’s, and different attractors’ curves.

Effect of hydrodynamic slip on the rotational dynamics of a thin Brownian platelet in shear flow

The classical theory by Jeffery predicts that, in the absence of Brownian fluctuations, a thin rigid platelet rotates continuously in a shear flow, performing periodic orbits. However, a stable orientation is possible if the surface of the platelet displays a hydrodynamic slip length$\lambda$comparable to or larger than the thickness of the platelet. In this article, by solving the Fokker–Plank equation for the orientation distribution function and corroborating the analysis with boundary integral simulations, we quantify a threshold Péclet number,${Pe}_{c}$, above which such alignment occurs. We found that for${Pe}$smaller than${Pe}_{c}$, but larger than a second threshold, a regime emerges where Brownian fluctuations are strong enough to break the platelet's alignment and induce rotations, but with a period of rotation that depends on the value of$\lambda$. For${Pe}$below this second threshold, slip has a negligible effect on the orientational dynamics. We use these thresholds to classify the dynamics of graphene-like nanoplatelets for realistic values of$\lambda$and apply our results to the quantification of the orientational contribution to the effective viscosity of a dilute suspension of nanoplatelets with slip. We find a non-monotonic variation of this term, with a minimum occurring when the slip length is comparable to the thickness of the particle.

Some generalized isospectral-nonisospectral integrable hierarchies

In this article, with the aid of the Lie algebra A1 composed of second order matrices and Lie algebra A2 composed of third order matrices, some new soliton hierarchies of evolution equations are deduced and the corresponding Hamiltonian structures are also worked out by utilizing the trace identity. Specially, one of the integrable soliton hierarchicy is reduced to the generalized Fokker–Plank equation (gFP) and special bond pricing equation. Next, the nonlinear self-adjointness of the generalized Fokker–Plank equation is verified and conservation laws are constructed with the aid of Ibragimov’ method. Moreover, we investigate the coverings and the nonlocal symmetries of the generalized Fokker–Plank equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product. Besides, we apply the Li’s method to obtain two basic Darboux transformations of (gFP) with two different forms of T.

Exact solutions and numerical simulations of time-fractional Fokker-Plank equation for special stochastic process

‎In this paper, a type of time-fractional Fokker-Planck equation (FPE) of the OrnsteinUhlenbeck process is solved via Riemann-Liouville and Caputo derivatives. An analytical method based on symmetry operators is used for finding reduced form and exact solutions of the equation. A numerical simulation based on the M¨untz-Legendre polynomials is applied in order to find some approximated solutions of the equation.

Trial of identification of Non-linear 1-dof system based on the maximum likelihood estimation using the analytical solution of Fokker-Plank equation

Trial of identification of Non-linear 1-dof system based on the maximum likelihood estimation using the analytical solution of Fokker-Plank equation

Thermal fluctuations in the Landau-Lifshitz-Bloch model

A formulation for thermal noise in the stochastic form of the Landau Lifshitz Bloch equation used for modeling the magnetization dynamics at elevated temperatures is presented. The diffusion coefficients for thermal fluctuations are obtained via the Fokker-Plank equation using the mean field approximation of the field in defining the free energy. The presented model leads to a mean magnetization consistent with the equilibrium magnetization for small and large particles. The distribution of the magnetization magnitude is of a Poisson-like type rather than the classical Boltzmann distribution. The presented model was tested by studying the equilibrium magnetization in macrospin particles at high temperatures. The model is appealing for multi-scale modeling, such as modeling heat assisted magnetic recording systems and all-optical magnetization reversal

A model for describing the velocity of a particle in Brownian motion by Robotnov function based fractional operator

This endeavour provides a new instance of understanding the velocity of a particle in Brownian motion, using the Fokker-Plank equation. Our treatment is based on the new mathematical tool invoked by Yang-Abdel-Aty-Cattani (YAC) to solve the fractional derivative on Fokker-Plank equation. The HPTM and RPSMs, are used to obtain the solution in terms of Prabhakar function. Numerical examples to extend these ideas and to establish the capability and truthfulness of the proposed method, which makes use of YAC-fractional derivative, is also briefly sketched.