Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces

We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, the existence and uniqueness of a measure valued solution, for the corresponding Fokker–Planck equation. In particular, we verify the Chapman–Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.     

A deterministic particle method for the Vlasov–Fokker–Planck equation in one dimension

The Vlasov–Fokker–Planck equation is a model for a collisional, electrostatic plasma. The approximation of this equation in one spatial dimension is studied. The equation under consideration is linear in that the electric field is given as a known function that is not internally consistent with the phase space distribution function. The approximation method applied is the deterministic particle method described in Wollman and Ozizmir [Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension, J. Comput. Phys. 202 (2005) 602–644]. For the present linear problem an analysis of the stability and convergence of the numerical method is carried out. In addition, computations are done that verify the convergence of the numerical solution. It is also shown that the long term asymptotics of the computed solution is in agreement with the steady state solution derived in Bouchut and Dolbeault [On long time asymptotics of the Vlasov–Fokker–Planck equation and of the Vlasov–Poisson–Fokker–Planck system with coulombic and Newtonian potentials, Differential Integral Equations 8(3) (1995) 487–514].     

Initial-Boundary Value Problems for the Korteweg-de Vries Equation

Exact and approximate solutions of the initial—boundaryvalue problem for the Korteweg—de Vries equation on thesemi-infinite line are found. These solutions are found forboth constant and time-dependent boundary values. The form ofthe solution is found to depend markedly on the specific boundaryand initial value. In particular, multiple solutions and nonsteadysolutions are possible. The analytical solutions are comparedwith numerical solutions of the Korteweg—de Vries equationand are found to be in good agreement.     

Dynamical Systems in the Variational Formulation of the Fokker—Planck Equation by the Wasserstein Metric

R. Jordan, D. Kinderlehrer, and F. Otto proposed the discrete-time approximation of the Fokker—Planck equation by the variational formulation. It is determined by the Wasserstein metric, an energy functional, and the Gibbs—Boltzmann entropy functional. In this paper we study the asymptotic behavior of the dynamical systems which describe their approximation of the Fokker—Planck equation and characterize the limit as a solution to a class of variational problems. Accepted 2 June 2000. Online publication 6 October 2000.     

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domainswith piecewise smooth boundaries is presented. It is intendedfor problems describing the Stokes flow in the situations whereone has corners or cusps formed by parts of the domain boundaryand, due to the nature of the boundary conditions on these partsof the boundary, these regions have a global effect on the shapeof the whole domain and hence have to be resolved with sufficientaccuracy. The algorithm combines the boundary integral equationmethod for the main part of the flow domain and the finite-elementmethod which is used to resolve the corner/cusp regions. Twoparts of the solution are matched along a numerical ‘internalinterface’ or, as a variant, two interfaces, and theyare determined simultaneously by inverting a combined matrixin the course of iterations. The algorithm is illustrated byconsidering the flow configuration of ‘curtain coating’,a flow where a sheet of liquid impinges onto a moving solidsubstrate, which is particularly sensitive to what happens inthe corner region formed, physically, by the free surface andthe solid boundary. The ‘moving contact line problem’is addressed in the framework of an earlier developed interfaceformation model which treats the dynamic contact angle as partof the solution, as opposed to it being a prescribed functionof the contact line speed, as in the so-called ‘slip models’.     

Mathematical Modelling of Heat Flow in Czochralski Crystal Pulling

A mathematical model of the heat flow in the holm region incrystal pulling by the Czochralski technique is developed. Thisis a moving boundary problem with two moving boundaries, thephase change surface and the air—liquid meniscus. Usingthe enthalpy method and co-ordinate transformation techniques,the problem is cast into a form suitable for numerical solution.A numerical scheme is outlined, and some results for the growthof germanium crystals are shown.     

The method of non-linear time transformation in boundary-value problems of potential theory with moving boundaries for the non-linear wave equation

A new method for solving boundary-value problems for the wave equation [1–3] with moving boundaries is used to obtain a solution of a boundary-value problem with boundary conditions of three types [4].     

Asymptotic Solution of a Model Non-linear Convective Diffusion Equation

In this paper we consider the large-time solution of the equation for initial data with compact support. With m = 4 and n = 3the equation models the flow of a thin viscous sheet on an inclinedbed while for n m > 1 it has application in porous mediaflow under gravity. The equation can also be regarded as ananalogue of Burgers equation in non-linear diffusion. It isknown that two moving boundaries exist along which certain boundaryconditions are required to hold. The paper extends earlier workand determines the analytic behaviour of the moving boundariesexist along which certain boundary conditions are required tohold. The paper extends earlier work and determines the analyticbehaviour of the moving boundaries together with the structureof the solution which at large times is shown to depend cruciallyon the location in (n, m) parameter space.     

Oscillations of a Stratified Liquid Partially Covered with Ice (General Case)

We study the problem of small motions of an ideal stratified liquid whose free surface consists of three regions: liquid surface without ice, a region of elastic ice, and a region of crumbled ice. The elastic ice is modeled by an elastic plate. The crumbled ice is understood as weighty particles of some matter floating on the free surface. Using the method of orthogonal projection of boundary conditions on a moving surface and the introduction of auxiliary problems, we reduce the original initial boundary value problem to an equivalent Cauchy problem for a second-order differential equation in a Hilbert space. We obtain conditions under which there exists a strong (with respect to time) solution of the initial boundary value problem describing the evolution of the hydrodynamic system under consideration.

     

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 One-step Method of Order 10 for y' = f(x, y)

In some situations, especially if one demands the solution ofthe differential equation with a great precision, it is preferableto use high-order methods. The methods considered here are similarto Runge—Kutta methods, but for the second-order equationy'= f(x, y). As for Runge—Kutta methods, the complexityof the order conditions grows rapidly with the order, so thatwe have to solve a non—linear system of 440 algebraicequations to obtain a tenth—order method. We demonstratehow this system can be solved. Finally we give the coefficients(20 decimals) of two methods with small local truncation errors.     

A Mildly Non-linear System of Parabolic Differential Equations Arising from Mass Transfer with Chemical Reaction in Electrochemistry

A system of three connected parabolic equations is studied.The first equation is the straight forward diffusion equationin one space dimension and the solution can be written down.The remaining two cannot be solved analytically but it is interestingto observe that a solution does exist for their difference.By considering the problem as a moving boundary value probleman approximate solution is obtained by a finite difference technique.An analysis of stability is performed and numerical resultsfor a specific chemical reaction are presented.     

The application of Hermite interpolation to the analysis of non-linear diffusive initial-boundary value problems

In this paper we examine the feasibility of using two-pointHermite interpolation as a systematic tool in the analysis ofinitial-boundary value problems for non-linear diffusion equations.We do this by considering a series of examples for the porousmedium equation involving both fixed and moving boundaries.Essentially, the idea is to construct polynomials which fitthe known and unknown function values and their derivativesat the two end points of a given interval. Systems of ordinarydifferential equations are then obtained for the unknown endpoint functions of time which need to be determined in orderto specify the polynomial representation—the initial conditionsfor such systems are related to the initial data for the originalproblem. As well as constructing approximate solutions, it emergesthat the method is particularly useful in identifying steadystates and similarity solutions together with their stabilityand other asymptotic properties. We believe that the techniqueprovides scientists and applied mathematicians with a valuablestrategy in the analysis of the qualitative and quantitativefeatures of solutions to initial-boundary value problems involvingnon-linear diffusion and related equations.     

High-order approximation of Pearson diffusion processes

This paper focuses on Pearson diffusions and the spectral high-order approximation of their related Fokker–Planck equations. The Pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient. They are widely used in the physical and chemical sciences, engineering, rheology, environmental sciences and financial mathematics. In recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations. Analytical solutions have been derived for some of the Pearson diffusions, including the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross and Jacobi processes. However, analytical investigations and computations for diffusions with so-called heavy-tailed ergodic distributions are more difficult to perform. The novelty of this research is the development of an accurate and efficient numerical method to solve the Fokker–Planck equations associated with Pearson diffusions with different boundary conditions. Comparisons between the numerical predictions and available time-dependent and equilibrium analytical solutions are made. The solution of the Fokker–Planck equation is approximated using a reduced basis spectral method. The advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving Fokker–Planck type equations.     

A Non-linear Stability Analysis of the Solidification of a Pure Substance

An investigation is made of the stability of the shape of amoving planar interface between the solid and liquid phasesduring the solidification of a pure substance. The prototypeproblem of controlled two-dimensional growth of a pure solidinto a thermally undercooled liquid bath is considered. Themodel employed postulates diffusion of heat with equal thermaldiffusivities in both phases under the simplifying assumptionsthat these phases are infinite in extent and there is no convectionin the liquid phase. In addition, modified versions of the conservationof heat boundary condition and the Gibbs—Thomson equation,developed by Wollkind and Maurer in earlier work, are imposedat the solid—liquid interface. The main results of ournon-linear analysis, which can be represented in a plot of undercoolingversus solidification speed are that the interface can be unstableto finite amplitude disturbances in some regions where lineartheory predicts stability to infinitesimal disturbances andthat it can exhibit finite amplitude equilibrium for other rangesof parameter values. These results are interpreted in relationto the growth and structure of dendrites.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.     

Some new approaches to the solution of the diffusion chaos problem

Using the solution of the Kuramoto–Tsuzuki equation as an example, we present the results of numerical investigations of diffusion chaos in the neighborhood of the thermodynamic branch of the “reaction–diffusion” equation system. Chaos onset scenarios are considered both in the small-mode approximation and for the solution of the second boundary-value problem for the original equation. In the phase space of the Kuramoto–Tsuzuki equation chaos sets in through period doubling bifurcation cascades and through subharmonic bifurcation cascades of two-dimensional tori by both internal and external frequency. Chaos onset scenarios in the Kuramoto–Tsuzuki equation phase space and in the Fourier coefficient space are compared both for the small-mode approximation and for direct numerical solution of the second boundary-value problem. Inappropriateness of the three-dimensional small-mode approximations is proved.     

The boundaries of the solutions of the linear Volterra integral equations with convolution kernel

Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of an equation in this type was developed in I. Özdemir and Ö. F. Temizer, 2002.

In this paper, by using the method in Özdemir and Temizer, it is shown that the boundary of the solution function of an equation in the same form can also be expanded under different conditions than those that they used.