Initial condition solutions of the generalized feller equation

Summary Solutions for given initial conditions are established for the generalized (autonomous parabolic) Feller equation in one positive space variable and one positive time variable. The coefficients of this equation are power functions of the space variable and depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the heat equation. Areas of application include biology, superradiant emission processes, heat propagation in solids (with special applications in the area of heat shield and ablation material design), and certain chemical reaction-diffusion processes. It is noteworthy that, for particular values of the parameters, the equation allows an evolution theoretic derivation of the fundamental distribution laws of Wien, Maxwell, Poisson, and Gauss. The general initial condition solution will be derived from a fundamental solution and will be given in terms of an integral transform for locally summable functions (singular integral). It is also shown that, for admissible parameter values, there always exist nontrivial solutions which approach zero as the time variable goes to zero and that, for particular parameter ranges, there exist singular solutions, conservative solutions, and delta function initial condition solutions.

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Zusammenfassung Es werden Lösungen für gegebene Anfangsbedingungen für die verallgemeinerte (autonome, parabolische) Fellersche Gleichung in einer positiven Raumvariablen und einer positiven Zeitvariablen aufgestellt. Die Koeffizienten dieser Gleichung sind Potenzen der Raumvariablen und hängen von vier Parametern ab. Die Gleichung ist im allgemeinen singulär am Ursprung und im Unendlichen. Sie umfasst als Spezialfälle die spezielle Fellersche Gleichung, die Kepinskische Gleichung und die Wärmegleichung. Anwendung findet sie in der Biologie, in superstrahlenden Emissionsprozessen, in der Theorie der Wärmeausbreitung in Festkörpern (besonders beim Entwurf von Hitzeschilden und Abschmelzmaterialien) und im Gebiet gewisser chemischer Reaktions-Diffusionsprozesse. Est ist bemerkenswert, dass die Gleichung für besondere Parameterwerte eine evolutionstheoretische Herleitung der grundlegenden Verteilungsgesetze von Wien, Maxwell, Poisson und Gauss ermöglicht. Die allgemeine Lösung für gegebene Anfangsbedingung wird aus einer Grundlösung entwickelt und in der Form einer Integraltransformation gegeben für lokal summierbare Funktionen (singuläres Integral). Es wird weiterhin gezeigt, dass für zulässige Parameterwerte stets nichtriviale Lösungen existeren, die nach Null streben, wenn die Zeitvariable nach Null geht, und dass es für besondere Parameterbereiche singuläre und konservative Lösungen gibt und solche, die einer Deltafunktion als Anfangsbedingung entsprechen.

     

Smoothness of bounded solutions of nonlinear evolution equations

It is shown that in many cases globally defined, bounded solutions of evolution equations are as smooth (in time) as the corresponding operator, even if a general solution of the initial-value problem is much less smooth; i.e., initial values for bounded solutions are selected in such a way that optimal smoothness is attained. In particular, solutions which bifurcate from certain steady states, such as periodic orbits, almost-periodic orbits and also homo- and heteroclinic orbits, have this property. As examples, a neutral functional differential equation, a slightly damped non-linear wave equation, and a heat equation are considered. In the latter case the space variable is included into the discussion of smoothness. Finally, generalized Hopf bifurcation in infinite dimensions is considered. Here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized.     

On a class of probability distributions

The application of a three parameter class of one-sided probability distributions is being discussed. For specific parameter values, this class contains as special cases a number of well-known distributions of statistics and statistical physics, namely, Gauss, Weibull, exponential, Rayleigh, Gamma, chi-square, Maxwell, and Wien (limiting case of Planck's distribution). One of the three parameters represents scale; the other two represent initial and terminal shape of the associated probability density function. A fourth parameter, shift, may be introduced. The distribution class discussed in this paper was introduced by L. Amoroso [2] in 1924. It is closely connected with a family of linear Fokker-Planck equations (generalized Feller equation). In fact, the class of probability density functions associated with the distribution class considered here is a special case of the set of all delta function initial condition solutions of the generalized Feller equation for a fixed value of the time variable. It will be shown that, as a function of the logarithm of the independent variable, the logarithm of the cumulative distribution function is asymptotically linear as the independent variable approaches zero from above. This fact leads to a general criterion for the applicability of the presented distribution family relative to given empirical data. The applicability criterion can be used to determine approximate values for the two shape parameters. They can subsequently be used as initial values in any of the established parameter estimation techniques.     

Existence of solutions and stability of a class of parabolic systems perturbed by generalized white noise on the boundary ∗

This paper is concerned with a class of stochastic boundary value problems and their stability questions. The system, we consider, is governed by a parabolic partial differential equation perturbed by generalized white noise on the boundary. Existence of weak solutions and their regularity properties are established. It is also shown that the solution of the autonomous system generates a Feller process in a Hilbert space, in case the spatial operator is time invariant. The questions of Lyapunov type stability of this class of systems are also examined. The system is shown to be almost surely globally asymptotically stable with respect to a ball centered at the origin. Further, it is shown that there exists a measure, supported on the attractor, which is invariant with respect to the adjoint Feller semigroup. An explicit expression for the generator of the semigroup is also given     

Smoothing properties of nonlinear stochastic equations in Hilbert spaces

A nonlinear stochastic equation in a Hilbert space is considered, with constant but possibly degenerate diffusion term. Some smoothing properties for the associated transition semigroup are studied. In particular, strong Feller property and irreducibility are proved. The main tools are Malliavin calculus and Girsanov transformation.Partially supported by Italian MURST 60% research funds.     

Ergodicity of stochastic 2D Navier–Stokes equation with Lévy noise

In this paper we deal with the 2D Navier-Stokes equation perturbed by a Lévy noise force whose white noise part is non-degenerate and that the intensity measure of Poisson measure is σ-finite. Existence and uniqueness of invariant measure for this equation is obtained, two main properties of the Markov semigroup associated with this equation are proved. In other words, strong Feller property and irreducibility hold in the same space.     

Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly (Ann. Math. 164:993–1032, 2006), we prove the uniqueness of invariant measures for the corresponding transition semigroup.     

Scale and time effects on mathematical models for transport in the environment

The purpose of this paper is to analyse mathematical models used in environmental modelling.Following a brief survey of the development in modelling scale-and time-dependent dispersion processes in the environment,this paper compares three similarity solutions,one of which is a solution of the generalized Feller equation(GF)with fractal parameters,and the other two for the newly-developed generalized Fokker-Planck equation(GFP).The three solutions are derived with parameters having physical significance.Data from field experiments are used to verify the solutions.The analyses indicate that the solutions of both GF and GFP represent the physically meaningful natural processes,and simulate the realistic shapes of tracer breakthrough curves.     

Markov selections for the 3D stochastic Navier–Stokes equations

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier–Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier–Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.      

Long Time Stability in Perturbations of Completely Resonant PDE's

In this paper we will present some results concerning long time stability in nonlinear perturbations of resonant linear PDE's with discrete spectrum. In particular we will prove that if the perturbation satisfies a suitable nondegeneracy condition then there exists a periodic like trajectory, i.e. a closed curve in the phase space with the property that solutions starting close to it remain close to it for very long times. Secondly, in the special case where the average of the main part of the perturbation is integrable we will prove that if the energy is initially essentially concentrated on finitely many modes, then along the corresponding solutions all the actions are approximatively constant for very long times. Applications to nonlinear wave and Schrödinger equations on a segment will also be given.     

Numerical treatment of the spherically symmetric solutions of a generalized Fisher-Kolmogorov-Petrovsky-Piscounov equation

In the present work, the connection of the generalized Fisher-KPP equation to physical and biological fields is noted. Radially symmetric solutions to the generalized Fisher-KPP equation are considered, and analytical results for the positivity and asymptotic stability of solutions to the corresponding time-independent elliptic differential equation are quoted. An energy analysis of the generalized theory is carried out with further physical applications in mind, and a numerical method that consistently approximates the energy of the system and its rate of change is presented. The method is thoroughly tested against analytical and numerical results on the classical Fisher-KPP equation, the Heaviside equation, and the generalized Fisher-KPP equation with logistic nonlinearity and Heaviside initial profile, obtaining as a result that our method is highly stable and accurate, even in the presence of discontinuities. As an application, we establish numerically that, under the presence of suitable initial conditions, there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value. An analytical prediction is provided for the Heaviside equation, against which we verify the validity of our computational code, and numerical approximations are provided for several generalized Fisher-KPP problems.     

Irreducibility and strong Feller property for non-linear SPDEs

Under some non-degeneracy condition, the strong Feller property and irreducibility are studied for non-linear stochastic partial differential equations driven by multiplicative noise within the framework called ‘variational approach’. Our result for irreducibility can be applied to equations with locally monotone coefficients. In some special cases, we discuss the Hölder continuity of the associated Markov semigroups. The main results are applied to several examples such as stochastic Burgers equation, stochastic porous media equation and stochastic fast diffusion equation.     

MOMENT GENERATING FUNCTIONS OF RANDOM VARIABLES AND ASYMPTOTIC BEHAVIOUR FOR GENERALIZED FELLER OPERATORS

1. Introduction and NotationsThe generalized Feller operators which include many famous operators, such ajsBernstein, Szasz-Mirakjan, BaskakoV, Meyer--K5nig and Zeller operators, can be constructed by making use of the probabilistic method. In the paper [1][2], Xu JihuaPr(--lvjdetl a general scheme f(,r its construction, and Zhao Jillghui showed that theFeller type operators are of good approximations f'or unbounded functions.Our purpose is to present representation of moment generating …     

Exact solitary wave solutions for a generalized KdV–Burgers equation

Using the special truncated expansion method, the solitary wave solutions are constructed for the compound Korteweg–de Vries–Burgers (KdVB) equation. Exact and explicit solitary wave solutions for a generalized KdVB equation are obtained by introducing a suitable ansatz equation. The generalized two-dimensional KdVB equation is discussed. Some particular cases of the generalized KdVB equation are solved by using these methods.     

Construction of global‐in‐time solutions to Kolmogorov‐Feller pseudodifferential equations with a small parameter using characteristics

Using an idea going back to Madelung, we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov‐Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta‐shock solution of the continuity equation for a discontinuous velocity field. We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics).     

A new symmetry approach to solve space–time‐dependent KdV systems

In this paper, a new method to solve space–time‐dependent non‐linear equations is proposed. After considering the variable coefficient of a non‐linear equation as a new dependent variable, some special types of space–time‐dependent equations can be solved from corresponding space–time‐independent equations by using the general classical Lie approach. The rich soliton solutions of space–time‐dependent KdV equation and mKdV equation are given with the help of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Wang's Harnack inequalities for space–time white noises driven SPDEs with two reflecting walls and their applications

In this paper, we establish Wang's Harnack inequalities for Gaussian space–time white noises driven the stochastic partial differential equation with double reflecting walls, which is of the infinite dimensional Skorokhod equation. We first establish both the Harnack inequality with power and the log-Harnack inequality for the special case of additive noises by the coupling approach. Then we investigate the log-Harnack inequality for the Markov semigroup associated with the reflected SPDE driven by multiplicative noises using the penalization method and the comparison principle for SPDEs. As their applications, we study the strong Feller property, uniqueness of invariant measures, the entropy-cost inequality, and some other important properties of the transition density.     

On non-standard finite difference models of reaction–diffusion equations

Reaction–diffusion equations arise in many fields of science and engineering. Often, their solutions enjoy a number of physical properties. We design, in a systematic way, new non-standard finite difference schemes, which replicate three of these properties. The first property is the stability/instability of the fixed points of the associated space independent equation. This property is preserved by non-standard one- and two-stage theta methods, presented in the general setting of stiff or non-stiff systems of differential equations. Schemes, which preserve the principle of conservation of energy for the corresponding stationary equation (second property) are constructed by non-local approximation of nonlinear reactions. Assembling of theta-methods in the time variable with energy-preserving schemes in the space variable yields non-standard schemes which, under suitable functional relation between step sizes, display the boundedness and positivity of the solution (third property). A spectral method in the space variable coupled with a suitable non-standard scheme in the time variable is also presented. Numerical experiments are provided.     

m阶多时滞中立型微分方程的2T周期解

研究m阶常系数线性多时滞中立型方程的周期解，讨论了2T周期解的存在性和唯一性，得到了2T周期解存在唯一若干新的充分必要条件。所得主要结果适用性更广。它包括了许多相关献的结果为其特例，推广、改进了这些献的主要结果。可对所研究的中立型方程周期解存在的大量情形作出判断，而这些情形用其他献的结果是无法判断的。换言之，对方程（1），主要结果是最一般化的，用同样的方法已不可能得到更好的结果。     

构造辅助函数计算准Vandermonde行列式

通过构造线性方程组和一元高次方程,利用线性方程组的解与一元高次方程根与系数的关系推导出第一类准Vandermonde行列式的值.通过构造辅助函数计算一个特殊的第一类准Vandermonde行列式,并把这种方法推广于两类特殊第二类准Vandermonde行列式的计算.