The quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation

We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schrödinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the “master equation”) is derived from the boundary value problem for the Schrödinger equation.     

Metrical characterization of conformal capacity zero

The introduction of a parameter in the fixed point equation yields a family of operators with identical fixed points. A method is presented for minimization of the Lipschitz constant as a function of this parameter. This yields an operator with an optimal error estimate. The choice of a suitable weighting function is essential. As an example, a nonlinear boundary value problem is treated. A one-sided Lipschitz condition is sufficient.     

A problem of random choice and its deterministic structure

A new problem of random choice for pills consumption process is formulated and considered. We find a kind of the Law of Large Numbers associated with any ordinary differential equation. This generalized LLN says that a stochastic analog of the Euler broken lines converges in probability to solution of the initial value problem for the ODE. This approach is applied to a stochastic process of pills consumption, and shows that after a suitable scaling the consumption process is almost deterministic, provided that the initial number of pills is large.     

On the Inverse Scattering of the Time-Dependent Schrödinger Equation and the Associated Kadomtsev-Petviashvili (I) Equation

The Kadomtsev-Petviashvili equation, a two-spatial-dimensional analogue of the Korteweg-deVries equation, arises in physical situations in two different forms depending on a certain sign appearing in the evolution equation. Here we investigate one of the two cases. The initial-value problem, associated with initial data decaying sufficiently rapidly at infinity, is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation of a nonlocal Riemann-Hilbert problem in terms of scattering data expressible in closed form in terms of given initial data. The lump solutions, algebraically decaying solitons, are given a definite spectral characterization. Pure lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t. Many of the above results are also relevant to the problem of inverse scattering for the so-called time-dependent Schrödinger equation.     

A weak solvability of a system of thermoviscoelasticity for the Jeffreys model

We establish a weak solvability of the initial-boundary value problem for a dynamic model of thermoviscoelasticity. The problem under consideration is an extension of the Jeffreys model obtained with the help of a consequence of the energy balance equation. We study the corresponding initial-boundary value problem by splitting the problem and reducing it to an operator equation in a suitable Banach space.     

To the theory of boundary-value problems for elliptic equations with superposition operators in the boundary condition. I

We study the existence, uniqueness, and constant sign property of classical solutions to a nonlocal boundary-value problem for a second-order elliptic equation in a bounded domain of the Euclidean space. Using the system of maps that define superposition operators, we construct some subset of the domain boundary and establish the connection between the solvability of the problem under consideration and the solvability of the boundary value equation on the constructed subset.     

A Herglotz wavefunction method for solving the inverse Cauchy problem connected with the Helmholtz equation

This paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method.     

Global classical solutions to ut − Δ(u2m + 1) + λu = ƒ

The global existence of a classical solution of the initial-boundary value problem or the initial value problem for certain degenerating parabolic equations is established by constructing approximate solutions by the standard Galerkin procedure and applying some differential and integral inequalities when the initial value is smooth enough, has small norm in a suitable sense, and may change sign.     

数值天气模式参数选择的全局优化方法

１引言数值天气预报模式中关于参数的选择直接影响到天气预报的准确率,在建立一个数值天气预报系统时,为了得到好的预报效果,必须对模式参数进行优化．在这方面已有许多文献［１］－［７］作过有益的探讨,提供了许多有效的方法,在文献［２］中,给出了一种参数反演的方法．并应用广义线性反演,获得较稳定的计算格式．然而,此方法在每一次迭代时,至少需要解ｎ＋１个正问题（其中ｎ为参数的个数）．又在文献［６］中。引进了四维同化的共轭梯度法,适宜于求解高维问题．然而,共轭梯度法只能求得局部最优解,对初始参数的选取很敏感,…     

Instability of the stationary solutions of generalized dissipative Boussinesq equation

In this work we study the generalized Boussinesq equation with a dissipation term. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive sufficient conditions for the blow-up of the solution to the problem. Furthermore, the instability of the stationary solutions of this equation is established.     

Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter

In this work we consider the first boundary value problem for a parabolic equation of second order with a small parameter on a half-axis (i.e., we consider the one-dimensional case). We take the zero initial condition. We construct the global (that is, the caustic points are taken into account) asymptotics of a solution for the boundary value problem. The asymptotic solution of this problem has a different structure depending on the sign of the coefficient (the drift coefficient) at the derivative of first order at a boundary point. The constructed asymptotic solutions are justified.     

一类具有摄动边界的弱非线性反应扩散方程

研究了具有摄动边界的非线性反应扩散方程的问题.在适当的条件下,利用微分不等式理论,讨论了问题的渐近解.     

Asymptotic behavior of solutions of the heat equation in domains with curvilinear boundaries

The first boundary value problem with null boundary conditions is studied for the one-dimensional heat equation in a domain with curvilinear lateral boundaries. It is proved that for domains sufficiently close to a half-strip, solutions with a sign change for any time value, for unbounded time increase, tend to zero faster than positive solutions.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 57–65, 1989.     

A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem

In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. We propose a quasi-boundary value regularization method combined with an a posteriori regularization parameter choice rule to deal with the backward problem and give the corresponding convergence estimate.     

A variational-type method of fundamental solutions for a Cauchy problem of Laplace’s equation

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace’s equation in a simply-connected bounded domain. Based on a global conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a regularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.     

Greenʼs function for first-order multipoint boundary value problems and applications to the existence of solutions with constant sign

We consider a first-order linear differential equation subject to boundary value conditions which take into account the values of the function at multiple points in the interval of interest. For this problem, we calculate the Green?s function which allows to express in integral form the exact expression of the unique solution to the multipoint boundary value problem under the appropriate conditions. From this study, some results are derived concerning the existence of solutions with a constant sign (that is, some comparison results for first-order multipoint boundary value problems).     

On the Cauchy problem for a generalized Boussinesq equation

In this paper, we consider the Cauchy problem for a generalized Boussinesq equation. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

On a terminal value problem for a generalization of the fractional diffusion equation with hyper-Bessel operator

In this paper, we consider an inverse problem of recovering the initial value for a generalization of time-fractional diffusion equation, where the time derivative is replaced by a regularized hyper-Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.