On quantum stochastic differential equations

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.     

On one class of differential operators and their application

We use a generalized differentiation operator to construct a generalized shift operator, which makes it possible to define a generalized convolution operator in the space H(?). Next, we consider the characteristic function of this operator and introduce a generalized Laplace transform. We study the homogeneous equation of the generalized convolution operator, investigate its solvability, and consider the multipoint Vallée Poussin problem.     

On persistence of superoscillations for the Schrödinger equation with time-dependent quadratic Hamiltonians

In this work, we prove the persistence in time of superoscillations for the Schrödinger equation with time-dependent coefficients. In order to prove the persistence of superoscillations, we have conditioned the coefficients to satisfy a Riccati system, and we have expressed the solution as a convolution operator in terms of solutions of this Riccati system. Further, we have solved explicitly the Cauchy initial value problem with three different kinds of superoscillatory initial data. The operator is defined on a space of entire functions. Particular examples include Caldirola-Kanai and degenerate parametric harmonic oscillator Hamiltonians, and other examples could include Hamiltonians not self-adjoint. For these examples, we have illustrated numerically the convergence on real and imaginary parts.     

On unique solvability of one nonlinear nonlocal with respect to the solution gradient nonstationary problem

We consider a parabolic equation whose space operator is a product of a nonlinear bounded function which depends on a nonlocal characteristic with respect to a solution gradient and a strongly monotone potential operator. We prove the existence and uniqueness of the solution in the class of the vector-valued functions with values in the Sobolev space.     

Operator differential-algebraic equations with noise arising in fluid dynamics

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.     

Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.     

Time discretization of an evolution equation via Laplace transforms

Following earlier work by Sheen, Sloan, and Thomée concerningparabolic equations we study the discretization in time of aVolterra type integro-differential equation in which the integraloperator is a convolution of a weakly singular function andan elliptic differential operator in space. The time discretizationis accomplished by using a modified Laplace transform in timeto represent the solution as an integral along a smooth curveextending into the left half of the complex plane, which isthen evaluated by quadrature. This reduces the problem to afinite set of elliptic equations with complex coefficients,which may be solved in parallel. Stability and error boundsof high order are derived for two different choices of the quadraturerule. The method is combined with finite-element discretizationin the spatial variables.     

Justification of the direct projection method for solving an integral equation of the potential theory

We justify the direct projection method for solving an integral equation with a logarithmic singularity in the kernel. The equation is treated as a mapping of one Hilbert space into another Hilbert space. The spaces are chosen from conditions ensuring the solution of a broad class of mathematical modeling problems with the use of a simple layer potential. The idea of the projection method is to choose finite-dimensional subspaces into which the exact solution and the right-hand side of the equation are projected. In this case, the problem of finding an approximate solution does not require computing the convolution of kernels. We prove an estimate for the solution error in the norm of the original operator equation.     

The inverse problem for the characteristic functions of several commuting operators

In this paper we derive conditions for an operator valued function to be the characteristic function of several commuting operators in a Hilbert space. We use the connection of this problem to some problems in partial differential equation to get a solution for a class of operator valued functions.     

Multipoint Vallée Poussin problem for convolution operators with nodes defined inside an angle

We prove the solvability of the multipoint Vallée Poussin (interpolation) problem for the kernel of a convolution operator in the case where the zeros of the characteristic function and nodal points (zeros of an entire function) are inside an angle.     

On the compactness of convolution-type operators in Morrey spaces

In a Morrey space, the product of the convolution operator with summable kernel and the operator of multiplication by an essentially bounded function is considered. Sufficient conditions for such a product to be compact are obtained. In addition, it is shown that the commutator of the convolution operator and the operator of multiplication by a function of weakly oscillating type is compact in a Morrey space.     

Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem.     

Application of the dual operator method to an equation describing the behavior of a boundary function

The dual operator is an analogue of the conjugate operator in linear theory. In this study the dual operator is applied to a second-order differential equation describing the behavior of the zero-order boundary function in the boundary function method used to derive the asymptotic solution of the singularly perturbed integro-differential plasma-sheath equation. This approach produces is a three-point difference scheme. The results of a numerical solution of the Cauchy problem are reported. __________ Translated from Prikladnaya Matematika i Informatika, No. 26, pp. 49–60, 2007.     

An exponentially convergent algorithm for nonlinear differential equations in Banach spaces

An exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in Banach space is proposed. The algorithm is based on an equivalent Volterra integral equation including the operator exponential generated by the operator coefficient. The operator exponential is represented by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient, and then the integrals involved are approximated using the Chebyshev interpolation and an appropriate Sinc quadrature. Numerical examples are given which confirm theoretical results.

     

Stability Conditions for Abstract Functional Differential Equations in Hilbert Space

Abstract. Stability conditions for functional differential equations of the form: du (t)/ dt = Au(t)+ bAu(t-h)+(a^\ast Au)(t) are studied, where A is the infinitesimal generator of an analytic semigroup in a Hilbert space, b\neq 0 and the convolution term contains a square integrable real function a\neq 0 . Norm discontinuity of the solution semigroup of the equation with discrete delay is avoided by studying the inverse of the characteristic operator. Sufficient and necessary conditions for the uniform exponential stability of the solution semigroup are obtained. The results are applied to a retarded partial integrodifferential equation.     

A Multisensor Deconvolution Problem

Meisters and Peterson gave an equivalent condition under which the multisensor deconvolution problem has a solution when there are two convolvers, each the characteristic function of an interval. In this article we find additional conditions under which the deconvolution problem for multiple characteristic functions is solvable. We extend the result to the space of Gevrey distributions and prove that every linear operator S, fromthe space of Gevrey functions with compact support onto itself, which commutes with translations can be represented as convolution with a unique Gevrey distribution T of compact support. Finally, we find explicit formula for deconvolvers when the convolvers satisfy weaker conditions than the equivalence conditions using nonperiodic sampling method.     

A priori estimates for solution operators of diffusion equations

The diffusion equation [d]=Au is considered, where u=u(t,x), t>0, and A is a second order uniformly elliptic differential operator in R^m Whose coefficients are bounded. Other conditions are prescribed on A to generate known soiution operators. We derive growth estimates for these solution operators in certain function spaces together with estimates for their derivatives in t and also estimates on the products of the first two spatial derivatives with these solution operators. Bounds on the solution operators are given which depmd only upon the i.u.b.'s for the ternination coefficients of A and the formal adjoint A_ * of A : These estimates are best with respect to each function space considered in the sense that equality holds for a particular solution operator     

Method of investigating boundary-value problems in distribution spaces and boundary integral equations

A theorem for representation of the solution of a nonhomogeneous linear differential equation with constant coefficients in D in the form of a convolution of the right side and the fundamental function is generalized to the case of a linear nonhomogeneous differential equation with infinitely differentiable coefficients. Based on this theorem, a method is proposed for investigating both direct and inverse boundary-value problems in a distribution space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 632–639, May, 1991.     

Integro-Differential Equations over a Closed Circuit with Gaussian Function in the Kernel

Integro-differential equations with kernels including hypergeometric Gaussian function that depends on the arguments ratio are studied over a closed curve in the complex plane. Special cases of the equations considered are the special integro-differential equation with Cauchy kernel, equations with power and logarithmic kernels. By means of the curvilinear convolution operator with the kernel of special kind, the equations with derivatives are reduced to the equations without derivatives. We find out the connection between special cases of the above-mentioned convolution operator and the known integral representations of piecewise analytical functions applied in the study of boundary value problems of the Riemann type. The correct statement of Noetherian property for the investigated class of equations is given. In this case, the operators corresponding to the equations are considered acting from the space of summable functions into the space of fractional integrals of the curvilinear convolution type. Examples of integro-differential equations solvable in a closed form are given.