A Generalized Analytic Operator-Valued Function Space Integral and a Related Integral Equation

We introduce a generalized Wiener measure associated with a Gaussian Markov process and define a generalized analytic operator-valued function space integral as a bounded linear operator from L _p into L _p^\prime (1<p ≤ 2) by the analytic continuation of the generalized Wiener integral. We prove the existence of the integral for certain functionals which involve some Borel measures. Also we show that the generalized analytic operator-valued function space integral satisfies an integral equation related to the generalized Schr?dinger equation. The resulting theorems extend the theory of operator-valued function space integrals substantially and previous theorems about these integrals are generalized by our results.     

A Generalized Analytic Operator-Valued Function Space Integral and a Related Integral Equation

We introduce a generalized Wiener measure associated with a Gaussian Markov process and define a generalized analytic operator-valued function space integral as a bounded linear operator from L _p into L _p^\prime (1<p ≤ 2) by the analytic continuation of the generalized Wiener integral. We prove the existence of the integral for certain functionals which involve some Borel measures. Also we show that the generalized analytic operator-valued function space integral satisfies an integral equation related to the generalized Schr?dinger equation. The resulting theorems extend the theory of operator-valued function space integrals substantially and previous theorems about these integrals are generalized by our results.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

Generalized Dirichlet‐to‐Neumann Map in Time‐Dependent Domains

We study the heat, linear Schrödinger (LS), and linear KdV equations in the domain l(t) < x < ∞ , 0 < t < T , with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.     

Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation

We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u _t in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables.     

A Mixed Problem with Integral Condition for the Hyperbolic Equation

In this paper, we study a mixed problem for the hyperbolic equation with a boundary Neumann condition and a nonlocal integral condition. We justify the assertion that there exists a unique generalized solution of the problem under consideration. The proof of uniqueness is based on an estimate, derived a priori, in the function space introduced in the paper, while the existence of a generalized solution is proved by the Galerkin method.     

Inverse problem of finding the coefficient of a lower derivative in a parabolic equation on the plane

We study the existence and uniqueness of the solution of the inverse problem of finding an unknown coefficient b(x) multiplying the lower derivative in the nondivergence parabolic equation on the plane. The integral of the solution with respect to time with some given weight function is given as additional information. The coefficients of the equation depend on the time variable as well as the space variable.     

Linearization Ill-Posedness for 2.5-D Ware Equation Inversion Model

Abstract For the weakly inhomogeneous acoustic medium in Ω={(x,y,z):z>0},we consider the inverse problemof determining the density function ρ(x,y).The inversion input for our inverse problem is the wave field givenon a line.We get an integral equation for the 2-D density perturbation from the linearization.By virtue of theintegral transform,we prove the uniqueness and the instability of the solution to the integral equation.Thedegree of ill-posedness for this problem is also given.     

The exterior dirichlet problem for the Laplace equation

Using a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex-valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper.     

Branching of solutions for the problem of mean-square approximation of a real finite function of two variables by the modulus of double Fourier transformation

We investigate the branching of solutions of a Hammerstein-type nonlinear two-dimensional integral equation that arises in the problems of mean-square approximation of a real finite nonnegative function of two variables by the modulus of double Fourier integral, depending on two parameters [Mat. Metody Fiz.-Mekh. Polya, 51, No. 1, 53–64; No. 4, 80–85 (2008)]. We have derived analytical expressions for eigenfunctions of the corresponding linear homogeneous integral equation, necessary for the construction of branched-off solutions, and obtained systems of transcendental equations for finding the points of their branching. We also present, in the first approximation, the analytical representations of complex solutions branched-off from the real solution for the two-dimensional case of branching.     

Integral equations of a two-dimensional lightguide

An integral equation describing oscillations of a two-dimensional periodic lightguide is studied. The integral operator is an analytic function of a spectral parameter. The homogenous integral equation has only the zero solution for all values of the spectral parameter expect for some isolated values. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 78–84.     

Existence of solution in elastic wave scattering by unbounded rough surfaces

We consider the two‐dimensional problem of the scattering of a time‐harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function ?∈C^1,1(?). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all L^p spaces, p∈[1, ∞] and hence existence of solution to the elastic wave scattering problem. Copyright © 2002 John Wiley & Sons, Ltd.     

A free boundary problem for three-dimensional harmonic vector fields

We study in this paper a free boundary value problem ( FB ), where a region G_o in R ³ is determined by the condition that there exists a vector field v_o in G_o which satisfies div v_o = e_o, curl v_o = g_o in G_o and v_o = E on the boundary ?G_o with a given scalar function e_o and given vector fields g_o and E. We give two equivalent formulations for this problem. Then we characterize the solutions by a non-linear integral equation. In order to solve the latter by a Newton method we linearize this equation. We investigate the ensuing linear integral equation. In case of axisymmetric configurations this is a singular integral equation whose index can be easily determined from the given data. We obtain a related equation, if we try to construct a field v in a region G which is on the boundary perpendicular to a given field B . Finally we use this method to investigate an astrophysical problem, which arises in the theory of pulsar magnetospheres.     

On the best mean-square approximation of a real nonnegative finite continuous function of two variables by the modulus of a double Fourier integral. II

We continue the investigation of the nonlinear problem of mean-square approximation of a real finite nonnegative continuous function of two variables by the modulus of a double Fourier integral depending on two parameters begun in the first part of this work [J. Math. Sci., 160, No. 3, 343–356 (2009)]. Finding the solutions of this problem is reduced to the solution of a nonlinear two-dimensional integral equation of the Hammerstein type. We construct and justify numerical algorithms for determination of branching lines and branched solutions of this equation. Numerical examples are presented.     

The double layer potential operator over polyhedral domains i: solvability in weighted sobolev spaces

We consider the integral equation, (λ-K)u=f, where K is the double layer (harmonic) potential operator on the boundary of a bounded polyhedron in R³ and λ∣λ∣≥1 is a complex constant. We study the mapping properties of λ - K in weighted Sobolev spaces, applying Mellin transformation techniques directly to the integral equation.     

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

We prove that the standard second‐kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign‐definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so‐called combined potential, or combined field, operator.) This coercivity result yields k‐explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical‐asymptotic methods developed recently). Coercivity also gives k‐explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise‐polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation. © 2015 Wiley Periodicals, Inc.     

Existence of Solutions for Impulsive Parabolic Partial Differential Equations

We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions.     

The Feynman-Kac Formula with a Lebesgue-Stieltjes Measure and Feynman's Operational Calculus

We investigate what happens if in the Feynman-Kac functional, we perform the time integration with respect to a Borel measure η rather than ordinary Lebesgue measure l. Let u(t) be the operator associated with this functional through path integration. We show that u(t), considered as a function of time t, satisfies a certain Volterra-Stieltjes integral equation. This result establishes a “FeynmanKac formula with Lebesgue-Stieltjes measure η.” One recovers the classical Feynman-Kac formula by letting η = l. We deduce from the integral equation that u(t) satisfies a differential equation associated with the continuous part μ of η when η = μ = l, this differential equation reduces to the heat or the Schrödinger equation in the probabilistic or quantum-mechanical case, respectively. Moreover, we observe a new phenomenon, due to the discrete part v of η: the function u(t) undergoes a discontinuity at every point in the support of v, assumed here to be finite. Further, one obtains an explicit expression for u(t) in terms of operators alternatively associated with μ and v. Our results are new even in the probabilistic or “imaginary time” case and allow us to unify various concepts. The derivation of our integral equation has an interesting combinatorial structure and makes essential use of the “generalized Dyson series”— recently introduced by G. W. Johnson and the author—that “disentangle” the operator u(t). We provide natural physical interpretations of our results in both the diffusion and quantum-mechanical cases. We also suggest further connections with Feynman?s operational calculus for noncommuting operators.     

On existence of integrable solutions of a functional integral equation under Carathéodory conditions

We study the solvability of a functional integral equation in the space of Lebesgue integrable functions on an unbounded interval. Using the conjunction of the technique of measures of weak noncompactness with the classical Schauder fixed point principle we show that the equation in question is solvable in the mentioned function space. Our existence result is obtained under the assumption that functions involved in the investigated functional integral equation satisfy Carathéodory conditions. Moreover, that result generalizes several ones obtained earlier in many research papers and monographs.