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P. A. Krutitskii - One of the best experts on this subject based on the ideXlab platform.
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the 2d dirichlet problem for the propagative helmholtz Equation in an exterior domain with cracks and singularities at the edges
International Journal of Mathematics and Mathematical Sciences, 2012Co-Authors: P. A. KrutitskiiAbstract:The Dirichlet problem for the 2D Helmholtz Equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm Equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.
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Acoustic scattering by a thin cylindrical screen with the Dirichlet boundary condition and the impedance boundary condition on opposite sides of the screen
2008 Proceedings of the International Conference Days on Diffraction, 2008Co-Authors: Valentina V. Kolybasova, P. A. KrutitskiiAbstract:A problem on scattering acoustic waves by a thin cylindrical screen is studied. In doing so, the Dirichlet condition is specified on one side of the screen, while the impedance boundary condition is specified on the other side of the screen. The solution of the problem is subject to the radiating condition at infinity and to the propagative Helmholtz Equation. By using the potential theory the scattering problem is reduced to a system of singular integral Equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm Equation of the second kind and index zero. It is proved that the obtained vector Fredholm Equation is uniquely solvable. Therefore the integral representation for a solution of the original scattering problem is obtained.
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the impedance problem for the propagative helmholtz Equation in interior multiply connected domain
Computers & Mathematics With Applications, 2003Co-Authors: P. A. KrutitskiiAbstract:Abstract The impedance problem for the propagative Helmholtz Equation in the interior multiply connected domain is studied in two and three dimensions by a special modification of a boundary integral Equation method. Additional boundaries are introduced inside interior parts of the boundary of the domain. The solution of the problem is obtained in the form of a single layer potential on the whole boundary. The density in the potential satisfies the uniquely solvable Fredholm Equation of the second kind and can be computed by standard codes. In fact, our method holds for any positive wave numbers.
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the neumann problem in a 2 d exterior domain with cuts and singularities at the tips
Journal of Differential Equations, 2001Co-Authors: P. A. KrutitskiiAbstract:Abstract The Neumann problem for the harmonic functions in an exterior connected plane region with cuts is studied. The problem is considered with different conditions at infinity, which lead to different theorems on uniqueness and solvability. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm Equation of the second kind, which is uniquely solvable. Explicit formulas for singularities of a gradient of the solution at the tips of the cuts are obtained. The results of the paper can be used to model the flow of an ideal fluid over several obstacles, including wings.
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wave scattering in a 2 d exterior domain with cuts the neumann problem
Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik, 2000Co-Authors: P. A. KrutitskiiAbstract:The Neumann problem in the exterior of several obstacles and double sided screens (cracks) is studied for a propagative Helmholtz Equation in 2 dimensions. Previously the unique solvability of this problem has been obtained if obstacles are non-resonance domains [9]. In the present paper the solvability of this problem is proved in the general case by the method of interior boundaries. In doing so additional boundaries are introduced inside obstacles. The Neumann problem is reduced to integral Equations containing Cauchy kernels on screens and next to a uniquely solvable Fredholm Equation of the second kind on the whole boundary.
M A Lyalinov - One of the best experts on this subject based on the ideXlab platform.
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diffraction of a skew incident plane electromagnetic wave by a wedge with axially anisotropic impedance faces
Radio Science, 2007Co-Authors: M A LyalinovAbstract:[1] This paper presents, as an extension of the authors' recent work, an exact solution to diffraction of a skew incident plane electromagnetic wave by a wedge with axially anisotropic impedance faces. Applying the Sommerfeld-Malyuzhinets technique to the boundary-value problem yields a coupled system of difference Equations for the spectra; on elimination, a functional difference (FD) Equation of higher order for one spectrum arises; after simplification in terms of a generalized Malyuzhinets function and accounting for the Meixner's edge condition as well as the poles and residues of the spectrum in the basic strip of the complex plane, the FD Equation is converted, via the so-called S integrals, to an integral equivalent; for points on the imaginary axis which belong to the basic strip the integral equivalent becomes a Fredholm Equation of the second kind with a nonsingular, wave number–free and exponentially decreasing kernel; solving this integral Equation by the quadrature method the spectrum can be determined by integral extrapolation and by analytical continuation; a first-order uniform asymptotic solution follows from evaluating the Sommerfeld integrals with the saddle-point method. Comparison with available exact solutions in several special cases shows that this approach leads to a fast and accurate solution of the problem under study.
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diffraction of a normally incident plane wave by an impedance wedge with its exterior bisected by a semi infinite impedance sheet
IEEE Transactions on Antennas and Propagation, 2004Co-Authors: Ning Yan Zhu, M A LyalinovAbstract:This paper reports an application of a previously proposed procedure to diffraction of a normally incident, arbitrarily polarized plane electromagnetic wave by a canonical structure which consists of a wedge with different face impedances and a semi-infinite impedance sheet bisecting the exterior of the wedge. The use of the Sommerfeld-Malyuzhinets technique converts the original boundary value problem into a system of linear Equations for two coupled spectral functions. Eliminating one of them, we get a second-order difference Equation for the other spectral function. From this function and the boundary condition on the upper wedge face we construct an even and in the basic strip regular new spectral function. Then we transform the second-order Equation into a simpler one by means of a generalized Malyuzhinets function /spl chi//sub /spl Phi//(/spl alpha/), and express the solution to the latter in an integral form with help of the so-called S-integrals. Solving a Fredholm Equation of the second kind for points on the imaginary axis of the complex plane, which follows from the integral representation, enables one to compute the sought-for function. The second spectral function is obtained via its dependence upon the first one. We present a first-order uniform asymptotic solution, as well as numerical results.
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a solution procedure for second order difference Equations and its application to electromagnetic wave diffraction in a wedge shaped region
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2003Co-Authors: M A LyalinovAbstract:This paper proposes an efficient solution procedure for second–order functional difference Equations, and outlines this procedure through investigating electromagnetic–wave diffraction by a canonical structure comprising an impedance wedge and an impedance sheet bisecting the exterior region of the wedge. Applying the Sommerfeld–Malyuzhinets technique to the original boundary–value problem yields a linear system of Equations for the two coupled spectral functions. Eliminating one spectral function leads to a second-order difference Equation for the other. The chief steps in this work consist of transforming the second–order Equation into a simpler one by making use of a generalized Malyuzhinets function χϕ(α), and in expressing the solution to the latter in an integral form with help of the so–called S–integrals. From this integral expression one immediately obtains a Fredholm Equation of the second kind for points on the imaginary axis of the complex plane. Solving this integral Equation by means of the well–known quadrature method enables us to calculate the sought–for spectral function inside the basic strip via an interpolation formula and outside it via an analytic extension. The second spectral function is obtained through its dependence upon the first. The uniform asymptotic solution, which is of particular interest in the geometrical theory of diffraction, follows, by evaluating the Sommerfeld integrals in the far field from the exact one. Several examples demonstrate the efficiency and accuracy of the proposed procedure as well as typical behaviour of the far–field solutions for such a canonical problem of diffraction theory.
Nikolai Tarkhanov - One of the best experts on this subject based on the ideXlab platform.
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An Open Mapping Theorem for the Navier-Stokes Equations
arXiv: Analysis of PDEs, 2019Co-Authors: Alexander Shlapunov, Nikolai TarkhanovAbstract:We consider the Navier-Stokes Equations in the layer ${\mathbb R}^n \times [0,T]$ over $\mathbb{R}^n$ with finite $T > 0$. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes Equations to a nonlinear Fredholm Equation of the form $(I+K) u = f$, where $K$ is a compact continuous operator in anisotropic normed Holder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes Equations for all $t \in [0,T]$. On using the particular properties of the de Rham complex we conclude that the Frechet derivative $(I+K)'$ is continuously invertible at each point of the Banach space under consideration and the map $I+K$ is open and injective in the space. In this way the Navier-Stokes Equations prove to induce an open one-to-one mapping in the scale of Holder spaces.
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An open mapping theorem for the Navier-Stokes Equations
Advances and Applications in Fluid Mechanics, 2018Co-Authors: Alexander Shlapunov, Nikolai TarkhanovAbstract:We consider the Navier-Stokes Equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes Equations to a nonlinear Fredholm Equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Holder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes Equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Frechet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes Equations prove to induce an open one-to-one mapping in the scale of Holder spaces.
Kentaro Iwasaki - One of the best experts on this subject based on the ideXlab platform.
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a fast volume integral Equation method for the direct inverse problem in elastic wave scattering phenomena
International Journal of Solids and Structures, 2009Co-Authors: Terumi Touhei, Taku Kiuchi, Kentaro IwasakiAbstract:A fast method for solving the volume integral Equation is introduced for the solution of forward and inverse multiple scattering problems in an elastic 3-D full space. For both forward and inverse scattering analysis, the volume integral Equation in the wavenumber domain is used. By means of the discrete Fourier transform, the volume integral Equation in the wavenumber domain can be dealt with as a Fredholm Equation of the 2nd kind with respect to a non-Hermitian operator on a finite dimensional vector space. The Bi-CGSTAB method is employed to construct the Krylov subspace in the wavenumber domain. The current procedure establishes a fast and simplified method without requiring the derivation of a coefficient matrix. Several numerical results validate the accuracy and effectiveness of the current method for both forward and inverse scattering analysis. According to the numerical results, the reconstruction of inhomogeneities of the wave field is successful, even for multiple scattering of several cubes.
Malham, Simon J. A. - One of the best experts on this subject based on the ideXlab platform.
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Integrability of local and nonlocal non-commutative fourth order quintic nonlinear Schrodinger Equations
2021Co-Authors: Malham, Simon J. A.Abstract:We prove integrability of a generalised non-commutative fourth order quintic nonlinear Schrodinger Equation. The proof is relatively succinct and rooted in the linearisation method pioneered by Ch. Poppe. It is based on solving the corresponding linearised partial differential system to generate an evolutionary Hankel operator for the `scattering data'. The time-evolutionary solution to the non-commutative nonlinear partial differential system is then generated by solving a linear Fredholm Equation which corresponds to the Marchenko Equation. The integrability of reverse space-time and reverse time nonlocal versions, in the sense of Ablowitz and Musslimani, of the fourth order quintic nonlinear Schrodinger Equation are proved contiguously by the approach adopted. Further, we implement a numerical integration scheme based on the analytical approach above which involves solving the linearised partial differential system followed by numerically solving the linear Fredholm Equation to generate the solution at any given time.Comment: 27 pages, 1 Figure, More background information, examples and citations added to the previous version. Clarified the text in a few places and removed abbreviations for consistenc
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Integrability of local and nonlocal non-commutative fourth order quintic NLS Equations
2020Co-Authors: Malham, Simon J. A.Abstract:We prove integrability of a generalised non-commutative fourth order quintic NLS Equation. The proof is relatively succinct and rooted in the linearisation method pioneered by Ch. Poppe. It is based on solving the corresponding linearised partial differential system to generate an evolutionary Hankel operator for the `scattering data'. The time-evolutionary solution to the non-commutative nonlinear partial differential system is then generated by solving a linear Fredholm Equation which corresponds to the Marchenko Equation. The integrability of reverse space-time and reverse time nonlocal versions, in the sense of Ablowitz and Musslimani, of the fourth order quintic NLS Equation are proved contiguously by the approach adopted. Further, we implement a numerical integration scheme based on the analytical approach above which involves solving the linearised partial differential system followed by numerically solving the linear Fredholm Equation to generate the solution at any given time.Comment: 25 pages, 1 Figure, More background information, numerical simulations and a figure added to the original versio
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Grassmannian flows and applications to non-commutative non-local and local integrable systems
'Elsevier BV', 2020Co-Authors: Doikou Anastasia, Malham, Simon J. A., Stylianidis IoannisAbstract:We present a method for linearising classes of matrix-valued nonlinear partial differential Equations with local and nonlocal nonlinearities. Indeed we generalise a linearisation procedure originally developed by P\"oppe based on solving the corresponding underlying linear partial differential Equation to generate an evolutionary Hankel operator for the `scattering data', and then solving a linear Fredholm Equation akin to the Marchenko Equation to generate the evolutionary solution to the nonlinear partial differential system. Our generalisation involves inflating the underlying linear partial differential system for the scattering data to incorporate corresponding adjoint, reverse time or reverse space-time data, and it also allows for Hankel operators with matrix-valued kernels. With this approach we show how to linearise the matrix nonlinear Schr\"odinger and modified Korteweg de Vries Equations as well as nonlocal reverse time and/or reverse space-time versions of these systems. Further, we formulate a unified linearisation procedure that incorporates all these systems as special cases. Further still, we demonstrate all such systems are example Fredholm Grassmannian flows.Comment: 31 page
