The cauchy problem for the helmholtz equation in a domain with a piecewise-smooth boundary

The Cauchy problem for the Helmholtz equation is investigated in the case when a piecewise-smooth boundary of a domain is determined in a parametric form. The condition at infinity is taken in the form of the absence of in-coming waves from infinity. The formula that expresses a Fourier image solution through its boundary values is obtained by using the Fourier transformation method in a class of slowly-growing generalized functions. The conditions that the boundary values of the functions have to satisfy are derived. It is proven that these conditions are the necessary and sufficient solvability conditions for the original problem. © 2006 IEEE

On optimal frequencies for reconstruction of a one-dimensional profile of gradient layer's refractive index

© 2014 Dmitrii Tumakov.The problem of reconstruction of a one-dimensional profile of gradient layer's refractive index is investigated. An algorithm for choosing a right frequency, at which a scattered field ismeasured, is proposed. It is concluded that at the correct choice of frequency onemeasurementmust be sufficient. Moreover, in this case, regularization parameters of the residual functional are chosen as zero. It is shown that in case of measurements being carried out with errors, residual terms must be added to the functional

On optimal frequencies for reconstruction of a one-dimensional profile of gradient layer's refractive index

© 2014 Dmitrii Tumakov. The problem of reconstruction of a one-dimensional profile of gradient layer's refractive index is investigated. An algorithm for choosing a right frequency, at which a scattered field ismeasured, is proposed. It is concluded that at the correct choice of frequency onemeasurementmust be sufficient. Moreover, in this case, regularization parameters of the residual functional are chosen as zero. It is shown that in case of measurements being carried out with errors, residual terms must be added to the functional

An over-determined boundary problem for the Helmholtz equation in a semiinfinite domain with a curvilinear boundary

In this paper we consider an over-determined Cauchy problem for the Helmholtz equation in a semiinfinite domain with a piecewise smooth curvilinear boundary. Applying the Fourier transform method in the space of distributions of slow growth, we establish the necessary and sufficient solvability conditions which connect the boundary functions. We construct integral representations of a solution. © 2010 Allerton Press, Inc

The classes of solving of the helmholz equation in halfplane

The estimation for a trace is received functions which is the solution of the Helmholz equation in halfplane to Sobolev space Hs. It is shown, that solutions of the Helmholz equation in space HS + at s≥1/2 does not exist. And the solution, appropriate to a flat wave, does not belong to Sobolev space HS + at any index s

Relativistic mask method for electron momentum distributions after ionization of hydrogen-like ions in strong laser fields

Wavefunction-splitting or mask method, widely used in the non-relativistic calculations of the photoelectron angular distributions, is extended to the relativistic domain within the dipole approximation. Since the closed-form expressions for the relativistic Volkov states are not available within the dipole approximation, we build such states numerically solving a single second-order differential equation. We calculate the photoelectron energy spectra and angular distributions for highly charged ions under different ionization regimes with both the direct and the relativistic mask methods. We show that the relativistic mask method works very well and reproduces the electron energy and angular distributions calculated by the direct method in the energy range where both methods can be used. On the other hand, the relativistic mask method can be applied for longer laser pulses and/or higher photoelectron energies where the direct method may have difficulties

Iterative method for solving the problem of scattering of an Electromagnetic wave by a partially shielded conducting sphere

© 2014 Dmitrii Tumakov. The problem of diffraction of an electromagnetic wave of parallel polarization by an perfectly conducting sphere sitting on top of a conducting surface is considered. The wave is supposed to fall onto the surface at a right angle. The original problem is solved by an iterative method involving consecutive solutions of the problems of diffraction by the sphere and by the conducting screen. The criteria for terminating the iterative process is a small amount of energy reflected off the sphere

Analysis of electromagnetic wave propagation through a layer with graded-index distribution of refraction index

The problem of plane electromagnetic harmonic wave diffraction on a graded-refractive-index layer of some thickness is considered. It is assumed that refractive index of a layer monotonically increases and then monotonically decreases. Cases of the linear, parabolic, sinusoidal, exponential and logarithmic refractive index profiles of the layer are investigated. The diffraction problem is reduced to an ordinary differential equation with appropriate boundary conditions. The problem for the linear profile is solved analytically; for the other profiles it is investigated numerically. The method of approximating an integral identity is applied to increase accuracy of the grid solution of the boundary value problem. Emphasis is given to the cases, in which wave energy, either reflected or transited, reaches maxima

Forced oscillations of the elastic strip with a longitudinal crack

© 2015 Kristina Stekhina and Dmitrii Tumakov. The problem of forced oscillations of an elastic strip containing a longitudinal crack of a finite length is considered. The diffraction problem is reduced to the system of paired integral equations. The system of integral equations is reduced to the system of linear algebraic equations by using the Galerkin method. Singularities of integrand functions, through which coeffcients of the system matrix are calculated, are determined