System Of Two-dimensional Integral Equations Research Articles

The impact of Legendre wavelet collocation method on the solutions of nonlinear system of two-dimensional integral equations

Numerical solutions of nonlinear system of two-dimensional integral equations have been rarely investigated in the literature. In this study, we suggest a numerically practical algorithm to approximate the solutions of nonlinear system of two-dimensional Volterra–Fredholm and Volterra integral equations. This scheme is based on two-dimensional Legendre wavelet to reduce these nonlinear systems of integral equations to a system of nonlinear algebraic equations. The main characteristic of this approach is high accuracy and computational efficiency of performing which are the consequences of Legendre wavelet properties. The main benefit of this basic function is their ability to detect singularities and their efficiency in dealing with non-sufficiently smooth function in comparison with Legendre polynomials and they minimize the error. The convergence analysis and error bound of the proposed Legendre wavelet method is investigated. Numerical examples confirm that the Legendre wavelet collocation method is accurate, reliable for solving nonlinear system of two-dimensional integral equations.

Depolarization of the electromagnetic field in a locally inhomogeneous earth-ionosphere waveguide

This paper presents a further development of the numerical-analytical method for the solution of three-dimensional problems in the theory of radio wave propagation. We consider a vector problem of the electromagnetic field of a vertical electric dipole in a plane Earth-ionosphere waveguide with a local large-scale irregularity on the anisotropic ionosphere wall. The possibility of lowering (elevating) of the local region of the upper waveguide wall with respect to the regular ionosphere level is taken into account. The field components on the boundary surfaces obey the Leontovich impedance conditions. The problem is reduced to a system of two-dimensional integral equations taking into account the overexcitation and depolarization of the field scattered by the irregularity. Using asymptotic (with respect to the parameter kr ≫1) integration along the direction perpendicular to the ray path, we transform this system to a system of one-dimensional integral equations. The system is solved numerically in the diagonal approximation, combining direct inversion of the Volterra integral operator and the subsequent iterations. The proposed method reduces the computer time required for solving the problem and is useful for the study of both small-scale and large-scale irregularities. We obtained estimates of the TE field components that are not excited by the source considered and originate entirely from field scattering by a three-dimensional irregularity disturbing the geometric regularity of the ionospheric waveguide wall.

Depolarization of the electromagnetic field scattered by a three-dimensional large-scale irregularity of the lower ionosphere

This paper develops analytical and numerical methods for the solution of three-dimensional problems of radio wave propagation. We consider a three-dimensional vector problem for the electromagnetic field of a vertical electric dipole in a planar Earth-ionosphere waveguide with a largescale local irregularity of negative characteristics at the anisotropic ionospheric boundary. The field components at the boundary surfaces obey the Leontovich boundary conditions. The problem is reduced to a system of two-dimensional integral equations taking into account the overexcitation and depolarization of the field scattered by the irregularity. Using asymptotic (with respect to the parameter kr≫1, where r is the distance from the source or receiver to the nearest point of the irregularity, k=2π/λ, and λ is the radio wavelength) integration over the direction perpendicular to the ray path, we transform this system to one-dimensional integral equations where integration contours represent the geometric contour of the irregularity. The system is numerically solved in the diagonal approximation, combining direct inversion of the Volterra integral operator and subsequent iterations. The proposed numerical algorithm reduces the computer time required for the solution of this problem and is applicable for studying both small-scale and large-scale irregularities. We obtained novel estimates for the field components that are not excited by the source but result entirely from scattering by the sample three-dimensional ionospheric irregularity.

On the use of jump conditions at a thin inclusion in solving problems of elasticity theory for a half-space with protuberances

We propose a method of approximate solution of problems of elasticity theory for a half-space with protuberances based on the use of jump conditions in the stresses and displacements at a thin elastic element. The problem of determining the stresses reduces to a system of two-dimensional integral equations of Newtonian potential type for determining the contact stresses between the protuberances and the half-space. We consider the case when the elastic characteristics of the material of the protuberances are different from the material of the half-space.

The method of potentials in the three-dimensional electromagnetic-wave diffraction problem

The problem of the diffraction of a stationary electromagnetic wave travelling in an infinite horizontally stratified medium in a three-dimensional inclusion is considered. The method of potentials is used to obtain a family of systems of two-dimensional integral equations over the boundary of the inclusion which are suitable for numerical solution. The correct solvability of the equations and their equivalence to the initial problem are studied.