Mykhaylo Andriychuk

By M. I. Andriychuk and A. G. Ramm

Part of the book: Numerical Simulations of Physical and Engineering Processes

By Mykhaylo Andriychuk

Part of the book: Numerical Simulation

By Mykhaylo Andriychuk

In the process of solving the problems of generalized phase optimization the necessity to apply an eigenvalue approach often appears. The practical statement of the optimization problems consists of using the amplitude characteristics of functions that are sought. The usual way of optimization is deriving the Euler equation of the functional, which is used as criterion of optimization. As a rule, such equation is an integral one. It is worth pointing out that the integral equations of the generalized phase optimization are nonlinear ones. The characteristic property of such equations is non-uniqueness of solutions and their branching or bifurcation. The determination of branching solutions leads to the investigation of the corresponding homogeneous equations and the respective eigenvalue problem. This problem is nonlinear because of specificity of the statement of the optimization problem. The study of the above problem allows us to determine a set of points, in which the respective eigenvalues are equal to unity that determines the branching points of solutions. The data of calculations testify to the ability of the approach proposed to determine the solutions of nonlinear equations numerically with not large computations.

Part of the book: Matrix Theory

By Mykhaylo Andriychuk

Part of the book: Chaos Theory

By Mykhaylo Andriychuk and Borys Yevstyhneiev

Solution to the problem of electromagnetic (EM) wave scattering on a set of small size impedance particles of arbitrary shape with the chaotic rule of their distribution is sought for by the asymptotic approach. The particles are distributed in a homogeneous volume with the constant material parameters. Solution to the problem is derived under the condition that the characteristic size of particles tends to zero; besides, the quantity of particles approaches to infinity at a specific principle. The solving procedure is reduced to derivation of an explicit form of solution that avoids the need to solve the governing integral equation, which is used to determine the fields in the particle’s surfaces. This allows to keep out of integration of the derivatives of Green function, which are presented in a kernel of the derived integral equation. The practical importance of approach consists of creating the media or materials with the close to desired inhomogeneous value of the effective refractive index or magnetic permeability. The explicit analytical relations are reduced for the above physical parameters, and they are verified by computations. It is substantiated that the chaotic distribution of particles in the initial medium makes possible to obtain more contrast material parameters comparing with the regular distribution of particles.

Part of the book: Chaos Theory

By Mykhaylo Andriychuk

The restriction on the optimized function in the problem of the antenna synthesis leads to the characteristic form of the nonlinear integral equations, which are equivalent to the respective functionals of synthesis problem. The kind of the reduced equation depends on the operator, which is used for the calculation of antenna’s directivity pattern (DP). The two types of restrictions are considered, the first one is that the amplitude of the objective function is prescribed. The second one is that the restrictions on the phase of the objective function are imposed. The problem is actual because it deals with the analytical investigation of integral Hammerstein equations of a new type, on the one hand, and gives a possibility to get new types of solutions to above equations that can be applied at the modelling of antennas of the different appointment, on the other hand. The peculiarity of such equations is the non-uniqueness of their solutions. The accuracy of analytical approach is supported by the numerical data. The method of successive approximations is much effective for solving the reduced equations. The advantage of the approach proposed is a fast convergence of iterative procedures, which are applied for solving the obtained equations.

Part of the book: Optimization Algorithms - Classics and Last Advances [Working title]