On the Construction and Classification of the Common Invariant Solutions for Some P(1,4) -Invariant Partial Differential Equations

We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.

Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transform

The numerical algorithms for finding the lines of branching and branching-off solutions of nonlinear problem on mean-square approximation of a real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on two parameters, are constructed and justified.

Quasi-Static Problem of Thermoelasticity for Thermosensitive Infinite Circular Cylinder of Complex Heat Exchange

Nonlinear nonstationary heat conduction problem for infinite circular cylinder under a complex heat transfer taking into account the temperature dependence of thermophysical characteristics of materials is solved numerically by the method of lines. Directing it to the Cauchy’s problem for systems of ordinary differential equations studied feature which takes place on the cylinder axis. Taken into account the dependence on the temperature coefficient of heat transfer that the different interpretation of its physical content makes it possible to consider both convective and convective-ray or heat ray. Using the perturbation method, the corresponding thermoelasticity problem taking into account the temperature dependence of mechanical properties of the material is construed. The influence of the temperature dependence of the material on the distribution of temperature field and thermoelastic state of infinite circular cylinder made of titanium alloy Ti-6Al-4V by radiant heat transfer through the outer surface has been analyzed.

Asymptotic Solution to Scattering Problem on a Set of Small Particles and Application to Creating the Materials with a Peculiar Refractive Index

The asymptotic solution to the scattering problem on a set of small particles, supplemented into homogeneous material, is used for modeling the materials with the desired refractive index. The consideration concerns the case of acoustic scalar scattering and the solution to initial scattering problem is built using an asymptotic approach. The closed form solution is reduced for the scattering problem. This is significant advantage of approach because there is no need to solve the respective system of boundary integral equations. High accuracy of solving the scattering problem is achieved by choosing the optimal parameters of the domain with small particles. The approach allows obtaining an explicit formula for the refractive index of the resulting inhomogeneous material. The numerical calculations show the possibility to get the specific values of refractive index including its negative values.

On Symmetry Reduction of the (1 + 3)-Dimensional Inhomogeneous Monge-Ampère Equation to the First-Order ODEs

We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous Monge-Ampère equation to first-order ODEs. Some classes of the invariant solutions are constructed.

Modeling of Plane Arrays Using a Variational Approach

The variational statement of synthesis problem is generalized in order to account the additional requirements to the synthesized radiation pattern (RP) and field distribution in the specified points of near zone. For this aim, the minimizing functional is supplemented by term providing the possibility to minimize the values of field in these points;creating the deep zeros in the RP for the certain angular coordinates is realized too. The approach foresees reduction of an explicit formula for field values in a near zone. The results of computational modeling testify the possibility to create zeros in the given RP and to minimize the values of field in a near zone of plane arrays in a great extent.

The Radiation System Synthesis by the Power Criterion as a Problem of Optimization with Restrictions

A nonlinear synthesis problem of antennas according to the prescribed power (squared amplitude) radiation pattern (RP) is considered in the variational statement that yields in the possibility to take into account an additional restriction to the synthesized power RP. The problem of synthesis consists of finding such currents in antenna, which generates the RP with the best approximation to the given one. The respective Euler’s equation is reduced on the basis of used functional. This is nonlinear integral equation of Hammerstein’s type. The effective numerical methods are elaborated and applied for its solving. The computational results verify the effectiveness of approach proposed.

Modeling the Physical Parameters of Solenoids for MEMS Applications

The paper is devoted to study of the electrical parameters of the motion parts of the MEMS such as solenoids. The analytical background is given in order to describe the influence of the electrical field components on the forces, which are result of interaction of the electromagnetic (EM) field components with the parts of motion devices of MEMS. The given analytical formulas open the ability to calculate the self-inductance of the microsolenoids of the different kind, as well as the stored energy of such motion devices, that could be used for the modeling and optimization of parameters of running devices of MEMS such as actuators, sensors etc.