Diffusion-Wave Type Solutions to the Second-Order Evolutionary Equation with Power Nonlinearities

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous medium equation and its generalizations. We construct solutions that describe perturbations propagating over a zero background with a finite velocity. Such effects are known to be atypical for parabolic equations and appear as a consequence of the degeneration of the equation at the points where the desired function vanishes. Previously, we have constructed it, but here the case of power nonlinearity is considered. It allows for conducting a more detailed analysis. We prove a new theorem for the existence of solutions of this type in the class of piecewise analytical functions, which generalizes and specifies the earlier statements. We find and study exact solutions having the diffusion wave type, the construction of which is reduced to the second-order Cauchy problem for an ordinary differential equation (ODE) that inherits singularities from the original formulation. Statements that ensure the existence of global continuously differentiable solutions are proved for the Cauchy problems. The properties of the constructed solutions are studied by the methods of the qualitative theory of differential equations. Phase portraits are obtained, and quantitative estimates are determined by constructing and analyzing finite difference schemes. The most significant result is that we have shown that all the special cases for incomplete equations take place for the complete equation, and other configurations of diffusion waves do not arise.

Motion of liquid Droplets in gas channels in a SO2 Module

Understanding of liquid droplets dynamics in gas channels is critical for improvement of performance and durability of the catalysts made of a dense porous material. This report describes a mathematical model for studying how different surface properties and operating conditions affect the dynamics of liquid droplets. We present multiple numerical simulations of a single droplet dynamics for different sizes of droplets and different choices of contact angles. We also study influence of an air flow to a thin liquid film and obtained travelling wave type solutions.

Tensor Product and Certain Solutions of Fractional Wave Type Equation

In this paper we Önd certain solutions of some fractional partial di§erential equations. Tensor product of Banach spaces is used to Önd some solutions where separation of variables does not work. We solve the fractional wave type equation using fractional Fourier Series

Analytic rogue wave-type solutions for the generalized (2+1)-dimensional Boussinesq Equation

Based on the Hirota bilinear form of the generalized (2+1)-dimensional Boussinesq equation, which can be expressed as the shallow water wave mechanism appearing in fluid mechanics, we applied the new polynomial functions to construct the rational solutions and rogue wave-type solutions. Next, the system parameters control on the rational solutions and rogue wave-type solutions were also shown. As a result, we found the following basic facts: (i) these parameters may affect the wave shapes, amplitude, and bright/dark for this considered equation, (ii) the solitary wave interaction rogue waves and triplet rogue wave-type solutions can be viewed on [Formula: see text], [Formula: see text], and [Formula: see text] planes, respectively. Their nonlinear dynamic behaviors were presented by numerical simulation of the 2D- and 3D-plots.

Применение технологии Plane Wave Imaging в ультразвуковом неразрушающем контроле

Image recovery of reflectors by digital antenna focusing (DFA), along with such advantages as high resolution over the entire image recovery area of reflectors, the ability to obtain images taking into account the reflection and transformation of the wave type from the boundaries of the object of control, has several disadvantages: a large volume of measured echo signals, a long image recovery time and insufficient energy of ultrasonic waves introduced into the object of control. The Plane Wave Imaging (PWI) method allows you to combine the advantages of phased array antenna technology (PHAR) and CFA technology. In the PWI mode, when a plane wave is emitted, all elements of the antenna array (AP) work, as in the FAR mode, which allows you to increase the energy introduced into the control object, and echo signals are recorded by all elements of the AP, as in the CFA mode. The image of the reflectors is restored by the raman SAFT method. To obtain an image, you can use the number of radiated plane waves less than the number of elements of the antenna array, which reduces the volume of measured echo signals. Translation of calculations to the area of spatial sectors allows you to increase the speed of recovery of the presentation of reflectors. Model experiments have shown the positive and negative aspects of obtaining images of reflectors by the PWI method in comparison with the CFA method, both for the case of using a prism and without a prism.