Integral Transformation Method Research Articles

Dynamic Axisymmetric Problem of Direct Piezoelectric Effect for A Bimorphic Plate of Stepwise Variable Thickness

The dynamic axisymmetric problem of direct piezoelectric effect for a circular two-layer transducer consisting of a metal substrate and a piezoceramic plate of smaller diameter is considered. The oscillations of the electroelastic system are carried out due to the action of a mechanical load on its front surface, which is an arbitrary function of time and radial coordinate. The presented mathematical model of calculation allows to take into account the connectedness of longitudinal-transverse oscillations of a structure that is asymmetric in height. A new analytical solution was obtained in the framework of the refined theory of Timoshenko by the method of finite integral transformations. The calculated ratios make it possible to choose the geometric dimensions of the bimorphic plate, which most effectively allow to convert the mechanical load into the generated electrical pulse. The considered designs are widely used as touch switches that regulate the operation of valves for various purposes.

Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness

This paper deals with the construction of numerical solutions of random hyperbolic models with a finite degree of randomness that make manageable the computation of its expectation and variance. The approach is based on the combination of the random Fourier transforms, the random Gaussian quadratures and the Monte Carlo method. The recovery of the solution of the original random partial differential problem throughout the inverse integral transform allows its numerical approximation using Gaussian quadratures involving the evaluation of the solution of the random ordinary differential problem at certain concrete values, which are approximated using Monte Carlo method. Numerical experiments illustrating the numerical convergence of the method are included.

Development and Investigation of Mathematical Models of Thermoelastic Stresses and Strains in Production of Multilayer Structures of Spherical Shape for Aircraft

The adequate mathematical models are developed that allow investigating the thermoelastic stresses and strains arising in production of multilayer structures of spherical shape for aircraft. The method of finite integral transformations was applied to solve the model tasks stated.

Integral Transform Method to Solve the Problem of Porous Slider without Velocity Slip

This study is about the lubrication of a long porous slider in which the fluid is injected into the porous bottom. The similarity transformation reduces the Navier-Stokes equations to couple nonlinear, ordinary differential equations, which are solved by a new algorithm. The proposed technique is based on integral transformation. Apparently, there is great symmetry between proposed method and variation iteration method, Adomian decomposition method but in integral transform method all the boundary conditions are applied, then a recursive scheme is used for the analytical solutions, which is unlike the Variational Iteration Method, Adomian Decomposition Method, and other existing analytical methods. Solutions are obtained for much larger Reynolds numbers, and they are compared with analytical and numerical methods. Effects of Reynolds number on velocity components are presented.

Analytical solution for three-dimensional radial heat transfer in a cold-region tunnel

The study of the temperature field of cold-region tunnels is fundamental for preventing frost heave. At present, considerable knowledge has been obtained by numerical simulation and physical experimentation. In this paper a computational model for three-dimensional radial heat transfer was established according to some reasonable assumptions, and the complex boundary conditions inside the tunnel surface including axial coordinate z and time t were obtained based on the analysis of the field temperature monitoring data. The z and t terms in the boundary were split to simplify the problem. Laplace and Fourier integral transformation methods were combined to solve this problem, and the Den Iseger method was adopted to solve the Laplace inverse transformation. Yuximolegai tunnel was introduced to study its temperature distribution. The freezing length in z direction is reduced from 143 m to 70 m and the maximum radial freezing depth at the entrance is also shortened from 4.7 m to 2.4 m due to 5 cm insulation layer, which reveals that the effect of 5 cm insulation layer was remarkable. Considering the efficiency of insulation layer, it needs to be combined with other measures to prevent frost damage of surrounding rock. The analysis of convective heat transfer coefficient, formation temperature, and thickness of the insulation layer was performed to obtain their influence on the lining temperature. The results show that the influence of the convection heat transfer coefficient is small. The effect of the insulation layer improves with increasing formation temperature and worsens as the thickness of the insulation layer increases. These achievements can enrich the research of temperature fields of tunnels in cold regions and are significant for guiding tunnel design in cold regions.

Improved Solutions to the Linearized Boussinesq Equation with Temporally Varied Rainfall Recharge for a Sloping Aquifer

Sloping unconfined aquifers are commonly seen and well investigated in the literature. In this study, we propose a generalized integral transformation method to solve the linearized Boussinesq equation that governs the groundwater level in a sloping unconfined aquifer with an impermeable bottom. The groundwater level responses of this unconfined aquifer under temporally uniform recharge or nonuniform recharge events are discussed. After comparing with a numerical solution to the nonlinear Boussinesq equation, the proposed solution appears better than that proposed in a previous study. Besides, we found that the proposed solutions reached the convergence criterion much faster than the Laplace transform solution did. Moreover, the application of the proposed solution to temporally changing rainfall recharge is also proposed to improve on the previous quasi-steady state treatment of an unsteady recharge rate.

Developing high-performance software for studying radon transport processes

A mathematical model of the three-dimensional problem of diffusion-advection of radon in piecewise homogeneous layered mediums with inclusions is proposed. The model takes into account anisotropy of diffusion properties of subregions of geological mediums. A combined method of solving the problem with the methods of integral transformations, integral representations, and boundary integral equations is proposed. Results of computer modeling with known models for the case of homogeneous piecewise layered mediums are presented. Results of computational experiments for studying processes of radon transport in anisotropic layered mediums with inclusions are presented. Additionally, the use of OpenMaple technology for developing high-performance software with language C# for studying radon transport processes is discussed.

Ramp Type Heating in a Semiconductor Medium under Photothermal Theory

The present investigation is concerned with study the two-dimensional deformation in a semi-infinite semiconducting medium subjected to ramp type heating. The deformation in the medium is caused subjected to a thermal source of ramp-type nature applied along the free surface of a semiconducting medium. The semiconductor thermoelastic medium is under the influence of an internal heat source of constant magnitude. Integral Transform method has been used to obtain the transformed expression of displacement components, stress components, temperature distribution, and carrier density. The transformed expressions are then inverted using numerical inversion technique. The effect of ramp-type source and thermoelectric coupling parameter on the components of displacement, force stress, temperature distribution, and carrier density has been shown graphically.

Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems.

Embedding of active substances in supramolecular systems has as the main goal to ensure the controlled release of the active ingredients. Whatever the final architecture or entrapment mechanism, modeling of release is challenging due to the moving boundary conditions and complex initial conditions. Despite huge diversity of formulations, diffusion phenomena are involved in practically all release processes. The approach in this paper starts, therefore, from mathematical methods for solving the diffusion equation in initial and boundary conditions, which are further connected with phenomenological conditions, simplified and idealized in order to lead to problems which can be analytically solved. Consequently, the release models are classified starting from the geometry of diffusion domain, initial conditions, and conditions on frontiers. Taking into account that practically all solutions of the models use the separation of variables method and integral transformation method, two specific applications of these methods are included. This paper suggests that “good modeling practice” of release kinetics consists essentially of identifying the most appropriate mathematical conditions corresponding to implied physicochemical phenomena. However, in most of the cases, models can be written but analytical solutions for these models cannot be obtained. Consequently, empiric models remain the first choice, and they receive an important place in the review.

Diagnostics of VVER-1000 Coolant Temperature Pulsations

The temperature pulsations arising in coolant flow at the entry into FA are examined from the standpoint of the possibility of their diagnostics by thermocouples of the protective-tube block are examined. It is shown that the temperature pulsations remain in the FA and can be recorded at the exit from FA by means of the in-reactor monitoring system. The aim of this work is to call the attention of specialists working in the field of temperature diagnostics of nuclear power facilities to the application of the elements of the theory of generalized functions, specifically, the fundamental solutions of heat-conduction operators and the methods of integral transformations of generalized functions, which are powerful means for solving applied problems of the temperature diagnostics of NPF in order to increase their operational safety.

About uniqueness of solutions of Fredholm linear integral equations of the first kind in the axis

In this work, we apply the method of integral transformation to prove uniqueness theorems for the new class of Fredholm linear integral equations of the first kind in the axis.

Analysis of the thermoelastic state of the inhomogeneous through the thickness isotropic cylindrical shell

The stress-deformed state of a functionally graded isotropic circular closed cylindrical shell subjected to local heating of the outer surface of the environment by means of convective heat transfer is investigated. For the study, the mathematical model of the shear theory of inhomogeneous shells of the Tymoshenko type is used. The two-dimensional heat equation is deduced under the linear distribution of temperature over the thickness. By methods of Fourier and Laplace integral transformations the solution of the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a finite simply supported circle cylindrical shell is obtained. Numerical results are presented for metal–ceramic composites. Cite as: U. V. Zhydyk, “Analysis of the thermoelastic state of the inhomogeneous through the thickness isotropic cylindrical shell,” Prykl. Probl. Mekh. Mat. , Issue 16, 119–125 (2018) (in Ukrainian), http://doi.org/10.15407/apmm2018.16.119-125

Physical and mathematical modelling of the mass transfer process in heterogeneous systems under corrosion destruction of reinforced concrete structures

The article presents the equations of "free calcium hydroxide" mass transfer describing the liquid corrosion of cement concretes. The boundary value problem of "free calcium hydroxide" mass conductivity in dimensionless variables is obtained. Also, in order to obtain a solution to the boundary value problem of mass conductivity, it is proposed to use a combined approach, which consists in the fact that at the beginning of each microprocess an analytical solution is obtained, and then the whole process is gradually calculated on a computer. It should be noted that the method gives good results if it is carried out in conjunction with the Laplace integral transformation method. In addition, the Laplace transform often makes it possible to obtain a solution in two forms: at Fom ˃0,1 and Fom ˂˂ 0,1, this is the advantage of using this transformation. The solution of the boundary value problem of mass conductivity is obtained, which allows to find not only the profiles of dimensionless and dimensional concentrations of “free calcium hydroxide” in concrete thickness, but also makes it possible to determine the value of this value gradient at the phases interface. The obtained solution allows further determining the time of reaching the critical concentration of “free calcium hydroxide” at the phases interface, corresponding to the beginning of high-basic components decomposition, and hence moving on to the further stages of study and corrosion processes modeling, which is confirmed by the graphical dependencies presented in the paper.

Density Forecast Evaluations via a Simulation-Based Dynamic Probability Integral Transformation*

Abstract This paper presents simulation-based density forecast evaluation methods using particle filters. The simulation-based dynamic probability integral transformation or log-likelihood evaluation method is combined with the existing density forecast evaluation methods. This methodology is applicable to various density forecast models, such as log stochastic volatility models and affine jump diffusion (AJD) models, for which the probability integral transform or likelihood computation is difficult due to the presence of latent stochastic volatilities. This methodology is applied to the daily S&P 500 stock index. The empirical results show that the AJD models with jumps perform the best for out-of-sample evaluations.

The Use of Integral Transform in Calculating Structured Interaction of Layered Soil Caused by Dynamic Loads

This study applies the Integral Transform Method (ITM) to solve the problems caused by dynamic loads that work on structures and its interactions with the soil. The structure and part of the soil from the half space layer are modeled with Finite Elements (FEM) where at the boundary must be coupled with the dynamic stiffness matrix from the half space calculating used Integral Transform. To obtain the dynamic stiffness matrix of the half space, the wave equation in the elastic half space which is a coupled partial differential equations must be uncoupled first with Helmholtz’s potential then it is used integral transform to find the wave equation in the transform domain. To get a response to the original domain, reverse transformation is performed using Inverse Integral Transform. This method is known as ITM - FEM Coupling.

State variable mapping method for studying initial-dependent dynamics in memristive hyper-jerk system with line equilibrium

Memristive hyper-jerk system has been reported to exhibit the initial-dependent multistability phenomenon, whose dynamical mechanism and control strategy deserve further in-depth investigations. However, a more effective method is required for the existence of line equilibrium. Inspired by the dimensionality decreasing flux-charge analysis method, we handle these problems via state variable mapping (SVM) method of integral transformation. With this method, the hyper-jerk system characterized by four autonomous differential equations can be reconstructed and a dimensionality decreasing model is obtained. In the newly formulated system, the initial conditions of the original system are mapped as standalone system parameters. Thus, the initial-dependent multi-stability phenomenon is transformed into the easily controlled parameter-relied dynamics, which facilitates physical direction of multi-stable system's operation mode in hardware circuit. Specially, the line equilibrium point exhibited in the voltage-current model is mapped as two determined equilibria. Thus, the dynamics of coexisting infinitely many attractors and their forming mechanisms are readily investigated in the newly constructed system. Moreover, a hardware circuit is designed and fabricated, whose operation mode is rigorously controlled by applying additional DC signals which are proportional to the specified initial conditions. Through circuit simulations and experimental measurements, it is demonstrated that the SVM method offers a direction for interpretation and control of multistability in memristive systems without any determined equilibrium.

The discontinuous solutions of Lame’s equations for a conical defect

In this article the discontinuous solutions of Lame’s equations are constructed for the case of a conical defect. Under a defect one considers a part of a surface (mathematical cut on the surface) when passing through which function and its normal derivative have discontinuities of continuity of the first kind. A discontinuous solution of a certain differential equation in the partial derivatives is a solution that satisfies this equation throughout the region of determining an unknown function, with the exception of the defect points. To construct such a solution the method of integral transformations is used with a generalized scheme. Here this approach is applied to construct the discontinuous solution of Helmholtz’s equation for a conical defect. On the base of it the discontinuous solutions of Lame’s equations are derived for a case of steady state loading of a medium.

Diffraction of wave on the conical defect in the acoustic environment

The discontinuous solution of the wave equation for a conical defect in acoustic environment under the quasistatic dynamic load is constructed in the article. The defect is the part of the surface when passing through it the wave potential and its normal to the defect's surface derivative are discontinuous with the given jumps. The integral transformation method by the classic scheme and generalized scheme relatively to the variable on which the function is discontinuous was applied for the construction of the solution. The boundary values of the wave potential are obtained. The representations for the discontinuous solutions of the dynamic movement equations for the conical defect in the case of quasistatic fluctuations could be derived by the known scheme of the discontinuous solutions's method based on the obtained relations for the wave potential and its normal derivative.

Mathematical Models of Heat Exchange in Multilayer Constructions with Various Thermalphysic Characteristics in Industrial Installations

The article presents the developed mathematical models of heat exchange in multilayer flatconstructions with honeycomb filler. The mathematical description of temperature process in composite materials polymerization at three production stages is made. This work continues researches in the field of thermalphysic processes modeling in multilayered designs. The application features of the method of finite integral transformations with Green's function and the isothermal surfaces method are considered. Describing the product cooling process the Laplace finite integral transformation has been used. Heat emission is measured in the course of polymerization. The universality of the developed mathematical apparatus allows to create the software for experimentations on the computer. The authors offer also further directions of the research.

MATHEMATICAL MODELS OF MECHANISMS FOR ROLLED PRODUCTS ACCELERATED COOLING

The tasks of increasing strength of heat-strengthened rolled products require thorough search for technical solutions, determined by level of understanding of processes occurring in thermal strengthening devices, of which the main is the process of water interaction with hot rolled product. This complex set of phenomena includes movement of water flows relative to moving rolled metal, emergence of vapor gap between water and rolled metal, generation of nanosized droplets and their movement through the layer of vapor, the droplets impact on surface, excitation of elastic waves in rolled material. Analysis of previously derived dispersion equation for Kelvin-Helmholtz instability of vapor-water interface was carried out. It is shown that for 30–60 m/s difference in velocities of liquid and vapor layers, the maximum increment in nanoscale range of wavelengths is observed. Average size of generated drops is determined by wavelength at which the maximum increment dependence is reached. Thus, mechanism of accelerated cooling of rolled steel proposed earlier is confirmed by quantitative calculations. Drops, reaching rolled metal, excite thermoelastic wave which being distributed along its section promotes increase in impact strength. To reveal regularities of propagation of elastic waves created by drops in rolled metal, problem of theory of thermoelasticity in cooling was solved by method of Fourier and Laplace integral transformations. It turned out that the problem posed is analogous to the problem of heating the surface with triangular temperature profile. Evolution of stress waves was investigated. At initial moments, the front of wave appeared to be a stretching wave. When reflected from the free end, the wave becomes a compression wave with stresses leading to cracks closure, and thus it leads to increase in toughness. Obtained results can be used in search for optimal modes of heat treatment of rolled products, providing high mechanical properties.