Integral Transformation Method Research Articles

Photothermal impacts induced by laser pulse in a 2D semiconducting medium with temperature-dependent properties under strain–temperature rate-dependent theory

The purpose of this research is to introduce a new model for studying the photothermal process in a 2D semiconducting material with temperature-dependent properties using the recently modified Green and Lindsay theory. The surface absorption fashion generated by a laser pulse is used to heat a medium whose surface is subjected to cooling impact and considered traction free. The integral transformation method is used to attain a general solution via Fourier and Laplace transforms, and the inverse Laplace transform is performed numerically using Riemann sum approximation. A silicon element is utilized to get findings that are graphically displayed as an application in which the consistency of outcomes may be deduced.

Analytical simulation of the simultaneous adsorption process of methane and water vapor in shales

ABSTRACT Investigations on the diffusion – adsorption of methane and water vapour in shale nanopores are essential for developing shale gas extraction. A kinetic model considering diffusion and different adsorption mechanisms was proposed. An analytical solution to the model was obtained through the integral transformation method and used to predict the distributions of methane and water within a shale particle. The results suggest that the reversible adsorption caused by the dispersion force is not the only adsorption mechanism; irreversible adsorption caused by hydrogen bonding or other types of intermolecular forces is more prominent, lengthening the delay to reach equilibrium. The presence of water strongly affects shale, considerably decreasing the diffusivity and adsorption capacity as well as the swelling of the shale matrix caused by the increase in the methane pressure. The diffusion and adsorption process expands to the shale particle surface after the pores near the particle centre are filled. The transport and storage of water vapour are more sensitive to the pore structure than those of methane. In the initial stage of the methane diffusion‒adsorption process, a large number of water molecules enter the particles, and the adsorbed water hinders methane molecules from contacting the shale surface.

Constructing mathematical models of thermal conductivity in individual elements and units of electronic devices at local heating considering thermosensitivity

This paper considers a heat conduction process for an isotropic medium with local external and internal thermal heating. It was necessary to construct linear and non-linear mathematical models for determining the temperature field, and consequently, for the analysis of temperature regimes in these heat-active environments. To solve the linear boundary value problems and the resulting linearized boundary value problems with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which the analytical solutions to these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in this environment were established. To determine the numerical values of the temperature and analyze the heat exchange processes in the given structure, caused by the external heat load, a geometric image of the temperature distribution was constructed depending on spatial coordinates. The resulting linear and non-linear mathematical models testify to their adequacy to the real physical process. They make it possible to analyze heat-active media regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements but the entire structure as well

ACTION OF A SOLID BODY ON THE INNER SURFACE OF A THICK-WALLED BIMETAL CYLINDER

Purpose. The goal of the work is to obtain an exact solution to the problem of the stress-strain state of a long thick-walled bimetallic cylinder within the framework of the classical theory of elastic materials. Then, using the results obtained with this formulation, it is necessary to propose simpler engineering approaches and explore the possibilities of using asymptotic formulas for express analyzes at the design stage of such structural elements. Research methods. For the main body of the cylinder, the classical equations of the theory of elasticity in displacements are used. For the outer coating (sputtering), the shell theory equations are written based on the Kirchhoff-Love hypotheses. The complex integral Fourier transform and the Failon method are used to approximately find the original stresses and displacements. Asymptotic representations of cylindrical Bessel functions for large values of the argument and representations of improper integrals in the form of combinations of elementary and special tabulated functions are also used. Results. A mathematical model has been constructed to analyze the stress-strain state of a bimetallic cylinder with a thin outer layer of a material different from that of the inner layer. Various boundary conditions are recorded on the inner surface of the cylinder, describing the transmission from a rigid body of either specified forces or specified displacements. For all considered options, using the method of integral transformations, the results were obtained in the form of improper integrals, for the calculation of which a special method was used, aimed at calculating integrals with highly oscillating functions. Examples of specific graphs of changes in the components of the stress-strain state in the cylinder material are given. Depending on the conditions on the inner surface of the cylinder, simpler models are proposed to describe the main body, which are based, depending on the nature of the description of the interaction of the rigid body and the cylinder, on one equation of the theory of elasticity. With this approach, in some important cases it was possible to obtain improper inversion integrals using the asymptotic approach in closed form as a combination of elementary and special tabulated functions. Comparison with the exact approach allowed us to prove the possibility of using approximate models. Scientific novelty. A model of the behavior of a bimetallic cylinder as a body is constructed, the main layer of which is described by the equations of the theory of elasticity, and the theory of shells is used for the outer coating. Various methods of describing the transfer of forces and displacements from a rigid body to the inner surface of the cylinder are considered. The possibility of using the asymptotic approach to obtain relatively simple formulas for carrying out preliminary calculations at the design stage of such structural elements is shown.

Triple Mixed Integral Transformation and Applications for Initial-Boundary Value Problems

In this paper, we establish a new integral transformation method for solving the initial-boundary value problems of the partial differential equations. The method is a combined expression mainly based on the triple integral transformation of Laplace and Sumudu. We study the basic properties of the triple Laplace-Sumudu transform, and give the triple Laplace-Sumudu transforms of some basic functions. And as applications, we solve some heat flow equations and wave equations with initial-boundary value conditions by using the method. The results show that the triple Laplace-Sumudu transform is more efficient and useful to deal with these problems.

Vertical vibration of a rigid disc over a transversely isotropic and layered poroelastic half-space

A novel semi-analytical method is proposed for solving the vertical vibration of a rigid disc over a transversely isotropic and layered poroelastic half-space. The fundamental solutions in the layered poroelastic half-space under a vertical patch load on the surface are solved in terms of the newly developed Fourier-Bessel series (FBS) system of vector functions and the unconditionally stable dual-variable and position method. The expansion coefficients in FBS system are discrete, as such the computational efficiency is greatly improved and the accuracy is better than the commonly used integral-transform methods. By virtue of the superposition method, the Green’s function due to vertical ring load is derived in the transform domain. By making use of the integral least-square scheme, the densities of a series of ring loads discretized within the disc area are determined. Finally, the dynamic vertical compliance is derived via the balance between the external applied force and the induced total contact traction. After verifying the reliability of the developed solution, selected numerical examples are presented to investigate the influence of material properties and excitation frequency on the forced vibration of the rigid disc.

SCTBEM: A scaled coordinate transformation boundary element method with 99-line MATLAB code for solving Poisson's equation

This study introduces the scaled coordinate transformation boundary element method (SCTBEM), a novel boundary-type method for solving three-dimensional potential problems. To apply the BEM to complex problems, it is common practice to use the fundamental solution corresponding to the partial governing equation operator to establish the integral equation. However, this approach introduces the domain integral, which may jeopardize the dimensionality reduction advantages of BEM. To preserve the benefits of dimensionality reduction, this study proposes a novel domain integral transformation method named SCT. The SCT is purely a mathematical operation that does not rely on a particular solution of the operators, which requires only discretization on the surface of the structure while remaining analytical in the radial direction. An even more significant advantage is that the lower-order singularity can be eliminated by the coordinate translation technique. To facilitate the wider adoption of BEM, the authors present an open-source 99-line MATLAB code. Numerical results demonstrate that the SCTBEM exhibits high numerical accuracy even when dealing with complex models.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Analytical solution of a pollutant transport model for unsaturated soil considering the effects of consolidation and temperature

The migration behavior of pollutants is affected by consolidation and temperature when using thermal desorption technology to clean contaminated sites. Based on a one-dimensional consolidation model for unsaturated soil and the traditional heat conduction equation, a pollutant transport model accounting for the combined effects of consolidation and temperature was established in this paper. An analytical solution was obtained by using the separation of variables method and the integral transformation method. In addition, the correctness of the proposed model was verified via a comparison between the existing analytical solution and the theoretical model. Finally, adopting benzene as the research object, the influence of different factors on pollutant migration was studied. It was found that the growth rate of the pollutant concentration increased with increasing consolidation pressure, and the final pollutant concentration decreased with increasing consolidation pressure. The pollutant concentration increment due to temperature first increased and then decreased with increasing migration distance. The higher the Soret coefficient and volumetric moisture content are, the higher the pollutant concentration.

Математичні моделі локального нагрівання елементів електронних пристроїв

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media sub-jected to internal and external local heat load, have been developed. In the case of nonlinear boundary-value problems, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equations and nonlinear boundary conditions were linearized, and as a result, linearized second-order differential equations with partial derivatives and a discontinu-ous right-hand side and partially linearized boundary conditions were obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the ap-proximation of the temperature according to one of the spatial coordinates on the boundary sur-faces of the inclusion was performed by piecewise constant functions. To solve linear bounda-ry-value problems, as well as obtained linearized boundary-value problems with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in this envi-ronment were obtained. Numerical analysis of temperature behavior as a function of spatial co-ordinates for given values of geometric and thermophysical parameters was performed. The in-fluence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal and external heat load, software tools were developed, using which a ge-ometric image of the temperature distribution depending on the spatial coordinates was made.

Mathematical models for analysis of temperature regimes in vehicle braking systems

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in the braking systems of vehicles, which are geometrically depicted as isotropic spatial heat-active media that are subject to internal local thermal heating, have been developed. With the use of classical methods, it is not possible to obtain analytical solutions of linear and nonlinear boundary value problems of mathematical physics in a closed form. This is especially the case when the right-hand sides of differential equations with partial derivatives and boundary conditions are discontinuous functions. The given approach is based on the application of the apparatus of generalized functions to describe the local concentration of thermal influence. This made it possible to apply the integral transformation and, on this basis, to obtain analytical solutions of both linear and nonlinear boundary value problems. In the case of a nonlinear boundary value problem, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a linearized second-order differential equation with partial derivatives and boundary conditions with a discontinuous right-hand side were obtained. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. On the basis of the developed mathematical models, a computational algorithm was created and on this basis, software tools were created, using which the heat exchange processes in the middle of the brake structures for the selected materials of the brake pads were analyzed in terms of their effectiveness, as well as the determination of the optimal temperature values for the effective operation of the braking system of vehicles. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active media with internal heating make it possible to analyze their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and individual elements, but also the entire structure.

Uticaj temperaturskog gradijenta na savijanje tankih ploča

Within the theory of thermo-elasticity, the temperature field of thin plates is commonly defined via two parameters: temperature in the mid-plane and linear temperature gradient normal to the mid-plane. First, the paper analytically proves the justification of that assumption in machine structures. Then, in an analytical closed form, applying the integral transformation method, the thin plate deflection caused by a constant temperature gradient is defined. It is shown that, in that case, the plate deflection does not depend on its thickness but only on the plate dimensions in the mid-plane. Analytically defined values are compared to corresponding values obtained by applying the thin plate finite element, where the temperature field is described using the two mentioned parameters. This finite element is defined and programmed within the Komips program package. The influence of the temperature gradient on the behavior of constructions mostly depends on the type of material. That is why the behavior of some structural elements made of brass, steel, and concrete is analyzed in this paper.

Indentation over a transversely isotropic, poroelastic, and layered half-space

Poroelastic materials are common in nature and have applications in many engineering fields. In this paper, we derive a general solution of indentation over a multilayered half-space consisting of transversely isotropic and poroelastic materials. The rigid disc-shaped indenter is subjected to a vertical force of Heaviside time-variation. The solution is expressed in terms of the recently introduced powerful Fourier-Bessel series (FBS) system of vector functions combined with the unconditionally stable dual-variable and position method for dealing with layering. Since the problem is a mixed boundary-value one, the Green's functions due to a vertical ring-load are first derived which are then utilized in the integral least-square formulation to derive the solution. In terms of the FBS method, the expansion coefficients, which are further called Love numbers, are discrete, and therefore can be pre-calculated and used repeatedly for different field points on the surface. As such, the solution based on the new FBS method is more efficient and accurate than previous integral-transform methods. This new FBS method is particularly attractive when dealing with mixed boundary-value problems where time-variation is further involved. Numerical examples are conducted to validate the accuracy of the proposed solution and to demonstrate the effects of material layering, geometry, and hydraulic boundary conditions on the contact performance of the material system.

Comparative study on curing kinetics of MDI-based polyurethanes with different chain length diol curing agents

The difference in the reaction between thermosetting polyurethane (CPU) and binary alcohol system curing agents is crucial for obtaining accurate curing kinetics models and selecting curing agents. Based on a systematic comparison of the reaction activation energies and kinetic models of three different chain length fatty alcohol curing agents, ethylene glycol, 1,4-butanediol, and 1,6-hexanediol, the effect of curing agent chain length on the curing reaction was elucidated. Using differential and integral transformation methods to process non-isothermal DSC tests results, the effect of different chain lengths curing agents on the activation energy of MDI based polyurethane were analyzed. Based on the self-catalytic model, the segmented curing mechanism of the MDI based polyurethane systems were studied, and applicable high-precision segmented kinetic models’ parameters were calibrated. The optimal curing conditions for these systems were determined using extrapolation. The research results indicate that when the curing degree increases to 95 %, the overall activation energy of the MDI/1,4-butanediol system decreases by 47.1 %, accompanied by a faster self-catalytic. However, the activation energies of the other two systems, MDI/propylene glycol system and MDI/1,6-hexanediol system increased by 62. % and 295.0 %, respectively. This slows down the reaction rate of the two systems, which was 7.8 % and 10.9 % slower than the MDI/1,4 butanediol. heir molecular cross-linking network limits the diffusion ability of their chains and reduces the self-catalysis, resulting in slow reactions. These results will provide theoretical guidance and reference for the material formulation, process optimization of MDI based polyurethane actual curing production.

An Operator Method for Investigation of the Stability of Time-Dependent Source Identification Telegraph Type Differential Problems

This article is devoted to the study of the stability of time-dependent source identification telegraph type differential problems with dependent coefficients. Time-dependent source identification problems (SIPs) for telegraph differential equations (TDEs) with constant coefficients can be solved by classical integral-transform methods. However, these classical methods can be used, basically, in cases where the differential equation has constant coefficients. We establish the basic theorem of the stability of the time-dependent SIPs for the second-order linear differential equation (DE) in a Hilbert space with a self-adjoint positive definite operator (SAPDO) and damping term. In practice, stability estimates for the solution of the three types of SIPs for one-dimensional and for multidimensional TDEs with dependent coefficients and classic and non-classic conditions are obtained.

A Composite Framework Model for Transient Pressure Dynamics in Tight Gas Reservoirs Incorporating Stress Sensitivity

Natural gas is an eco-friendly energy source with low carbon emissions, making it attractive globally. Understanding gas reservoirs is crucial for sustainable extraction and optimizing potential. However, the complicated fluid flow and production dynamics within intricate gas reservoirs, particularly those characterized by abnormally high pressures and tight porous media, remain partially understood and demand further investigation. In a tight porous medium subjected to high pressure, the assumption of constant permeability is no longer valid. Consequently, a novel composite seepage model has been developed in this study, which considers the responsiveness of permeability to stress. Perturbation theory is employed to address the inherent non-linearity demonstrated by the permeability modulus. The solution of dimensionless pressure responses under constant production conditions is accomplished in the Laplace domain by implementing integral transformation methods. Overall, a comprehensive model is provided to understand the production behaviors of tight gas reservoirs. Moreover, in order to comprehend the transient flow characteristics of tight gas reservoirs, log–log plots are generated through the Stehfest numerical inversion approach, with the flow regimes categorized based on the normalized time phases of the pressure curves. Parametric investigations reveal that stress sensitivity detrimentally affects permeability, resulting in more pronounced pressure declines during the intermediate and late flow phases. The transient seepage model elaborated in this study is able to consider the pertinent formation and well parameters. These interpreted parameters bear significance in designing hydraulic fracturing operations, assessing the potential of tight gas reservoirs, and ultimately enhancing gas production. The presented model not only enhances our understanding of the behavior of horizontal wells in stress-sensitive tight gas reservoirs but also makes a valuable contribution to the broader discussion on transient flow phenomena in petroleum engineering.

Математичні моделі температурного поля у елементах електронних пристроїв із чужорідним включенням

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic media with semi-transparent foreign inclusions that are also subject to external heat load, have been developed. To solve the nonlinear boundary-value problem, a linearizing function was introduced, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a partially linearized second-order differential equation with partial derivatives and discontinuous and singular coefficients relative to the linearizing function and partially linearized boundary conditions was obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the approximation of the temperature according to one of the spatial coordinates on the boundary surfaces of the inclusion was performed by piecewise constant functions. To solve the obtained linear boundary value problem, the Hankel integral transformation method was used, as a result of which an analytical solution was obtained, which determines the introduced linearizing function. For the numerical analysis of temperature behavior and heat exchange processes caused by external heat load, software tools have been developed, which are used to create a geometric image of temperature distribution depending on spatial coordinates. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The developed models make it possible to analyze environments with foreign elements under external thermal loads regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual elements, but also the entire structure.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

Straightforward Free Vibration Solutions of Open Cylindrical Shells by the Finite Integral Transform Method

Straightforward analytical free vibration solutions of non-Lévy-type open cylindrical shells are obtained in this study by the finite integral transform method, which was not achieved by conventional semi-inverse methods. Three double finite integral transforms are imposed on the high-order partial differential equations (PDEs) for the problems, with some boundary conditions input, by which the relations between the transformed quantities and specific unknowns are given. Substituting the corresponding inverse transforms into the other boundary conditions provides systems of linear algebraic equations for determining the final analytical solutions. Comprehensive benchmark results for representative open cylindrical shells under different boundary conditions are shown. The results are well verified by other solution methods. With the present solutions, the effects of geometric parameters and boundary conditions on the vibration behaviors are quantitatively revealed by a parametric study. Due to its rigorous and straightforward solution procedure, this study presents a solid easy-to-implement way to explore new analytical solutions of similar problems.

Analytic solutions to the two-dimensional solute advection-dispersion equation coupled with heat diffusion equation in a vertical aquifer section

Analytical and semi-analytical solutions to solute transport equations are not only used to predict the fate and transport of contaminants in soil or groundwater systems, but also to provide reference standards for validation of numerical models. However, when the solute transport equation is coupled with heat, the corresponding analytical solutions are rarely reported, perhaps due to the complexity of the coupled models. The main purpose of this paper is to present an analytical solution to the solute transport equation coupled with heat for the case of modeling solute transport in a vertical section of a homogenous infinite aquifer with steady uniform groundwater flow under constant-flux solute source. In earlier work (Shan and Javandel, 1997), analytical solutions to the solute transport equation in both finite and infinite aquifer thickness were presented, influence of heat (Soret effect) was generally not considered. Heat can, however, have a significant impact on movement of concentration front. In the present work, we first couple the solute equation with a generalized classic heat diffusion equation. To do this, we assume that the heat is not influenced by the solute (Dufour effect) such that only a one-way coupled model is considered. The repeated integral transformation method (RITM) is utilized to obtain the analytical solutions to the proposed coupled model. As a limiting case, a comparison with analytical solution (Shan and Javandel, 1997), equation (21) and other literature findings is done verify the new proposed solutions and validate our results. The influences of the thermopheresis effect and other model parameters are pictured graphically by the use of MATLAB R2015a. In comparison with the earlier work, the present results show that movement of contaminant in the medium is affected by the presence of heat, in particular along the direction of flow, consistent with the literature. The analysis of longitudinal and transverse concentration profiles suggest that the proposed solutions are applicable for monitoring of groundwater contaminants and risk assessment. Also, the solutions should be useful for comparison with numerical models.