Problems Of Mathematical Physics Research Articles

Mathematical model for heat transfer during underground coal gasification process

Purpose. To develop a mathematical model of the “coal-gas” medium heat transfer during underground coal gasification to predict the combustion face advance velocity and the duration of gasification column mining. Methodology. To detect the temperature fields in the coal seam and gas, depending on the displacement length of the phase transition boundary, a boundary-value problem of mathematical physics has been developed. To solve this boundary-value problem, Boltzmann transformations, as well as methods for solving ordinary differential equations, are used. Newton-Raphson method, which has quadratic convergence, is used to find the roots of the transcendental equation. Findings. Tendencies in the formation of mathematical models when studying temperature fields around an underground gasifier have been analyzed, with highlighting their disadvantages. A mathematical model of heat transfer during underground gasification has been developed, taking into account the phase transition boundaries of the “coal-gas” medium. A computational experiment was conducted to determine the temperature of the phase transition boundary based on the length of the gasification column and the duration of the process. Originality. A mathematical model for heat transfer during coal gasification in the form of a boundary-value problem of mathematical physics, which consists of parabolic heat-transfer equations, the Stefan condition at the phase transition boundary, and the Dirichlet boundary conditions, has been constructed. As a result of solving the boundary-value problem, a self-similar solution has been obtained for the distribution of the coal seam and gas temperature fields, as well as the position of the phase transition boundary depending on the gasification duration and on the medium density, thermal conductivity coefficients, specific heat capacity of gas and coal, specific calorific value and temperature of coal combustion, initial coal temperature and constant temperature of the gasification process. The conducted analysis of numerical calculations provides a deeper understanding of the dynamics of underground coal gasification process and makes necessary corrections to achieve the maximum process efficiency. Practical value. A methodology for determining the displacement length of the phase transition boundary of the “coal-gas” medium has been developed, taking into account the change in the combustion face temperature along the gasification zone length depending on the duration of this process. Application of the methodology makes it possible to predict the time of mining the gasified coal column for drawing up a calendar plan for mining operations.

Метод R-функцій для розв’язання задачі масообміну циліндричного тіла з твірною у вигляді кривої Ламе з в’язкою нестисливою рідиною

The problem of stationary mass exchange of a cylindrical body with viscous incompressible fluid flow is reduced to solving the equation for concentration with the corresponding boundary conditions on the surface of the body and at infinity. Applying the constructive apparatus of the R-functions theory allows us to accurately take into account the geometry of the domain, as well as the boundary conditions together with the condition at infinity. The R-function method was proposed by V. L. Rvachev, Ukraine National Academy of Sciences academician. For boundary value problems of mathematical physics, this method allows constructing the so-called structure of the solution of the boundary value problem, i.e. a bundle of functions that accurately takes into account all boundary conditions of the problem and depends on some uncertain components. The choice of these uncertain components is performed in such a way as to satisfy (in a certain sense) the equation of the problem. For this, one can use standard numerical methods of mathematical physics (the Ritz method, the Galerkin method, the least squares method, collocations method, etc.). It should be noted that the geometry of the region is taken into account exactly, i.e. in particular, there is no replacement of curvilinear sections of the boundary with polygonal lines inscribed in them, as it is done, for example, in the finite element method. The purpose of this article is to apply the R-functions method to the problem of mass exchange of a cylindrical body formed by the Lame curve, with a viscous incompressible fluid flowing around it. To achieve this goal, a complete structure of the solution of a linear boundary value problem for concentration is constructed by the R-functions theory methods, and a numerical algorithm based on the Galerkin method for approximating indefinite components in the problem structure is used. The degree of rigor and the conditions for using the considered equations of fluid motion are not considered in the work; these equations are considered as mathematical models subject to numerical algorithmization

Solution of Inverse Photoacoustic Problem for Semiconductors via Phase Neural Network

The inverse photoacoustic problem is an ill-posed mathematical physics problem. There are many methods of solving the inverse photoacoustic problem, from parameter reduction to the development of complex regularization algorithms. The idea of this work is that semiconductor physical properties are determined from phase characteristic measurements because phase measurements are more sensitive than amplitude measurements. To solve the inverse photoacoustic problem, the thermoelastic properties and thickness of the sample are estimated using a neural network approach. The neural network was trained on a large database of photoacoustic phases calculated from a theoretical Si n-type model in the range of 20 Hz to 20 kHz, to which random Gaussian noise was applied. It is shown that in solving the inverse photoacoustic problem, high accuracy and precision can be achieved by applying phase measurement and neural network approaches. This study showed that a multi-parameter inverse problem can be solved using phase networks.

The overdetermined Cauchy problem for the hyperbolic Gellerstedt equation

Abstract Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.

Asymptotic Criticality of the Navier–Stokes Regularity Problem

The problem of global-in-time regularity for the 3D Navier-Stokes equations, i.e., the question of whether a smooth flow can exhibit spontaneous formation of singularities, is a fundamental open problem in mathematical physics. Due to the super-criticality of the equations, the problem has been super-critical in the sense that there has been a ‘scaling gap’ between any regularity criterion and the corresponding a priori bound (regardless of the functional setup utilized). The purpose of this work is to present a mathematical framework-based on a suitably defined ‘scale of sparseness’ of the super-level sets of the positive and negative parts of the components of the higher-order spatial derivatives of the velocity field—in which the scaling gap between the regularity class and the corresponding a priori bound vanishes as the order of the derivative goes to infinity.

A novel study of analytical solutions of some important nonlinear fractional differential equations in fluid dynamics

The space-time fractional Burger-like equation and the space-time coupled Boussinesq equation with conformable derivative, both of which are significant in fluid dynamics, are investigated in this work using the improved [Formula: see text] method. The process works effectively and produces soliton solutions. The method was successfully and consistently implemented with Maple, a symbolic computing tool. The solutions also contain a few of graphics. Numerous novel exact solutions to these equations, distinct from those found earlier with the proposed approach, have been provided. The study’s findings add to the body of knowledge by offering insightful justifications for various types of nonlinear systems. The results demonstrated the value of the proposed method as a mathematical tool and the ease, dependability, and speed increases that result from carrying out these tasks using a symbolic computing program. Notably, it applies to many nonlinear evolution problems in mathematical physics.

Parabolic Boundary-Value Problems of Mathematical Physics in a Piecewise Homogeneous Wedge-Shaped Hollow Cylinder

Parabolic Boundary-Value Problems of Mathematical Physics in a Piecewise Homogeneous Wedge-Shaped Hollow Cylinder

Singular backward stochastic Volterra integral equations in infinite dimensional spaces

In this paper, the notion of singular backward stochastic Volterra integral equations (singular BSVIEs for short) in infinite dimensional space is introduced, and the corresponding well-posedness is carefully established. A class of singularity conditions are proposed, which not only cover that of fractional kernel, Volterra Heston model kernel, completely monotone kernels, to mention a few, but also happen to be used in the forward stochastic Volterra integral with new conclusions arising. Motivated by mathematical physics problem such as the viscoelasticity/thermoviscoelasticity of materials, heat conduction in materials with memory, optimal control problems of abstract stochastic Volterra integral equations (including fractional stochastic evolution equations and stochastic evolutionary integral equations) are presented. At last, our BSVIEs are surprisingly used in maximum principle of controlled stochastic delay evolution equations. One advantage of this new standpoint is that the final cost functional can naturally depend on the past state.

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.

Differential (Pukhov) transform method analysis of transient regimes in electrical circuits

The generalized classical method (GCM) was developed on the basis of the differential Taylor transform method, which was first used by the Ukrainian scientist G.E. Pukhov. In this study, some applications of the GCM to the analysis of transient events in simple electrical circuits are examined. It is shown that, if the solution can be decomposed into a steady-state part and a transient part, the use of the GCM becomes more effective, and the transient regime of the circuit solution can be analyzed without a full solving process of differential equations. The efficiency of the proposed approximation method is illustrated by comparing it with prior arts on similar problems. The results reveal that the proposed method is simple and can be applied successfully for the analysis of both linear and nonlinear problems in mathematical physics by similar solving procedures.

Inverse Problem of Determining the Unknown Coefficients in an Elliptic Equation

The inverse problem of determining the coefficients of an elliptic equation under different boundary conditions in a given rectangle is considered. These problems lead to the necessity of approximate solution of inverse problems of mathematical physics, which are incorrect in the classical sense. The existence, uniqueness, and stability theorems for the solution of the set inverse problem are proved and a regularizing algorithm for determining the coefficient is constructed.

The data-driven rogue waves of the Hirota equation by using Mix-training PINNs approach

Recently, solving differential equations by deep neural networks has attracted lots of attention, however, when the solution of differential equations has localized areas such as the rogue waves, the classical physics-informed neural networks (PINNs) cannot guarantee its prediction accuracy. In this paper, we propose two neural networks methods: mix-training physics-informed neural networks (MTPINNs) and prior information mix-training physics-informed neural networks (PMTPINNs), two deep learning models with more approximation ability based on PINNs. A series of numerical experiments show that the learning efficiency and absolute error accuracy of our proposed model can be improved significantly. The PMTPINNs model can so as to make up for the shortcomings of PINNs in this aspect. Furthermore, by testing the robustness of these two methods, for two kinds of rogue waves of the Hirota equation, it is surprising that these models also have good performance and thereby may provide more possibilities for solving mathematical physics problems better.

A New Economical Unconditional Stable Splitting Method for Numerical Solution of Problems of Mathematical Physics

A New Economical Unconditional Stable Splitting Method for Numerical Solution of Problems of Mathematical Physics

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Cluster modeling of the interaction of statyonary SH-waves with a system of curvilinear cracks in a half-space

A method is proposed for solving problems of mathematical physics for semi-infinite media containing systems of curvilinear cuts. Based on singular integral equations (SIE), a unified approach to solving the problem has been developed. Numerical implementation was carried out using parallelisation. A graph of the duration of the initial cluster hour is given for calculating an array of searched functions in the context of a parabolic form as a function of the number of processes for one variant of implementation. It is shown that the entire algorithm scales well and has an efficient number of processes. The expected result was obtained, because the structure of the calculation process in models based on SIE is mostly unchanged and built on well-defined procedures. It turned out that 150–200 processes are effective. An accuracy of 10–12 was achieved with the number of collocation points of the contour of each section N = 300, because the algorithm for numerical solution of the problem uses interpolation by Chebyshev polynomials in accordance with the fact that the unknown function has a key feature at the ends of the section, which causes a higher speed of convergence of the algorithm. The study of the question of the further increase in the accuracy of the result was not conducted. The corresponding dynamic boundary value problems for a restrained and force-free half-plane are studied. The influence of the curvature of defects, their interaction and proximity of the boundary on the magnitude and nature of the change in the dynamic stress intensity coefficients was studied. To check the reliability of the algorithm, two tests were carried out: removing three inhomogeneities at a distance of 106 of their length from each other and saturating the system by increasing the number of geometrically equal reflectors to 13–15. When removed, the characteristics of each reflector tend to the characteristics of a single one, and when saturated, they tend to the results of the corresponding periodic problem. The agreement between the results showed good reliability of the algorithm. The proposed method can be used to assess the influence of various mechanical or geometric factors on the strength of bodies with defects.

Debonding of an elastic layer with a cavity from a rigid substrate caused by rotation of a bonded rigid cylinder

Debonding of an elastic layer with a circular cylindrical cavity 0 ≤ ζ ≤ h , ρ ≥ 1 , from a rigid substrate under action of a rigid cylinder is the object of this study. The annular debonding zone 1 ≤ ρ < 1 + η is caused by rotation of a cylinder bonded to the cavity surface. The problem is reformulated as dual integral equations with Weber integral transforms kernels. A Volterra operator transforming a Weber transforms kernel into a Bessel function of the first kind and Hankel integral transforms allow us to reduce dual equations to a Fredholm integral equation of the second kind and then, by some transformation, to another Fredholm integral equation which is more suitable for approximate methods. As h ≥ 0 . 5 and 0 < η ≤ 1 , a highly accurate analytic approximate solution of the problem is suggested. The asymptotic solution of the problem is obtained as the width of debonding zone is very small while the thickness is not small. When the thickness is small, the Fredholm integral equations are computationally inefficient. A new method based on an operator transforming a Bessel function of the first kind into the kernel of Mehler–Fock integral transforms enabled us to convert one of the above-mentioned Fredholm equations into an equivalent Fredholm integral equation of the second kind that is effective for a small thickness. The asymptotic solution of the problem is obtained when both the layer thickness and the debonding zone width are small. Accurate mathematical methods, in particular investigations, and transformations of equations, developed in this study can be interesting to researchers employing dual integral equations technique in problems of mechanics and mathematical physics with mixed boundary conditions.

A mathematical model for the formation of the mineral composition of rocks removed to the surface during a volcano eruption

&#x0D; &#x0D; &#x0D; The article examines the origin of chemicals obtained as a result of cooling on the Earth’s surface during a volcanic eruption. For this, we propose a mathematical model of the formation of the mineral composition of rocks brought to the surface during a volcanic eruption, depending on the crystallization temperature. To do this, we consider a certain inverse problem for the heat equation and prove its solvability under certain conditions. The proposed method can also be used in case of an earthquake. Thus, by studying the composition of rocks on the surface, we can determine at what depth the corresponding layer of rocks is located. Mathematically, this means that you need to find the unknown functions f (x) and u (x, t), where the function f (x) characterizes the source of the formation of rocks, and the function u (x, t) characterizes the crystallization of rocks depending on tem- perature because each rock has its crystallization temperature depending on temperature. We assume that the functions h (x, t), g (x, t) are known. Unknown functions f (x) and u (x, t). Using the inverse problem, we determine the function u (x, t) – the temperature in the lower layers of the earth, and knowing the temperature of crystallization of rocks, we can determine what depth the required rock is. The main purpose of the inverse problem is to determine the composition of rocks below the earth’s surface using data obtained at the earth’s surface. Experimentally, this mathematical model means that with the help of certain sensors on the surface, it is possible to give information about the subterranean depths. This information is determined by the nature of the object under study and the experimental complex used in this study. In such situations, for the diagnosis of objects and their internal structure, mathematical processing, and interpretation of observation results are required .We are talking about those tasks in which it is required to determine the causes if the consequences obtained as a result of observation are known. For example, determining the location and power of an earthquake from the vibrations measured on the earth’s surface. Or, by the structure of minerals on the surface of the earth, determining the depth of their location underground, etc. When processing the experimental data, a conclusion is made about the internal connections of the phenomenon or process. When the mathematical model of the process under study is known, the problem of identifying the mathematical model is posed. Namely, the determination of the coefficients of differential equations, their right-hand side, the boundary of the region, boundary, and initial conditions – these problems are related to inverse problems of mathematical physics. In general, inverse problems.&#x0D; &#x0D; &#x0D;

PREFACE — SPECIAL ISSUE ON FRACTALS AND LOCAL FRACTIONAL CALCULUS: RECENT ADVANCES AND FUTURE CHALLENGES

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems.

ON THE CORRECTNESS OF ONE CLASS OF FOURTH-ORDERSINGULAR DIFFERENTIAL EQUATIONS

The differential equation of fourth-order with variable smooth coefficients, given on the real axis, is investigated. The coefficients of the equation can be unbounded, and the sign of the potential (the lowest coefficient) is not defined. The case is considered that the intermediate term containing the second derivative of the desired function, as an operator, does not obey the operator of the equation. It is known that fourth-order differential equations with variable coefficients are used in various problems of mathematical physics. The simplest biharmonic equation with minor terms is important for its applications in the theory of elasticity, the mechanics of elastic plates and the slow flow of viscous liquids. However, it is a very special case of a general equation with variable coefficients. Some problems of stochastic analysis, oscillation theory, biology and mathematical finance lead to the equation we are studying. In addition, fourth–order singular differential equations are often used as regularizers in the study of lower-order equations, for example, reaction-diffusion equations. In the paper, sufficient solvability conditions of the equation and the maximal regularity inequality of a strong generalized solution are obtained. The restrictions are formulated in terms of the coefficients themselves and rather weak. They do not contain conditions for any derivatives of coefficients and represent relations between the orders of growth of coefficients of different orders at infinity. The weight estimation of the norms of a solution established by us allows us to additionally apply the methods of the theory of functional spaces to the study of further properties of the solution, for example, the approximation of its elements by finite-dimensional spaces.

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.