Problems Of Mathematical Physics Research Articles

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.

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.

АППРОКСИМАЦИЯ ПРОГИБОВ ПЛАСТИНОК, ЛЕЖАЩИХ НА ВИНКЛЕРОВОМ ОСНОВАНИИ

The purpose of this research is to develop the method of shape factor interpolation for calculating the maximum deflection of thin plates on an elastic Winkler base, which are widely used in modeling the operation of elements of building constructions of buildings and structures. The above calculation method allows to obtain solutions based on direct analytical dependences, the argument of which is an integral characteristic of a flat convex one-connected area - the shape factor. This characteristic has applications in a number of problems of mathematical physics and is known from the works of scientists G. Polia and G. Szegö. The shape factor was first applied to the calculation of plates by Professor V.I. Korobko. The method of interpolation by shape factor was developed by Professor A.V. Korobko. When determining the maximum deflection of thin plates on an elastic base, some parameters of the problem are considered as functions of the shape factor of the plate in consideration and are determined by the type of boundary conditions on its contour. The present study is devoted to the construction of approximating functions for continuous sets of plates of characteristic outlines and boundary conditions. The paper presents functions for calculating the maximum deflection of elastic plates in the form of isosceles triangles, rhombuses and rectangles. The plates with various combinations of hinged support and rigid pinch along their individual sides, loaded with a continuous uniformly distributed load, are considered. The established functional dependences are intended for direct use in the calculation of plates of the specified outlines, as well as for obtaining reference solutions during interpolation of the values of maximum deflections of plates of more complex outlines.

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ В СТАТИЧНИХ ТРИВИМІРНИХ КРАЙОВИХ ЗАДАЧАХ — КОСОСИМЕТРИЧНА ЗАДАЧА ДЛЯ ШАРУ, ПОСЛАБЛЕНОГО НАСКРІЗНИМ ОТВОРОМ І КОВЗНИМ ЗАЩЕМЛЕННЯМ ТОРЦІВ

Spatial static boundary-value problems of mathematical physics for a layer with a non-circular cylindrical through hole have hardly been solved using the method of singular integral equations (SIE) despite the fact that numerous fundamental theoretical issues have been developed. Methods for the calculation of spectral characteristics are also absent. The paper provides an overview of the methods for solving these problems. A new mathematical model has been constructed, and a new method based on a system of three SIEs has been developed and tested numerically. As a result of a high-precision numerical study, it was found that with an increase in the thickness of the layer, an increase in the relative circumferential stress occurs. In the case of a circular hole, a shift of the maximum relative circumferential stress from the ends to the depth of the layer is observed. In the case of an elliptical hole, with a decrease in one of the radii, an increase in the relative circumferential stress is also observed. Keywords: three-dimensional boundary-value problems, singular integral equations, numerical experiment, static bending, a through hole.

Параболічні крайові задачі математичної фізики в кусково-однорідному клиновидному циліндрично-круговому півпросторі з порожниною

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous by the radial variable r wedge-shaped by the angular variable φ cylindrical-circular half-space with a cavity were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of the 1st kind (Dirichlet) and the 2nd kind (Neumann) and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a finite integral Fourier transform on the Cartesian semiaxis (0; +∞) by an applicative variable z and a Weber hybrid integral transform type on the polar axis (R0; +∞) with n points of conjugation by a radial variable were used to construct solutions of investigated boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The consistent application of inverse integral transforms to the obtained solution in the space of images restores the solutions of the considered parabolic boundary value problems through their integral image in an explicit form in the space of the originals. At the same time, the main solutions to the problems were obtained in an explicit form.

Numerical simulation of an external ballistic problem using analytical approach and atmosphere flow visualization by finite element method

Purpose. Analysis of the dynamic characteristics of the cargo in the presence of the speed of its carrier, taking into account the influence of the external environment and the determination of factors that affect the operational characteristics and parameters for the dynamic system control. Methodology. An approximate analytical approach to solving the nonlinear problem of external ballistics of a system with time-dependent parameters, which is implemented using the asymptotic perturbation method, and a numerical algorithm for modeling a dynamic process using a 3D software complex and the finite element method, which allows to visualize the nature of the flow around the object under wind load conditions. Findings. An analytical approach to solving the nonlinear problem of external ballistics of a system with time-varying parameters is proposed, as well as a numerical model, solution and visualization of a dynamic process that can be applied in problems of mathematical physics and engineering calculations. Originality. The use of an approximate analytical approach to solving the nonlinear problem of external ballistics, which includes a system of time-varying parameters, is a significant innovative step. For the first time, a three-dimensional distribution of the free fall parameters under atmospheric conditions was obtained depending on the time of free fall under the condition of an initial velocity different from zero. Obtaining the characteristics of the distribution, taking into account the aerodynamic quality of the object, made it possible to adapt the methods of analytical mechanics and differential equations with variable coefficients to the solution of the applied problem. The obtained characteristic three-dimensional surface allows to perform a study of the aerodynamic quality of the object under study using numerical methods and to visualize dynamic processes in three-dimensional space in order to obtain a qualitative picture of the perturbation of the object by the air flow. This integrated approach to the study of ballistic characteristics of cargo forms the scientific principle to perform engineering calculations to solve the problem of controlling dynamic systems. Practical value. An approximate analytical approach and a calculation model of the dynamic process of a mechatronic unmanned system make it possible to increase its efficiency in the presence of an initial speed and ensure the accuracy of the application of ballistic cargo delivery systems in real operating conditions.

Stability analysis and dynamics of solitary wave solutions of the (3+1)-dimensional generalized shallow water wave equation using the Ricatti equation mapping method

Our aim is to examine the dynamic characteristics of the (3+1)-dimensional generalized equation governing shallow water waves. When the horizontal extent of the fluid significantly surpasses the vertical dimension, the employment of shallow water equations becomes appropriate. By employing an inventive Ricatti equation mapping approach, we have obtained a range of solitary wave solutions in both explicit and generalized forms. Solitons are particularly useful in signal and energy transmission due to their ability to preserve their shape during propagation. By studying soliton solutions in nonlinear systems, we can gain valuable knowledge applicable to practical fields. To deepen our grasp of the equation’s physical implications, we have presented some solutions through 3D, 2D, and contour graphics. Employing a linear stability approach, we confirm the equation’s stability. The method used distinguishes out for its simplicity, dependability, and ability to generate novel solutions for nonlinear partial differential equations in the area of mathematical physics. The research findings reported here demonstrate the viability of the used method in studying nonlinear phenomena in the studied equation as well as other nonlinear problems in mathematical physics. By employing this tool, academics have the chance to deepen and increase their grasp of the complex mathematical concepts underlying real-world problems.

The partial-rogue ripple solutions of nonlocal Kadomtsev–Petviashvili equation

The derivation of new exact solutions for integrable systems has always been an open problem in mathematical physics and engineering. In this paper, a large class of partial-rogue ripple solutions of a nonlocal Kadomtsev–Petviashvili [KP] equation are derived, which means that these ripple waves are semi-localized in time. Firstly, the Wronskian and Grammian determinant expression of the nonlocal KP equation are constructed. Then, three kinds of partial-rogue ripplon solutions of nonlocal KP equations, including (i) partial-rogue ripple soliton solutions, (ii) partial-rogue ripple lump chain solutions, and (iii) partial-rogue ripple lump solutions, are proposed. These solutions approach to the zero background plane as t→∞, which is the characteristic of partial-rogue ripple waves. These solutions are rarely reported, and their dynamics have been analytically and numerically discussed. The results of this paper are more practical and closely match the attenuated ripple waves stirred up in water waves. Additionally, the research method in this paper provides new references for existing results.