Cauchy Conditions Research Articles

Study of multiscale mechanical properties of Al-Pd-Mn quasicrystals

Based on the quantum mechanics and molecular mechanics scales, the elastic and tensile mechanical properties of Al-Pd-Mn quasicrystals (QCs) are investigated by utilizing the first-principles and molecular dynamics (MD) methods. The density functional theory (DFT) is adopted to obtain all elastic constants, stability, elastic modulus, Poisson's ratio, anisotropy, Debye temperature of Al57Pd16Mn3 QCs. Simultaneously, the MD simulations are conducted to analyze the influence of temperature and strain rate on the stress-strain curves, atomic structures, and deformation mechanism. The results indicate that Al57Pd16Mn3 QCs are stable orthogonal structures and satisfy the Cauchy conditions. Additionally, the mechanical properties of Al-Pd-Mn QCs obtained by the first-principles are consistent with the MD and experimental results. The elastic properties of Al57Pd16Mn3 QCs are approximately isotropic. The elastic modulus and ultimate tensile strength of Al57Pd16Mn3 QCs always decrease with increasing temperature. An increase in the strain rate results in a negligible change in the elastic modulus but in a noticeable enhancement in the ultimate tensile strength and strain. The current results could provide essential theoretical insights for studying the mechanical properties of QCs, and extend the scope of multi-scale study in the field of QCs at microcosmic scale as well.

A feasible numerical computation based on matrix operations and collocation points to solve linear system of partial differential equations

In this work, a system of linear partial differential equations with constant and variable coefficients via Cauchy conditions is handled by applying the numerical algorithm based on operational matrices and equally-spaced collocation points. To demonstrate the applicability and efficiency of the method, four illustrative examples are tested along with absolute error, maximum absolute error, RMS error, and CPU times. The approximate solutions are compared with the analytical solutions and other numerical results in literature. The obtained numerical results are scrutinized by means of tables and graphics. These comparisons show accuracy and productivity of our method for the linear systems of partial differential equations. Besides, an algorithm is described that summarizes the formulation of the presented method. This algorithm can be adapted to well-known computer programs.

Classical solution to mixed problems from the theory of longitudinal impact on an elastic semi-infinite rod in the case of separation of the impacting body after the collision

In this work, we consider two coupled initial-boundary value problems, which, based on the Saint-Venant theory, model the longitudinal impact phenomena in a semi-infinite rod. The mathematical formulation of the problem is two mixed problems for the one-dimensional wave equation with conjugation conditions. The Cauchy conditions are specified on the spatial half-line. The initial condition for the partial derivative with respect to the time variable has a discontinuity of the first kind at one point. The boundary condition, which includes the unknown function and its first- and second-order partial derivatives, is specified on the time half-line. The solution is constructed by the method of characteristics in an explicit analytical form. The uniqueness of the solution is proved, and the conditions under which a piecewise-smooth solution exists are established. The classical solution to a mixed problem with matching conditions is considered.

Classical solution of the Cauchy problem for a semilinear hyperbolic equation in the case of two independent variables

In the upper half-plane, we consider a semilinear hyperbolic partial differential equation of order higher than two. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. The solution is constructed in an implicit analytical form as a solution of some integral equation. The local solvability of this equation is proved by the Banach fixed point theorem and/or the Schauder fixed point theorem. The global solvability of this equation is proved by the Leray-Schauder fixed point theorem. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established.

The Cauchy Problem for the General Telegraph Equation with Variable Coefficients under the Cauchy Conditions on a Curved Line in the Plane

The Riemann method is used to prove the global correctness theorem to Cauchy problem for a general telegraph equation with variable coefficients under Cauchy conditions on a curved line in the plane. The global correctness theorem consists of an explicit Riemann formula for a unique and stable classical solution and a Hadamard correctness criterion for this Cauchy problem. From the formulation of the Cauchy problem, the definition of its classical solutions and the established smoothness criterion of the right-hand side of the equation, its correctness criterion is derived. These results are obtained by Lomovtsev’s new implicit characteristics method which uses only two differential characteristics equations and twelve inversion identities of six implicit mappings. If the righthand side of general telegraph equation depends only on one of two independent variables, then it is necessary and sufficient that it be continuous with respect to this variable. If the right-hand side of this equation depends on two variables and is continuous, then in its integral smoothness requirements it is necessary and sufficient the continuity in one and continuous differentiability in the other variable. The correctness criterion represents the necessary and sufficient smoothness requirements of the right-hand side of the equation and the Cauchy data. From the established global correctness theorem, the well-known Riemann formulas for classical solutions and correctness criteria to Cauchy problems for the general and model telegraph equations in the upper half-plane are derived. In the works of other authors, there is no necessary (minimally sufficient) smoothness on the right-hand sides of the hyperbolic equations of real Cauchy problems for the set of classical (twice continuously differentiable) solutions.

Classical Solution of the Second Mixed Problem for the Telegraph Equation with a Nonlinear Potential

For the telegraph equation with a nonlinear potential, we consider a mixed problem in the first quadrant in which the Cauchy conditions are specified on the spatial semiaxis and the Neumann condition is set on the temporal semiaxis. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integral equations. The solvability of these equations is studied, as well as the dependence of the solutions on the smoothness of the initial data. For the problem under consideration, the uniqueness of the solution is proved and conditions are established under which a classical solution exists. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not smooth enough, then a mild solution is constructed.

Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential in a Curvilinear Quadrant

For the telegraph equation with a nonlinear potential in a curvilinear quadrant, we consider a mixed problem with the Cauchy conditions on a spatial half-line and the Dirichlet condition on a noncharacteristic curve. The solution of the problem is constructed by the method of characteristics in an implicit analytical form as a solution of integral equations. We study the solvability of these equations depending on the initial data and their smoothness. For the problem under consideration, the uniqueness of the solution is proved and conditions under which there exists a classical solution are established. A mild solution is constructed in the case of insufficiently smooth data of the problem.

Evolution equations with nonlocal multivalued Cauchy problems

We consider evolution equations in Banach spaces. Their linear parts generate a strongly continuous C0-semigroup of contractions. The nonlinear term is a Carathéodory function. When the semigroup is not compact the nonlinearity has an additional restriction, involving the Hausdorff measure of noncompactness. We provide solutions satisfying nonlocal, multivalued Cauchy conditions. Our approach involves a suitable degree argument. The duality mapping is used for guaranteeing the lack of fixed points of the associated homotopic fields along the boundary of their domain. We apply our results for the investigation of transport and diffusion equations for which we provide the existence of nonlocal solutions.

Adaptive Tikhonov regularization and dynamic control points for accurate shape parameter control of plasmas

Precise control of plasma shape parameters, such as elongation and triangularity is duly needed to achieve high-performance tokamak plasmas, for which we propose adaptive search schemes of (1) optimum regularization parameter for the Tikhonov regularization, and (2) control points to specify the key shape parameters such as elongation and triangularity. Many control points that exceed the number of actuator coils become an ill-conditioned problem, which is successfully resolved by Tikhonov regularization with adaptively optimized free parameters. Furthermore, we develop dynamically changed control points by using the Cauchy Condition Surface scheme, where elongation and triangularity become direct control values. By virtue of both the adaptive Tikhonov regularization scheme and the dynamic control point, we achieved accurate shape control with elongation up to 1.93 and triangularity up to 0.65 in JT-60SA, where the allowable speed for the change of the elongation is 0.1 s−1. We also verified the resilience of our developed our logics to the noises. The sequence of the result will contribute to enhance equilibrium controllability in upcoming JT-60SA experiments and provide the robust shape parameter control scheme for ITER and DEMO.

A Solution of the Navier-Stokes Problem for an Incompressible Fluid with Cauchy Condition

A Solution of the Navier-Stokes Problem for an Incompressible Fluid with Cauchy Condition

Thermal energy storage with tunnels in different subsurface conditions

Thermal energy storage with tunnels in different subsurface conditions

Variational quasi-harmonic maps for computing diffeomorphisms

Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation of variational quasi-harmonic maps to obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps.

Weakly Coupled System of Semi-Linear Fractional θ-Evolution Equations with Special Cauchy Conditions

In this paper, we consider a weakly system of fractional θ-evolution equations. Using the fixed-point theorem, a global-in-time existence of small data solutions to the Cauchy problem is proved for one single equation. Using these results, we prove the global existence for the system under some mixed symmetrical conditions that describe the interaction between the equations of the system.

STATISTICAL CONVERGENCE OF MATRIX SEQUENCES

This paper extends the statistical convergence of real or complex numbers to the real square matrices sequences . In this context, we investigate the relation between the statistical convergence, statistical Cauchy condition, strong Ces\`{a}ro summability of matrices sequences. This leads us to an initial analysis of the Tauberian conditions for the statistical convergence of matrix sequences.

Classical solution of a problem for a system of equations from mechanics with nonsmooth Cauchy conditions

In this article, we study a mixed problem in a quarter-plane for one system of differential equations, which describes vibrations in the string from viscoelastic material, which corresponds to the Maxwell model. At the bottom of the boundary, we pose the Cauchy conditions, and one of them has a discontinuity of the first kind at one point. We set a smooth boundary condition on the lateral boundary. We derive the Klein – Gordon – Fock equation for one function of the studied system. We use the method of characteristics to build the classical solution as a solution of some integral equation. We prove the uniqueness and establish conditions under which a piecewise smooth solution exists. The Cauchy problem is considered the system’s second function. We determine the conditions under which the solution of the system has sufficient smoothness

Automatic simplification of the geometry of a cartographic line using contractive self-mapping – illustrated with an example of a polyline band

Abstract The present article is another attempt to adapt map geometry to automatic digital cartography. The paper presents a method of digital polyline generalisation that uses contractive self-mapping. It is a method of simplification, not just an algorithm for simplification. This method in its 1996 version obtained a patent entitled “Method of Eliminating Points in the Process of Numerical Cartographic Generalisation” – Patent Office of the Republic of Poland, No. 181014, 1996. The first results of research conducted using the presented method, with clearly defined data (without singular points of their geometry), were published in the works of the authors in 2021 and 2022. This article presents a transition from the DLM (Digital Landscape Model) to the DCM (Digital Cartographic Model). It demonstrates an algorithm with independent solutions for the band axis and both its edges. The presented example was performed for the so-called polyline band, which can represent real topographic linear objects such as rivers and boundaries of closed areas (buildings, lakes, etc.). An unambiguous representation of both edges of the band is its axis, represented in DLM, which can be simplified to any scale. A direct consequence of this simplification is the shape of the band representing the actual shape of both edges of the object that is classified in the database as a linear object in DCM. The article presents an example performed for the so-called polyline band, which represents real topographic linear objects (roads, rivers) and area boundaries. The proposed method fulfils the following conditions: the Lipschitz condition, the Cauchy condition, the Banach theorem, and the Salichtchev’s standard for object recognition on the map. The presented method is objective in contrast to the previously used approximate methods, such as generalisations that use graph theory and fractal geometry, line smoothing and simplification algorithms, statistical methods with classification of object attributes, artificial intelligence, etc. The presented method for changing the geometry of objects by any scale of the map is 100% automatic, repeatable, and objective; that is, it does not require a cartographer’s intervention.

A particular solution of a problem for a system of equations from mechanics with nonsmooth Cauchy condition

In this article, we study a mixed problem in a quarter-plane for a system of differential equations, which describes vibrations in a string from viscoelastic material, which corresponds to Maxwell material. At the bottom of the boundary, the Cauchy conditions are specified, and one of them has a discontinuity of the first kind at one point. A smooth boundary condition is set at the side boundary. The Klein – Gordon – Fock equation is derived for one of the system’s functions. We find a particular solution in two ways. The first method builds it in an explicit analytical form (with a continuation of one function), and the second one constructs it as a solution of an integral equation using the method of characteristics (without continuation of one function). Conditions are established under which the solution has sufficient smoothness.

On the application of the Fourier method to solve the problem of correction of thermographic images

The work is devoted to the construction of computational algorithms implementing the method of correction of thermographic images. The correction is carried out on the basis of solving some ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. This problem corresponds to the problem of the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The cylindrical region is bounded by an arbitrary surface and plane. On an arbitrary surface, a temperature distribution is measured (and thus is known). It is called a thermogram and reproduces an image of the internal heat-generating structure. On this surface, which is the boundary of the object under study, convective heat exchange with the external environment of a given temperature takes place, which is described by Newton’s law. This is the third boundary condition, which together with the first boundary condition corresponds to the Cauchy conditions - the boundary values of the desired function and its normal derivative. The problem is ill-posed. In this paper, using the Tikhonov regularization method, an approximate solution of the problem was obtained, stable with respect to the error in the Cauchy data, and which can be used to build effective computational algorithms. The paper considers algorithms that can significantly reduce the amount of calculations.

Numerical Solution of an Inviscid Burger Equation with Cauchy Conditions

This article deals with the solution of the Cauchy problem for the Inviscid Burger equation. Various numerical techniques like Upwind non Conservative, Upwind Conservative, Lax Friedrich, Lax Wendorff, and Mac Cormack, are used to solve initial-value problems for the Inviscid Burger equation. Through various model problems, the efficiency and accuracy of the techniques have been shown via the graphical and tabulated form with the exact solution.

An analytical and numerical study of free boundary dynamics for an isolated mass of a self-gravitating gas

The paper considers an evolution of the free boundary of a finite volume of a self-gravitating ideal gas moving in the vacuum. Unsteady flows are described by a phenomenological mathematical model, which has the form of a system of nonlinear Volterra integro-differential equations written in Eulerian coordinates. The gas volume moves in a force field generated by the Newtonian potential in general form. The boundary conditions are specified on the free gas-vacuum boundary, which is a priori unknown and determined simultaneously with gas flow construction. Conversion to Lagrangian coordinates allows us to reduce the original problem to an equivalent one, which consists of Volterra integral equations and the continuity equation in Lagrangian form, with Cauchy conditions specified for all these equations. Therefore, the application of Lagrangian coordinates makes it possible, in particular, to eliminate the unknown boundary. The theorem of the existence and uniqueness of the solution in the space of infinitely differentiable functions is proved for this problem. The free boundary is determined as an image of the surface bounding the gas-filled region in reverse transition. Herewith, the method for studying the free boundary is similar to the approach that the authors apply to studying the dynamics of the rarefied mass of a self-gravitating gas. Nu-merical calculations of gas flow are made, including the construction of the free gas-vacuum boundary. The influence of gravitation and the initial gas particle velocity on the formation of gas cloud configuration in the vacuum and on cloud evolution is studied. The results are of interest in terms of solving relevant astrophysical and cosmogonic problems.