Solution Of Problem Research Articles

Exact and numerical solutions of a free boundary problem with a reciprocal growth law

Abstract A two-dimensional free boundary problem is formulated in which the normal velocity of the boundary is proportional to the inverse of the gradient of a harmonic function $T$. The field $T$ is defined in a simply connected region which includes the point at infinity where it has a logarithmic singularity. The growth problem in which the boundary expands outwards is formulated both in terms of the Schwarz function of the boundary and a Polubarinova–Galin equation for the conformal map of the region from the exterior of the unit disk. An expanding free boundary is shown to be stable and explicit solutions for growing ellipses and a class of polynomial lemniscates are derived. Numerical solution of the Polubarinova–Galin equation is used to compute the evolution of the boundary having other initial shapes.

Diastereo-Divergent Synthesis of Ring A-lactones Derived From Cholesterol and Diosgenin. A Convenient Solution for an Old Problem. NMR and X-Ray Characterization

A convenient protocol for the synthesis of A-ring lactones derived from cholesterol was developed. A lactone reduction-lactol separation-reoxidation sequence, applied to the inseparable mixture of diastereomeric lactones allows the production of multigram amounts of each lactone in pure form. The same sequence applied to A-ring lactones derived from diosgenin produced similar results. A detailed NMR characterization of all the obtained lactones is also provided. X-ray diffraction corroborated the structure of the obtained compounds.

Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations

Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations

The poroelastic layer with an axisymmetric cylindrical hole under different types of loading

The exact solution of the poroelastic axisymmetric problem for a layer with a cylinderical hole is constructed under assumptions of Biot's model. The calculations provided by derived explicit formulas for full stress and pore pressure allow to state some important dependencies between the poroelastic stress state of the layer and type of poroelastic materials, loading types and height of the layer.

Two-point boundary-value problems for differential equations with generalized piecewise-constant argument

UDC 517.9 We consider a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept of general solution. The interval is partitioned with regard for the singularities of the argument. The values of the solution at the interior points of the partition are regarded as additional parameters, and the differential equation is transformed into a system of Cauchy problems with parameters on subintervals of the partition. By using the solutions of these problems, we obtain a new general solution of the differential equation with piecewise-constant argument and establish its properties. The new general solution, boundary conditions, and the conditions of continuity of the solution at the interior points of the partition are used to construct a linear system of algebraic equations for the introduced parameters. The coefficients and the right-hand side of the system are found as a result of the solution of Cauchy problems for linear ordinary differential equations on the subintervals of the partition. It is shown that the solvability of the boundary-value problem is equivalent to the solvability of the constructed system. We propose algorithms of the parametrization method for solving the analyzed boundary-value problem and establish necessary and sufficient conditions for the well-posedness of this problem.

Input‐output linearization and decoupling of mechanical control systems

AbstractIn this article, we present the problem of simultaneous input‐output feedback linearization and decoupling (noninteracting) for mechanical control systems with outputs. We show that the natural requirement of preserving mechanical structure of the system and of transformations imposes supplementary conditions when compared to the classical solution of the same problem for general control systems. These conditions can be expressed using objects on the configuration space only. We illustrate our results with several examples of mechanical control systems.

On the rate–distortion–perception–semantics tradeoff in low-rate regime for lossy compression

This paper formulates a multi-letter information theoretical optimization problem to characterize the tradeoff between rate, distortion, perception, and semantics. It turns out that our formulation is related to existing works, such as the rate–distortion theory, utility-privacy tradeoff, and rate–distortion–perception tradeoff. In this paper, we solve this problem in the low-rate regime by applying a local geometric approach, which allows us to express the optimal solutions analytically. By using such an approach, we obtain the analytical solution of the multi-letter optimization problem, which demonstrates the optimal rate–distortion–perception–semantics tradeoff, and characterizes the gain from increasing the number of letters as well as the limiting performance. Moreover, our result also validates the single-letter optimality of the scenarios above, which reveals the information theoretical insights of system and algorithm designs.

Stable difference scheme for the numerical solution of the source identification problem for hyperbolic equations

UDC 517.9 We present a stable difference scheme of the second order of accuracy for a one-dimensional hyperbolic equation. The well-posedness of the difference scheme is established. Numerical results are presented.

Риск-ориентированный анализ безопасности и защищенности промышленных объектов на основе характеристик прочности и ресурса

The concept of computation and experimental determination of safety and protectability of industrial facilities on parameters of risk of initiation of refusals, accidents and disasters taking into account possibility of regular and emergency situations is considered. The scientifically substantiated, normalized and controlled parameters of risks estimated by criteria of strength, life-time and survivability are its basis. Risks at the same time are estimated on probabilities of initiation of emergency situations and losses accompanying them at their interdependence. It is shown that generally safety is defined by ability of installations not to pass into emergency and catastrophic states with drawing significant damages to the person, a technosphere and environment; the risk is defined by probability of initiation on installation of adverse situations and possible losses from these situations; protectability is defined by ability of installation to resist to initiation and an evolution of adverse situations (refusals, accidents, disasters) in regular and emergency conditions. The new perspective direction for maintenance of conditions of safe service of industrial facilities is that which initially forms the level of their protectability, and this protectability is defined by the formed and acceptable risks. In this case the solution of basic problems of protectability, safety and risk has to be guided surely by the solution of problems of strength, a life-time, survivability in their direct interaction This circumstance is based on a quantitative estimation of conditions of service damages accumulation and on the analysis of conditions of transition from regular to the damaged and limit states of installation taking into account criteria of strength, fatigue life and survivability. The solution of practical problems of maintenance of strength, life-time and safety within of strength, life-time and risk-focused design assumes determination of nominal and local stresses from service loadings in the most loaded zones with use of characteristics of mechanical properties of constructional materials. In the paper it is put and solved the new scientific and almost important problem about risk-focused definition, maintenance, increase and a standartization of complex safety and protectability of technosphere installations by criteria of the acceptable and controlled risks.

Regularization errors introduced by the one-fluid formulation in the solution of two-phase elliptic problems

This manuscript discusses the structure of the errors in the solution of elliptic problems introduced by the regularization of the fluid properties discontinuity in a small region of finite size. By using a multiscale approach, the problem of the error calculation is splitted into an outer problem, where the regularization region is replaced by a sharp interface, and an inner local one dimensional problem that eventually imposes effective jump conditions across the interface for the outer problem. Except in some particular cases, the use of regularization techniques introduces first order errors in the solution imposing an error flux jump that is proportional to the tangential Laplacian of the averaged solution and an error jump that is proportional to the normal flux, both multiplied by a prefactor that depends on the averaging rule used. In general, the optimal averaging procedure is shown to depend on the structure of the problem at hand. The errors introduced by standard arithmetic and harmonic averages are obtained for various analytical and numerical examples which are used to discuss the nature of the errors introduced and the importance of first order errors in the solution. The influence of the ratio between the regularization thickness and the grid size is also investigated in numerical implementations of the one-fluid model.

DETERMINATION OF A KERNEL IN A NONLOCAL PROBLEM FOR THE TIME-FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

In this work, we consider an inverse problem of determining the kernel of a fractional diffusion equation. The backward problem is the initial-boundary/value problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the kernel, an overdetermination condition of the integral form is specified for the solution of the backward problem. Using the Fourier method and an ordinary fractional differential equation with a non-local boundary condition, the inverse problem is reduced to an equivalent problem. Further by using the fixed point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.

The irregularity strength of dense graphs — on asymptotically optimal solutions of problems of Faudree, Jacobson, Kinch and Lehel

The irregularity strength of a graph G, s(G), is the least k such that there exists a {1,2,…,k}-weighting of the edges of G attributing distinct weighted degrees to all vertices, or equivalently the least k allowing the creation of a multigraph with nonrecurring degrees by blowing each edge e of G to at most k copies of e. In 1991 Faudree, Jacobson, Kinch and Lehel asked for the optimal lower bound for the minimum degree of a graph G of order n which implies that s(G)≤3. More generally, they also posed a similar question regarding the upper bound s(G)≤K for any given constant K. We provide asymptotically tight solutions of these problems by proving that such optimal lower bound is of order 1K−1n for every fixed integer K≥3.

АЭРОДИНАМИКА МОДЕЛИ ЖЕЛЕЗНОДОРОЖНОГО ГРУЗОВОГО ВАГОНА ПРИ РАЗНЫХ УГЛАХ АТАКИ ВОЗДУШНОГО ПОТОКА

The problem solution is considered for an airstream flow around a rectangular parallelepiped railway car model. The results of computer modeling in ANSYS CFX software system of air flow aerodynamics at its deviation from the longitudinal axis of the car are given. The k-ε turbulence model is used to close the Navier–Stokes equations averaged by Reynolds. The distribution diagrams of the flow velocities and pressures on the frontal and side surfaces of the vehicle are obtained. The values of aerodynamic drag coefficients of the car depending on the attack angle are determined. It is shown that when the attack angle increases from 0 to 10°, the aerodynamic coefficient changes nonlinearly, and this increase corresponds to the experimental values. The developed numerical modeling technique makes it possible to analyze the airflow around both railways rolling stock and automobiles.

Merging preferences into the best solution seeking for many-objective optimization problems

Finding out a best solution of the multi-objective optimization problem (MOP) or many-objective optimization problem (MaOP) in the user’s view, is a significant and practical problem. The multi-objective evolutionary algorithms (MOEAs) are applied to solve the MaOPs. Considering many conflicting objectives, they would generate numerous nondominated solutions. Merging the preferences from the user to objectives is an effective way to measure the nondominated solutions and find out the best one (a posterior approach). But the preference relations may be contradictory and highly sparse. In this paper, we propose a process of finding out a best solution with preference relations among attributes (BSPA). In which, we emphasize to test the contradiction and ease the high sparsity of preference relations, then an available preference degree matrix is obtained and merged into the selection of best solution. Based on the proposed process, three methods of NSGA-II-BSPA, P-NSGA-II and P-GA are proposed to solve the best solution of an MOP or MaOP. The experiments are conducted, and the metrics of contradiction probability, feasibility and availability are proposed to measure the BSPA and the vigorous results are obtained. At last, three case studies are adopted to verify the proposed methods’ effectiveness and efficiency. Meanwhile, the hypothesis tests are conducted and the results indicate that the obtained best solutions of proposed methods are satisfiable.

Asymptotics of the Solution of the Initial Boundary Value Problem for the One-Dimensional Klein–Gordon Equation with Variable Coefficients

Asymptotics of the Solution of the Initial Boundary Value Problem for the One-Dimensional Klein–Gordon Equation with Variable Coefficients

IDENTIFICATION OF POSITIONS OF SEPARATED RIGID INCLUSIONS BY BOUNDARY MEASUREMENTS FOR A SIGNORINI PROBLEM

A variational inequality describing an elastic body with a finite set of rigid inclusions is considered. The Signorini condition is imposed on a part of the boundary of the body. On the other part a homogeneous Dirichlet boundary condition is specified. The inclusions are arranged such that distance between any two inclusions is not less than a given positive number. All inclusions are located at a nonzero distance from the outer boundary. An inverse problem is investigated, which consists in identification of positions of the rigid inclusions from the measurement of displacements on an observation boundary. Continuous dependency of the solution of the forward problem on parameters of inclusions’ location and rotation is established. This provides existence of a solution for the inverse identification problem.

Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation

In this paper, an inverse problem of a two-dimensional diffusion equation is considered. The purpose here includes recovering not only the source term but also the initial value from given observations at two fixed times t=T1 and t=T2. The uniqueness of the solutions for the inverse problem is given. Instead of the classical Tikhonov regularization strategy, we propose an iterated fractional Tikhonov regularization method to solve this problem, an a priori and an a posterior parameter selection rules and corresponding convergence rates are derived. For verification of the theoretical estimates, several numerical examples are constructed and compared with the standard iterated Tikhonov regularization method.

Vascular Access: Going Back to Old Solutions for The Same Old Problems

Vascular Access: Going Back to Old Solutions for The Same Old Problems

Predictive modeling of shallow tunnel behavior: Leveraging machine learning for maximum convergence displacement estimation

Accurate prediction of maximum convergence in unsupported, shallow tunnel construction is crucial for optimizing the lining and ensuring tunnel safety. Machine learning (ML) algorithms, especially through boosting techniques, enable effective solution of complex engineering problems and demonstrate their capabilities in problem solving and optimization. In this study, the FLAC 3D package was used to create a robust and validated database of 954 datasets. Five tree-based ML algorithms, including extreme gradient boosting (XGBoost), adaptive boosting (AdaBoost), gradient boosting machine (GBM), histogram-based gradient boosting (HGB) and categorical boosting (CatBoost), were used to predict the maximum convergence displacement for unsupported shallow tunnels. For the test dataset, XGBoost outperformed the other models with an excellent coefficient of determination of 0.9633, a minimum mean absolute error of 0.0021 and a low root mean squared error of 0.00725. HGB followed closely behind, and GBM and CatBoost showed strong performances, while Adaboost was less effective. The superior performance of XGBoost highlights its effectiveness in predicting maximum convergence in shallow tunnels. An in-depth sensitivity analysis within the XGBoost model showed the significant influence of soil elastic modulus on the maximum convergence displacement in unsupported tunnels. The remarkable results achieved by the XGBoost algorithm on our complex tunnel convergence predictions illustrate the profound ability of ML to tackle complicated geotechnical challenges. This interdisciplinary collaboration demonstrates the potential of advanced algorithms to improve safety and efficiency in construction, underlining the crucial role of technology in tackling complex problems and establishing a new paradigm for innovation in the field.

Prospects on the detection of solar dark photons by the International Axion Observatory

Dark (hidden) photons are widely recognised as well motivated candidates for physics beyond the standard model, and have been invoked for the solution of several outstanding problems, including to account for the dark matter in the universe. In this paper, we consider a simple model for dark photons, which is coupled to ordinary matter only through kinetic mixing with ordinary photons. Within this framework, we calculate the flux of solar dark photons on Earth and revise the potential to detect it with the next generation of axion helioscopes, particularly with the International AXion Observatory (IAXO). This paper extends on previous theoretical analyses in two main ways. Firstly, it includes a more complete analysis of the possible sources of dark photons from the sun, including the contribution of the solar magnetic field and of nuclear processes, and secondly it includes predictions on the parameter space accessible in the gas-filled phase of IAXO.