One-dimensional Problem Research Articles

THE EXISTENCE OF GENERALIZED SOLUTION TO ONEDIMENSIONAL MIXED PROBLEM FOR ONE CLASS OF THIRD ORDER NONILINEAR PSEUDOPARABOLIC EQUATIONS

Abstract. This work is dedicated to the study of existence in small of generalized solution of onedimensional mixed problem for one class of third order nonlinear pseudoparabolic equations. Conception of generalized solution for mixed problem under consideration is introduced. After applying Fourier method, the solution of original problem is reduced to the solution of some countable system of nonlinear inteqro-differential equations in unknown Fourier coefficients of the sought solution. Besides, existence theorem in small of generalized solution of the mixed problem is proved by contracted mappings principle.

ДЕЯКІ КРАЙОВІ ЗАДАЧІ ДРОБОВО-ДИФЕРЕНЦІЙНОЇ ФІЛЬТРАЦІЙНОЇ ДИНАМІКИ ЩОДО БІПАРАБОЛІЧНОЇ МАТЕМАТИЧНОЇ МОДЕЛІ

Closed-form solutions are obtained to some one-dimensional boundary-value problems for modeling anomalous filtration dynamics in a layered geoporous medium, posed within the framework of the fractional-differential generalization of the biparabolic evolutionary partial differential equation of the fourth order. In particular, the formulation and solution of the direct and inverse model boundary-value problems of geofiltration dynamics based on the mathematical model with conjugation conditions are presented, and the conditions of the existence of regular solutions to these problems are defined. Keywords: mathematical modeling, fractional-differential dynamics of geofiltration processes, nonclassical models, biparabolic evolutionary equation, the fractional-differential analog of the biparabolic evolutionary equation, nonstationary boundary-value problems on a finite interval, direct and inverse problems, conjugation conditions, closed-form solutions.

A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results

A one-dimensional two-phase Stefan problem for the melting of a semi-infinite material with a power-type temperature-dependent thermal conductivity was considered. The assumption of taking thermal parameters as functions of temperature found its basis in physical and industries applications, allowing for a more precise and realistic description of phase change processes. By imposing a Dirichlet condition at the fixed face, a theoretical and approximate study was developed. Through a similarity transformation, an equivalent ordinary differential problem was obtained from which an integral problem was deduced. The existence of at least one analytical solution was guaranteed by using the Banach fixed point theorem. Due the unavailability of an analytical solution, a heat balance integral method was applied, assuming a quadratic temperature profile in space, to simulate temperature variations and the location of the interface during the melting process. For constant thermal conductivity, results can be compared with the exact solution available in the literature to check the accuracy of the approximate method.

Necessary and sufficient conditions for one-dimensional variational problems with applications to elasticity

This paper deals with necessary and sufficient conditions for weak and strong minimizers of functionals Φ ( u ) = ∫ a b f ( x , u ( x ) , u ′ ( x ) ) d x , where u ∈ C 1 ( [ a , b ] , R N ) . We first derive conditions which are simpler than the known ones, and then apply them to several particular problems, including stability problems in the elasticity theory. In particular, we solve some open problems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices and buckling solutions in three dimensions, Nonlinearity 27(2014), 2841–2867] by finding optimal conditions for the stability of a naturally straight Kirchhoff rod under various types of endpoint constraints.

One-dimensional packaging with method of modified ant colony

The study explores the use of the ant algorithm for solving one-dimensional packing problems. It discusses the structure of the decision search graph, the search process on the graph, methods of pheromone deposition and evaporation. The study employs a cyclic method of ant systems. A new strategy for finding effective solutions is proposed based on modification and hybridization of the canonical representations of the particle swarm algorithm. The paradigm of a swarm of transforming chromosomes is described, which provides the ability to represent solutions in the affine space of chromosomes with integer parameter values. The mechanisms of chromosome transformation in the affine space to increase the weight of affine bonds are considered. Experimental research was conducted on an IBM PC, showing improved results compared to existing algorithms. The proposed approach can be useful for solving packaging problems and optimizing the distribution of one-dimensional objects.

On a one-dimensional time-fractional Stefan problem

On a one-dimensional time-fractional Stefan problem

A mathematical theory of super-resolution and two-point resolution

Abstract This paper focuses on the fundamental aspects of super-resolution, particularly addressing the stability of super-resolution and the estimation of two-point resolution. Our first major contribution is the introduction of two location-amplitude identities that characterize the relationships between locations and amplitudes of true and recovered sources in the one-dimensional super-resolution problem. These identities facilitate direct derivations of the super-resolution capabilities for recovering the number, location, and amplitude of sources, significantly advancing existing estimations to levels of practical relevance. As a natural extension, we establish the stability of a specific $l_{0}$ minimization algorithm in the super-resolution problem. The second crucial contribution of this paper is the theoretical proof of a two-point resolution limit in multi-dimensional spaces. The resolution limit is expressed as $$\begin{align*}\mathscr R = \frac{4\arcsin \left(\left(\frac{\sigma}{m_{\min}}\right)^{\frac{1}{2}} \right)}{\Omega} \end{align*}$$ for ${\frac {\sigma }{m_{\min }}}{\leqslant }{\frac {1}{2}}$ , where ${\frac {\sigma }{m_{\min }}}$ represents the inverse of the signal-to-noise ratio ( ${\mathrm {SNR}}$ ) and $\Omega $ is the cutoff frequency. It also demonstrates that for resolving two point sources, the resolution can exceed the Rayleigh limit ${\frac {\pi }{\Omega }}$ when the signal-to-noise ratio (SNR) exceeds $2$ . Moreover, we find a tractable algorithm that achieves the resolution ${\mathscr {R}}$ when distinguishing two sources.

Analysis of a one-dimensional nonlocal thermoelastic problem

Analysis of a one-dimensional nonlocal thermoelastic problem

Convergence of digitized-counterdiabatic QAOA: circuit depth versus free parameters

Recently, digitized-counterdiabatic (CD) quantum approximate optimization algorithm (QAOA) has been proposed to make QAOA converge to the solution of an optimization problem in fewer steps, inspired by Trotterized CD driving in continuous-time quantum annealing. In this paper, we critically revisit this approach by focusing on the paradigmatic weighted and unweighted one-dimensional MaxCut problem. We study two variants of QAOA with first and second-order CD corrections. Our results show that, indeed, higher order CD corrections allow for a quicker convergence to the exact solution of the problem at hand by increasing the complexity of the variational cost function. Remarkably, however, the total number of free parameters needed to achieve this result is independent of the particular QAOA variant analyzed for the problems considered.

Mass Sensing by Symmetry Breaking and Mode Localization in a System of Parametrically Excited Microbeam Resonators

In this work, we study the nonlinear dynamics of a mode-localized mass detector. A system of equations is obtained for two weakly coupled beam resonators with an alternating electric current flowing through them and taking into account the point mass on one of the resonators. The one-dimensional problem of thermal conductivity is solved, and a steady-state harmonic temperature distribution in the volume of the resonators is obtained. Using the method of multiple scales, a system of equations in slow variables is obtained, on the basis of which instability zones of parametric resonance, amplitude-frequency characteristics, as well as zones of attraction of various branches, are found. It is shown that in a completely symmetrical system (without a deposited particle), the effect of branching of the antiphase branch of the frequency response is observed, which leads to the existence of an oscillation regime with different amplitudes in a certain frequency range. In the presence of a deposited particle, this effect is enhanced, and the branching point and the ratio of the amplitudes of oscillations of the resonators depend on the mass of the deposited particle.

On the collision of railcars as an interaction of nonlinear shock wave-like perturbations

The article examines the collision of railcars based on the solution of hydrodynamic equations using shock waves during the transition from weak nonlinearity to perturbations of arbitrary amplitude. Several new mathematical issues have been set for the development of high-speed transport, which can be solved in the framework of hydrodynamics to describe the process of hydrodynamics, creating effective rolling stock dampers. which requires the improvement and development of the corresponding mathematical apparatus. In this work, we use a hydrodynamic approach to find the density distributions of matter during railcar collisions at high speeds, which is important in light of the problems of high-speed transport. In our approach, we found an analytical solution to the obtained hydrodynamic equations for the one-dimensional case. The equations under study were obtained taking into account nonequilibrium processes. To find a solution to the hydrodynamic equations, the shock wave approximation is used, similar to the soliton solutions we considered earlier. Taking into account possible deviations from the results of a one-dimensional problem is considered. Such a reduction of solutions of hydrodynamic equations to shock waves has not been considered previously and may be of interest for a wide variety of applied problems. The resulting consideration of railcar collisions is important for solving problems of transport safety and technospheric safety.

A fundamental sequences method with time-reduction for one-dimensional lateral Cauchy problems

A fundamental sequences method is derived for the numerical solution of an ill-posed one-dimensional lateral Cauchy problem for a hyperbolic damped wave equation, including as a special case the parabolic heat equation. Either the Laguerre transform or the Houbolt finite difference scheme is applied to reduce the time-dependent lateral Cauchy problem to a sequence of second-order ordinary differential equations (ODEs) with function values and derivatives specified at the right endpoint of a finite space interval. A set of fundamental solutions is constructed, termed a fundamental sequence, to the differential equations. The solution of the obtained ODEs is approximated by a linear combination of elements in the fundamental sequence. Source points are placed outside of the solution interval in space, and by collocating at the endpoints of this interval a sequence of linear equations is obtained for finding the unknown coeficients. Tikhonov regularization is used to render a stable solution to the obtained systems of linear equations. Numerical results both for the parabolic and hyperbolic case confirm the efficiency of the proposed method including noisy data. The presented results complement the higher-dimensional case initiated in our previous researches.

A Dimension-Reduced Line-Element Method to Model Unsaturated Seepage Flow in Porous Media

Contrary to the continuum hypothesis, which averages water flow across the entire domain, including both grains and pores, the line-element model concentrates unsaturated flow in the pore space in the intermediate region of horizontal and vertical channels. The flux equivalent principle is used to deduce the equivalent unsaturated hydraulic conductivity, the flow velocity and the continuity equations. It is found that the relative hydraulic conductivities derived from the line-element model and the continuum model are identical. The continuity equations in the two models are also similar, except that the coefficient in the water content term is half that in the line-element model. Thus, the unsaturated flow problem in porous media is transformed into a one-dimensional problem. A dimension-reduced finite line-element method is proposed that includes a complementary algorithm for Signorini’s-type boundary conditions involving the seepage-face boundary and the infiltration boundary. The validity of the proposed model is then proved by good agreement with analytical, experimental and simulated results for one-dimensional infiltration in a vertical soil column, unsaturated flow in a sand flume with drainage tunnels, and transient unsaturated flow water-table recharge in a soil slab, respectively. In general, the proposed method has good computational efficiency, especially for smaller mesh sizes and short time intervals.

Studying the features of simulating fluid movement in porous media

The study of fluid flow in porous media, differing from traditional pipe flow is crucial for developing efficient methods in oil extraction and minerals, considering the challenges associated with ambiguous flows and diverse porous structures. This paper addresses the complexity of investigating fluid flow in porous media, a phenomenon significantly distinct from fluid movement in pipes. Emphasis is placed on the absence of clearly defined flow tubes in porous media, complicating measurements, and analysis. The study introduces a new approach incorporating both analytical and numerical methods, applied to various porous media. The research proposes a mathematical model based on laws and correlations to describe systems, including concepts such as permeability, flow velocity potential, characteristics of single-phase and multiphase systems, and fluid compressibility. Special attention is given to the characteristics of oil reservoirs, determined based on fluid properties in the reservoir, including porosity and saturation, as assessed by engineers. Numerical results represent fluid displacement in a flat channel and a onedimensional problem in a porous medium, performed using finite-difference approximation of equations with an explicit scheme. The numerical results of this model were implemented in the Matlab software.

An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition

Abstract In this article, we develop an efficient Legendre-Galerkin approximation based on a reduced-dimension scheme for the fourth-order equation with singular potential and simply supported plate (SSP) boundary conditions in a circular domain. First, we deduce the equivalent reduced-dimension scheme and essential pole condition associated with the original problem, based on which a class of weighted Sobolev spaces are defined and a weak formulation and its discrete scheme are also established for each reduced one-dimensional problem. Second, the existence and uniqueness of the weak solution and the approximation solutions are given using the Lax-Milgram theorem. Then, we construct a class of projection operators, give their approximation properties, and then prove the error estimates of the approximation solutions. In addition, we construct a set of effective basis functions in approximate space using orthogonal property of Legendre polynomials and derive the equivalent matrix form of the discrete scheme. Finally, a large number of numerical examples are performed, and the numerical results illustrate the validity and high accuracy of our algorithm.

DIRECT SOLUTION OF THE DYNAMICAL ELASTICITY PROBLEM FOR A QUARTER SPACE

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load is concentrated at the point on another face. The problem was solved by the direct application of the integral Laplace and Fourier transforms to the motion equations and the boundary conditions. This operation leads to the one-dimensional vector inhomogeneous boundary value problem with respect to unknown displacement’s transformants. The problem was solved using the matrix differential calculus. A fundamental matrix and a decreasing solution to the corresponding homogenous matrix equation were constructed with a basic residue theorem. A singular integral equation was obtained in the process by satisfying unrealized boundary condition. Weakly convergent part of the equation was summed up. The behavior of the unknown function had been analyzed based on its mechanical sense. The form of unknown function was expressed as a series on Laguerre polynomials. The original displacements’ field was found after an application the inverse integral transforms.

Boundary value problems of the dynamics of thermoelastic rods and their solutions

We consider spatially one-dimensional boundary value problems of uncoupled thermoelasticity, which can be used to study various rod structures under thermal heating conditions. As is known, rod structures are connecting and transmission links of various parts of the machines and mechanisms. Here we propose a unified technique for solving various boundary value problems typical for practical applications. The problems of determining the thermally stressed state of a thermoelastic rod under various boundary conditions at its ends and acting power and heat sources along the entire length of the rod are considered. Based on the method of generalized functions, generalized solutions of non-stationary and stationary direct and semi-inverse boundary value problems under the action of power and heat sources of various types, including stationary sources of periodic oscillations, are constructed. Acting sources can also be specified by singular generalized functions, under various boundary conditions at the ends of the rod. Shock elastic waves that arise in such structures under the action of shock loads are considered. Regular integral representations of generalized solutions are obtained, which provide an analytical solution to the posed boundary value problems. The peculiarity of the constructed solutions makes them convenient for studying network thermoelastic systems that can be modeled by thermoelastic graphs.

One-Dimensional Inverse Problem for Nonlinear Equations of Electrodynamics

For the system of nonlinear electrodynamics equations, we consider the problem of determining the medium conductivity coefficient multiplying the nonlinearity. It is assumed that the permittivity and permeability are constant and the conductivity depends only on one spatial variable, with this conductivity being zero on the half-line x . For a mode in which only two electromagnetic field components participate, the wave propagation process caused by the incidence of a plane wave with a constant amplitude from the domain x0 onto an inhomogeneity localized on the half-line x0 is considered. With a given conductivity coefficient, the conditions for the solvability of the direct problem and the properties of its solution are studied. To solve the inverse problem, the trace of the electrical component of the solution of the direct problem is specified on a finite segment of the axis x=0. A theorem on the local existence and uniqueness of the solution of the inverse problem is established, and a global estimate of the conditional stability of its solutions is found.

On a set of tests for numerical methods of integrating differential equations, based on the Calogero system

Based on the completely integrable Calogero dynamical system, which describes the one-dimensional many-body problem, a tool for testing difference schemes has been developed and implemented in the original fdm package integrated into the Sage computer algebra system. This work shows how the developed tools can be used to examine the behavior of numerical solutions near the collision point and how to study the conservatism of the difference scheme. When detecting singularities using Alshina’s method, a difficulty was discovered associated with false order fluctuations. One of the main advantages of this set of tests is the purely algebraic nature of the solutions and integrals of motion.

Minimizing the total waste in the one-dimensional cutting stock problem with the African buffalo optimization algorithm.

The one-dimensional cutting-stock problem (1D-CSP) consists of obtaining a set of items of different lengths from stocks of one or different lengths, where the minimization of waste is one of the main objectives to be achieved. This problem arises in several industries like wood, glass, and paper, among others similar. Different approaches have been designed to deal with this problem ranging from exact algorithms to hybrid methods of heuristics or metaheuristics. The African Buffalo Optimization (ABO) algorithm is used in this work to address the 1D-CSP. This algorithm has been recently introduced to solve combinatorial problems such as travel salesman and bin packing problems. A procedure was designed to improve the search by taking advantage of the location of the buffaloes just before it is needed to restart the herd, with the aim of not to losing the advance reached in the search. Different instances from the literature were used to test the algorithm. The results show that the developed method is competitive in waste minimization against other heuristics, metaheuristics, and hybrid approaches.