One-dimensional Boundary Value Problem Research Articles

A Boundary Value Problem with Strong Degeneracy and Local Splines

A new method for solving the singular one-dimensional boundary value problem with a strong degeneracy is proposed in this paper. In the case of the strong degeneration of the differential equation, the boundary condition is set only at one end of the interval. This method is based on the use of the polynomial and non-polynomial Lagrangian and Hermitian type local splines and the variational method. The use of splines of Hermitian type with the first level is convenient if it is needed to obtain simultaneously a solution and the first derivative of the solution at the grid nodes. Next, it is possible to construct the solution between the grid nodes using the same spline approximation formulas. The non-polynomial splines help us to construct a more accurate solution. The results of solving a one-dimensional boundary value problem with strong degeneracy are presented in this paper.

Generalized solutions of boundary value problems ofdynamics of thermoelastic rods

We consider spatially one-dimensional boundary value problems of uncoupled thermoelasticity, which can be used to study various rod structures under thermal heating conditions. This model well describes thermodynamic processes at low strain rates. Here, a unified methodology has been developed for solving various boundary value problems typical of practical applications. The task is set to determine the thermo-stressed state of a thermoelastic rod under various boundary conditions at its ends, as well as the acting forces and heat sources along the entire length of the rod.Shock elastic waves arising in such structures under the action of shock loads are considered. Based on the method of generalized functions, generalized solutions of non-stationary and stationary direct and semi-inverse boundary value problems in the class of generalized vector functions of slow growth are constructed. Regular integral representations of generalized solutions are also given, which provide analytical solutions to the posed boundary value problems. The peculiarity of the constructed solutions makes them convenient for engineering applications because make it possible to study the influence of each boundary condition, as well as the acting power and heat sources separately, which is very important when assessing the strength properties of rod structures.

High multiplicity of positive solutions in a superlinear problem of Moore–Nehari type

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in Moore and Nehari (1959). Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number κ≥1 of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter λ and on κ.Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as λ↓−∞, in the regions where the weight vanishes. Our result leads us to conjecture the existence of 2κ+1−1 solutions for sufficiently negative λ. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in λ and the profiles of positive solutions.Finally, we give additional numerical results suggesting that high multiplicity also holds true for a much larger class of weights, even arbitrarily close to situations where there is uniqueness of positive solutions.

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Stress state modeling of non-circular orthotropic hollow cylinders under different types of loading

Based on a spatial model of the linear theory of elasticity, using an unconventional approach of the reduction of the original three-dimensional boundary value problem described by a system of partial differential equations with variable coefficients to a one-dimensional boundary value problem for a system of ordinary differential equations with constant coefficients, the problem of finding the dimensional stress of hollow elliptic orthotropic cylinders under the influence of various types of loading has been solved under certain boundary conditions at the orientation plane. Reducing the dimensionality of the original problem is carried out using analytical methods of separating variables in two coordinate directions in combination with the method of approximating functions by discrete Fourier series. The one-dimensional boundary value problem is solved by the stable numerical method of discrete orthogonalization.

Robust error estimates of PINN in one-dimensional boundary value problems for linear elliptic equations

<p>In this paper, we study physics-informed neural networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of the coefficients. In particular, we rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach. Deriving $ L^2 $-contraction estimates, we show that the error, defined as the mean square of the differences between the true solution and our trial function at the sample points, is dominated by the training loss. Furthermore, we show that as the quantities of the coefficients for the differential equation increase, the error-to-loss ratio rapidly decreases. Our theoretical and experimental results confirm the robustness of the error regardless of the quantities of the coefficients.</p>

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.

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.

Визначення двовимірних нестаціонарних температурних полів у плитах та панелях з плоскопаралельними межами за наявності джерел тепла

Non-stationary two-dimensional problems of thermal conductivity for plates and panels with plane-parallel boundaries in the presence of volume-distributed non-stationary heat sources are formulated. A method of constructing a solution to the formulated heat conduction problems for the considered bodies is proposed. The technique uses the approximation of the temperature distribution in both elements by the thickness variable by a cubic polynomial. The coefficients of the approximation polynomial are given through the integral over the thickness variable temperature characteristics and the conditions for the boundary values of the temperature on the outer surfaces of the plate and panel. As a result, the initial two-dimensional initial boundary value problems for the temperature for the plate and panel are reduced to one-dimensional initial boundary value problems for the integral temperature characteristics. To construct the solution of the initial-boundary value problem for the integral characteristics of the temperature in the case of a plate infinite in longitudinal and transverse coordinates, the integral Laplace transform in time and the integral Fourier transform in the longitudinal coordinate were used. The solution of the problem on the integral temperature characteristics in the case of the panel is found using the integral Laplace transform in time and the finite integral transform in the transverse coordinate. Expressions of integral temperature characteristics for the plate and panel are obtained in the form of convolutions of functions corresponding to homogeneous solutions of initial-boundary value problems for integral temperature characteristics and functions describing the available non-stationary heat sources in these bodies and given surface temperature values. The general solutions of the two-dimensional initial boundary value problems of thermal conductivity for slabs and panels are recorded for the presence of arbitrarily variable spatial coordinates of non-stationary heat sources and the conditions of convective heat exchange with the external environment on the surfaces of the considered bodies.

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

Purpose This paper aims to discuss a new form of the Adomian decomposition technique for the numerical treatment of Bratu’s type one-dimensional boundary value problems (BVPs). Moreover, the author also addresses convergence and error analysis for the completeness of the proposed technique. Design/methodology/approach First, the author discusses the standard Adomian decomposition method and an algorithm based on Duan’s corollary and Rach’s rule for the fast calculation of the Adomian polynomials. Then, a new form of the Adomian decomposition technique is present for the numerical simulation of Bratu’s BVPs. Findings The reliability and validity of the proposed technique are examined by calculating the absolute errors of Bratu’s problem for some different values of Bratu parameter λ. Numerical simulation demonstrates that the proposed technique yields higher accuracy than the Bessel collocation and other known methods. Originality/value Unlike the other methods, the proposed technique does not need linearization, discretization or perturbation to handle the non-linear problems. So, the results obtained by the present technique are more physically realistic.

Investigation of the Stress-Strain State of a Rectangular Plate after a Temperature Shock

In this paper, the temperature shock phenomenon is considered. This phenomenon occurs during the operation of engineering structures on Earth and in outer space. A rectangular plate has been selected as a structural element exposed to temperature shock. It has a rigidly sealed edge and three free edges. A one-dimensional third initial boundary value problem of thermal conductivity was posed and solved to study the stress–strain state of the plate. Fourier’s law was used to solve this problem, taking into account the inertial term, since the temperature shock is a fairly fast-dynamic phenomenon. It was believed that all the thermophysical properties of the plate are constant and do not depend on its temperature. As a result, the temperature field of the plate was obtained after the temperature shock. This temperature field generates temperature stresses inside the plate, which lead to temperature deformations. To determine these deformations, the initial boundary value problem of thermoelasticity was posed and solved in this work. The Sophie Germain equation was used while solving this problem. To describe the plate, the theory of flexible plates was used, taking into account the stresses in the middle surface of the plate. Next, the accuracy of analytical solutions for the points displacement of a homogeneous plate subjected to a temperature shock was investigated. The temperature field was constructed using a numerical simulation. Functions of the displacement vector components were obtained using approximate analytical solutions. The accuracy of approximate analytical solutions for the components of the plate points deformation vector was estimated. The obtained results allow us to describe the stress–strain state of the plate after the temperature shock. The results of this work can be used in the design of engineering structures for both terrestrial and space purposes in terms of stability calculations and the implementation of deformation constraints.

Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems

In this paper, third-order compact and fourth-order finite element methods (FEMs) based on simple modifications of traditional FEMs are proposed for solving one-dimensional Sturm-Liouville boundary value problems (BVPs). The key idea is based on interpolation error estimates. A simple posterior error analysis of the original piecewise linear finite element space leads to a third-order accurate solution in the L2 norm, second-order in the H1, and the energy norm. The novel idea is also applied to obtain a fourth-order FEM based on the quadratic finite element space. The basis functions in the new fourth-order FEM are more compact compared with that of the classic cubic basis functions. Numerical examples presented in this paper have confirmed the convergence order and analysis. A generalization to a class of nonlinear two-point BVPs is also discussed and tested.

Дослідження поронапруженого стану півнескінченного порожнистого циліндру під дією осесиметричного навантаження

The exact solution of the poroelasticity problem for a semi-infinite hollow cylinder under the axisymmetric load is derived in the paper. The original problem was reduced to a one-dimensional problem by applying the integral Fourier transform. The one-dimensional boundary value problem in the transform domain is formulated in a vector form, its solution is found using the matrix differential calculation apparatus. The derived analytical formulas allow to investigate the change of displacements, stresses and pore pressure depending on the type of porous material, the applied load, the size of outer and inner radii of the cylinder.

One-Dimensional Spatial Boundary Value Problems of the Connected Thermoelasticity. Generalized Functions Method

Problems of definition of thermotension of a thermoelastic core with use of model of the connected thermoelasticity are considered. In this case in the equation of heat conductivity there is the divergence of speed of the movement of material points of the rod, and the elasticity equations contains the temperature gradient. On the basis of generalized functions method the generalized solutions of non-stationary and stationary boundary value problems have been solved at action of the power and thermal sources of various type including ones described singular generalized functions, under various boundary conditions on the ends of a core. Thermoshock waves which arise in such designs at action of impact loads and heat fluxes are considered, conditions on their fronts are received. The uniqueness of the set boundary tasks, including taking into account shock waves has been proved. Regular integral representation of the generalized solutions are given, which give the analytical solution of the tasks. Numerical implementation of solutions of a number of direct, return and semi-return boundary value problems of stationary fluctuations is carried out and results of computer experiments are presented

The dynamical problem on acting distributed load on the elastic layer

The wave field of an elastic half-layer is constructed, when a dynamic normal load distributed over a rectangular area acts on upper face at the initial moment of time. The lower face of the half-layer is rigidly fixed to the foundation, and the side border is in the conditions of a smooth contact. The method of decomposing the system of motion equations into a system of equations and an independently solvable equation is used, this approach was proposed by Popov~G.~Ya. Laplace and Fourier integral transformations are applied directly to the motion equations and boundary conditions, which reduces the problem to a vector one-dimensional boundary value problem, which is solved by the matrix differential calculus method. The output displacements are obtained using inverse integral transformations. The case of steady oscillations was considered and the amplitude of vertical displacement occurring in the layer was analyzed depending on the shape of the distributed load section, the material of the layer medium and the values of the natural frequency of the layer oscillations.

On Adaptive Grid Approximations in the Weight Norm

The purpose of this paper is to develop an algorithm construction of adaptive variation-grid methods for classes of one-dimensional boundary value problems of the second order. Classes of non-degenerate problems are considered, as well as classes of problems with weak and strong degeneracy. The results obtained are suitable for learning computer systems designed to solve problems of the aforementioned classes. To achieve the set goals, the corresponding approximation theorems are established with degeneration. Ways of adaptive choice approximation space at a variation-grid method in a one-dimensional boundary value problem are considered. The locality of the approximation is substantially used. The considerations are reduced to an iterative process, while building an adaptive grid. Numerical examples illustrating the effectiveness of the proposed approach are given.

Spatially One-Dimensional Boundary Value Problems of Coupled Thermoelasticity: Generalized Functions Method

Spatially One-Dimensional Boundary Value Problems of Coupled Thermoelasticity: Generalized Functions Method

On One Approximate Method of a Boundary Value Problem for a One-Dimensional Advection–Diffusion Equation

This article discusses the author’s version of the technology for solving a one-dimensional boundary value problem for a one-dimensional advection–diffusion equation based on the method of separation of variables, as well as the theory of eigenvalues and eigenfunctions when constructing a solution to a differential equation. This problem is solved in two stages. Firstly, we illustrate the technology of separating variables for equations with fractional derivatives, and then apply the theory of eigenvalues and eigenfunctions to obtain an exact solution in the form of an infinite series. Since this series converges very quickly, it is natural to replace it with the sum of the first few terms. The approximate solution obtained in this way is quite suitable for numerical calculations in practice. The article provides a listing of the program for performing calculations, as well as the results of calculations themselves.

Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence.