Solution Of Boundary Value Problem Research Articles

The boundary value problem for the two-dimensional Laplace equation with transmission conditions given on open Lipschitz curve

We consider boundary value problem for Laplace equation in bounded two-dimensional Lipschitz domain with transmission boundary condition given upon open curve. This conditions includes itself the jump of solution of boundary value problem and the meaning of boundary value of its normal derivative. We prove the equivalence of considered boundary value problem and obtained variational problem. As a result we prove existence and uniqueness of solution of the posed problems in appropriate functional spaces. Based on the integral representation of the solution the considered boundary value problem is reduced to the system of boundary integral equation which has unique solution.

Formulation and Numerical Solution of Plane Problems of the Theory of Elasticity in Strains

This article is devoted to the formulation and numerical solution of boundary-value problems in the theory of elasticity with respect to deformations. Similar to the well-known Beltrami–Michell stress equations, the Saint-Venant compatibility conditions are written in the form of differential equations for strains. A new version of plane boundary-value problems in strains is formulated. It is shown that for the correctness of plane boundary value problems, in addition to the usual conditions, one more special boundary condition is required using the equilibrium equation. To discretize additional boundary conditions and differential equations, it is convenient to use the finite difference method. By resolving grid equations and additional boundary conditions with respect to the desired quantities at the diagonal nodal points, we obtained convergent iterative relations for the internal and boundary nodes. To solve grid equations, the elimination method was also used. By comparing with the Timoshenko–Goodyear solution on the tension of a rectangular plate with a parabolic load, the validity of the formulated boundary value problems in strains and the reliability of the numerical results are shown. The accuracy of the results has been increased by an average of 15%.

On the reduction of an autonomous nonlinear boundary value problem, unsolved with respect to the derivative, to the first-order critical case

Constructive solvability conditions and a scheme for constructing solutions of an autonomous nonlinear boundary value problem unsolved with respect to the derivative have been found. A convergent iterative scheme for finding approximations to the solutions of a nonlinear autonomous nonlinear boundary value problem unsolved with respect to the derivative was constructed by reducing the problem to the first-order critical case. As an example of application of the constructed iterative scheme, approximations to the solutions of a periodic boundary value problem for the autonomous Lotka-Volterra equation were determined.

On divergence type linear and quasilinear equations in the complex plane

This is a survay of our recent results concerning divergence type linear and quasilinear equations in the complex plane. It contains a number of existence, representation and regularity theorems for solutions of fundamental boundary value problems for such equations. The degeneration case of uniform ellipticity is also covered by means of BMO, VMO, Calderon-Zigmund, Lehto and Orlicz technic.

Approximate Solutions of Nonlinear Boundary Value Problems by Collocation Methods Compared to Newer Methods

A large variety of new methods are being developed for fast and efficient solutions of nonlinear boundary value problems. Some of these methods are, Adomian decomposition (ADM), differential transform (DTM), least squares vector machines (LSSVMM), and multiple variational iteration (MVIM). A natural question arises as to how efficient and simple to use these newer methods are compared to classical methods. One of the simplest and widely applicable classical methods is the collocation method. The overall performance of collocation method and the newer methods are compared on a number of problems, which were previously used to benchmark the newer methods. It is concluded that, at least for the problems considered, the collocation method performs as successfully as the newer methods.

Raznostnye skhemy dekompozitsii na osnove rasshchepleniya resheniya i operatora zadachi

Domain decomposition methods are used for the approximate solution of boundary value problems for partial differential equations on parallel computing systems. The specifics of nonstationary problems is most completely taken into account when using noniterative domain decomposition schemes. Regionally additive schemes are constructed on the basis of various classes of splitting schemes. A new class of domain decomposition schemes with an additive representation of the solution on a new time level is distinguished that is based on splitting the domain into subdomains based on a partition of unity. An example of the Cauchy problem for first-order evolution equations with a positive self-adjoint operator in a finite-dimensional Hilbert space is considered. Unconditionally stable two- and three-level splitting schemes are constructed for the corresponding system of equations.

Estimation of the stress intensity factor of an interface crack by the scattered SH-far field

The diffraction of the plane elastic SH-wave from a semi-infinite interface crack on junction of two elastic semi-infinite media is studied. The crack is modelled by the mathematical cut with no stress on its faces. The displacement and the stress fields are continuous outside of the crack. The wave diffraction problem is reduced to the solution of the mixed boundary value problem for Helmholtz equation. We search the solution which satisfies the Neumann boundary condition on the crack faces and the continuity condition for the stress and displacement fields outside of the crack. The radiation condition at the infinity and the Meixners condition at the crack tip must be also satisfied. Using the Fourier integral transform, the mixed boundary value problem is reduced to the Wiener-Hopf functional equation which is valid in the given strip of regularity in the complex plane. The method of factorization and decomposition, as well as the Liouvilles theorem, are used to solve this equation. Its kernel function is factorized and represented as a product of two split functions that are regular in the overlapping half-planes. These ones allow for the simple poles outside of the regularity regions. The solution of the Wiener-Hopf equation is presented in analytical form. The scattered displacement field is found for an arbitrary frequency and sounding angle by applying an inverse Fourier integral transform to the solution. The asympŹtotic formula for the stress intensity factor (SIF) at the crack tip is obtained. The correlation between complex amplitude of the SH-wave far scattered displacement field and the SIF caused by this field is obtained for an arbitrary radiation angle, frequency and medium parameters. This allows us to express the SIF through the amplitude of the far field if the radiation frequency, sounding angle as well as the physical characteristics of materials are known. It is shown that in the plane that is normal to the tip of the crack the ratio of SIF for the given junction and two fixed values of the sounding angle or two sounding frequencies is proportional to the ratio of the scattered fields. It is found that under above mentioned condition the proportionality rate does not depend on material properties. The obtained relations can be applied for estimation of the SIF with changing frequency and sounding angle.

Some characteristic properties of the solutions in the three-phase-lag heat conduction

In this paper we consider the three-phase-lag model of heat conduction that involves second-order effects in phase lag of the heat flux vector. This model leads to a fourth-order in time equation of Moore–Gibson–Thompson type. We use the thermodynamic restrictions derived from the compatibility of the constitutive equation with the Second Law of Thermodynamics to study the properties of the solutions of the initial boundary value problems associated with the model in concern. In this connection we establish a series of well-posedness results concerning the related solutions like: uniqueness, continuous data dependence, exponentially stability or domain of influence. Furthermore, based on the thermodynamic restrictions, we show that the thermal model in question admits damped in time propagating waves as well as exponentially decaying standing modes. We also show that when the thermodynamic restrictions are not fulfilled, then wave solutions appear that cause the energy blows up as time goes to infinity.

Existence and multiplicity of solutions for boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity

In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo–Krasnosel’skii fixed point theorem and the Leray–Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively.

Multi-level neural networks for accurate solutions of boundary-value problems

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy, classical discretization methods such as the finite element methods, remains a significant challenge. Deep learning methods usually struggle to reliably decrease the error in their approximate solution. A new methodology to better control the error for deep learning methods is presented here. The main idea consists in computing an initial approximation to the problem using a simple neural network and in estimating, in an iterative manner, a correction by solving the problem for the residual error with a new network of increasing complexity. This sequential reduction of the residual of the partial differential equation allows one to decrease the solution error, which, in some cases, can be reduced to machine precision. The underlying explanation is that the method is able to capture at each level smaller scales of the solution using a new network. Numerical examples in 1D and 2D dealing with linear and non-linear problems are presented to demonstrate the effectiveness of the proposed approach.

Finite difference spectral collocation schemes for the solutions of boundary value problems

This article introduces novel numerical approaches utilizing both standard and nonstandard finite difference methods to solve one-dimensional Bratu's problems. Using the quasilinearization technique, the original problem is converted into a sequence of linear problems. Chebyshev polynomials are employed to approximate the second derivative of the function y(x), after which Sumudu transform is applied to obtain a new form of trial function. The obtained trial function is then substituted into a linearized and discretized Bratu's equations. We discuss the convergence of the schemes and compare the numerical outcomes to those derived using other relevant methods. We further modify one of the new schemes and apply it to solve boundary value problem with associated Robin conditions. The results show that the proposed schemes yield accurate approximations to the solutions of the problems considered.

Development of an Optimal Control Algorithm for the Course Movement of an Unmanned Vehicle

The article presents an algorithm for controlling a ground unmanned vehicle. The main task is to obtain a solution to control problems that allows you to transfer an unmanned vehicle from the initial position to a given final position at a certain time. Currently, autonomous vehicles are being actively introduced in all areas. Including in Russia, you can find unmanned trucks on the federal highway. Given the development of microprocessor technology and the significant economic benefits of using unmanned vehicles, this task is relevant. A lot of research in the field of control of unmanned vehicles concerns the issue of their use in urban environments and on rough terrain. In this paper, we present a solution to the problem of optimal control of an unmanned vehicle using the maximum principle. The problem of optimal control is solved in a deterministic setting with an integro-terminal criterion. The solution of the two-point boundary value problem arising from the maximum principle was carried out using Newton’s method. The ranges of initial values of conjugate variables are obtained, which ensure the convergence of calculations. For the chosen mathematical model of the course movement of the car, solutions to the problem were obtained. The results of numerical simulation are presented, showing the possibility of using the proposed algorithm to control an unmanned vehicle under various initial and final conditions. The developed algorithm has been successfully applied in the presence of a penalty zone. The algorithm can be used when applying the concept of " flexible trajectories" in the tasks of controlling moving objects.

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.

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.

Distributed Swarm Optimization for the Solution of Boundary Value Problems in Astrodynamics

Distributed Swarm Optimization for the Solution of Boundary Value Problems in Astrodynamics

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Abstract We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin’s maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi’s theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation.

On the characteristic functions and Dirchlet-integrable solutions of singular left-definite Hamiltonian systems

In this work, a singular left-definite Hamiltonian system is considered and the characteristic-matrix theory for this Hamiltonian system is constructed. Using the results of this theory we introduce a lower bound for the number of Dirichlet-integrable solutions. Moreover we share a relation between the kernel of the solution of the nonhomogeneous boundary value problem and the characteristic-matrix.

Exploring Integral ϝ-Contractions with Applications to Integral Equations and Fractional BVPs

In this article, two types of contractive conditions are introduced, namely extended integral Ϝ-contraction and (ϰ,Ω-Ϝ)-contraction. For the case of two mappings and their coincidence point theorems, a variant of (ϰ,Ω-Ϝ)-contraction has been introduced, which is called (ϰ,Γ1,2,Ω-Ϝ)-contraction. In the end, the applications of an extended integral Ϝ-contraction and (ϰ,Ω-Ϝ)-contraction are given by providing an existence result in the solution of a fractional order multi-point boundary value problem involving the Riemann–Liouville fractional derivative. An interesting existence result for the solution of the nonlinear Fredholm integral equation of the second kind using the (ϰ,Γ1,2,Ω-Ϝ)-contraction has been proven. Herein, an example is established that explains how the Picard–Jungck sequence converges to the solution of the nonlinear integral equation. Examples are given for almost all the main results and some graphs are plotted where required.

Sparse series solutions of random boundary and initial value problems

The study investigates the utilization of sparse representations through dictionary elements to address the classical Dirichlet and Cauchy types of stochastic boundary value problems (BVPs) and initial value problems (IVPs). A novel approach is introduced based on the recently developed stochastic pre-orthogonal adaptive Fourier decomposition (SPOAFD) technique. By employing SPOAFD, both analytic and numerical solutions for the stochastic BVPs and IVPs are formulated. Furthermore, the scope of the study is extended to include BVPs and IVPs associated with a specific class of fractional heat equations and fractional Poisson equations. In addition to establishing the theoretical framework, important computational aspects are thoroughly discussed to enable the implementation of practical algorithms. The proposed methodology is validated through numerical examples, demonstrating its effectiveness and computational efficiency.

Existence and Uniqueness Results of Fractional Differential Inclusions and Equations in Sobolev Fractional Spaces

In this work, by using the iterative method, we discuss the existence and uniqueness of solutions for multiterm fractional boundary value problems. Next, we examine some existence and uniqueness returns for semilinear fractional differential inclusions and equations for multiterm problems by using some notions and properties on set-valued maps and give some examples to explain our main results. We explore and use in this paper the fundamental properties of set-valued maps, which are needed for the study of differential inclusions. It began only in the mid-1900s, when mathematicians realized that their uses go far beyond a mere generalization of single-valued maps.