System Of Fredholm Integral Equations Research Articles

$C^*$-algebra valued symmetric spaces and fixed point results with an application

In this paper, we firstly introduce the class of $C^*$-algebra valued symmetric spaces and utilize the same to prove our fixed point results. We furnish an example to highlight the utility of our main result. Finally, we apply our result to examine the existence and uniqueness of a solution for a system of Fredholm integral equations.

Light field reconstruction from scattered light using plenoptic data.

We investigate the utility of plenoptic data for extracting information from a scene where the light from objects in the scene is viewable only after scattering from a surface such as a wall. We derive the rigorous relationship between the object and the scattered light, which is cast in terms of a system of Fredholm integral equations of the first kind with the bidirectional reflectance distribution function (BRDF) of the scattering surface; further, the object information is reconstructed by solving the equations. Based on the Fourier transformation, we propose a simple BRDF model and analyze the reconstruction errors by introducing newly defined parameters reflecting the BRDF's characteristics, i.e., the degree of specularity and the effective SNR, and obtain optimal regularized solutions under a variety of surface BRDFs. Moreover, we provide a fundamental limit of retrievable information content from the scattered light. A comparison with experimental results is reported.

Triangular functions based method for the solution of system of linear Fredholm integral equations via an efficient finite iterative algorithm

Triangular functions based method for the solution of system of linear Fredholm integral equations via an efficient finite iterative algorithm

Method for Measuring the Absolute Spectral Response of Infrared Photodetector Arrays

Physical and technical aspects of the implementation of an alternative method for measuring the absolute spectral response of infrared photodetector arrays (IR PDA) (current sensitivity spectrum, voltage sensitivity, and quantum efficiency) without the use of spectral instruments are considered. The method is based on multiple measurements of the output signal of all IR array’s photosensitive elements (PSEs) generated by modulated black-body radiation at different temperatures of the black body (BB). The signal is measured against the sum of constant signals from the background radiation, the detector input optical window, the BB radiation modulator, the PSE dark current, and the constant signal of the LSI multiplexer. A system of Fredholm integral equations of the first kind is constructed based on the measured PSE signals. Its left side includes measured BB signals and, the right side, analytical expressions describing these signals. The solution of the system are the absolute values of the above-mentioned spectral components of all PSEs of PDAs. The block diagram of the measurement setup is considered, functional features of its operation are analyzed, and requirements for its blocks are substantiated. Additional advantages of the new method in comparison with the existing methods are presented.

Modeling of magnetic field geometry for high precision executive devices

The paper presents the theoretical foundations of constructing a mathematical model for modeling the geometry of the magnetic field in the working area of special devices with complex geometry of the magnetic system.It is proposed to use the method of secondary sources as a method for solving the problem, in the presence of a number of different homogeneous and isotropic media in space. This allows us to divide the solution of the problem of finding the field distribution geometry into two stages: at the first stage, we find the distribution of secondary sources, the action of which is equivalent to the influence of an inhomogeneous medium, and then the total field from the primary and secondary sources is calculated.Thanks to this approach, for a piecewise-homogeneous medium, the problem of calculating the field pattern can be formulated in the form of a system of linear Fredholm integral equations of the second kind.To solve the integral equation, a transition was made to a finite system of linear algebraic equations. In this case, the geometry features of the magnetic system of the structure (boundary conditions) were taken into account. The solution of the system of algebraic equations was obtained by several methods to evaluate the simulation results.Based on the proposed method, a mathematical model was obtained and a software package was developed that implements a numerical algorithm for calculating the magnetic field geometry in the working area of a special device.

Detailed Solution of a System of Singular Integral Equations for Mixed Mode Fracture in Functionally Graded Materials

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).

Haar Wavelet Collocation Method for Solving the System of Linear Fredholm Integral Equations with Constant Coefficients

In this paper, we propose different procedure to solve the problem of linear system of Fredholm integral equations with constant coefficients. The proposed method gives the rapidly convergent successive approximation to the analytic solution. The reliability and efficiency of the method are illustrated by investigating the convergence results for some problems. In fact, comparisons are made between the contributed scheme and the generated results.

О новом подходе в оценке поведения большой совокупности параллельных штолен

The problem of assessing the behavior of a large number of underground structures such as parallel tunnels, characteristic of mines for the extraction of minerals, including coal, is considered. Traditionally, studies are performed for a single object, and then the parameters found are accepted for all other objects. At the same time, the multiplicity of such objects can lead to another factor of strength impairment associated with the possibility of localization of the stress-strain state in one of the zones of the structure, leading to excess of the planned strength parameters. In this paper, the theory of calculation of strength properties of such objects is based on the example of underground structures. The research is based on the block element method based on factorization approaches. The problem is reduced to a system of integral equations of the first kind with a difference kernel, which is reduced to a system of Fredholm integral equations of the second kind. By calculating the integrals describing the nuclei of these equations according to the theory of deductions, it is possible to reduce the integral equations to a system of algebraic equations available for analytical analysis, which allows to identify the localization of stresses or displacements. The long-range approach made it possible to find a connection between these equations with the Riemann problem for a set of pairs of analytic functions, the number of which coincides with the number of galleries. In earlier works, the resulting stress-strain state was characterized by calculating only the vertical contact stresses on the supports and the vertical displacements of roof slack charges. In this paper, it is possible to construct an approach that allows us to calculate all three components of the stresses on the supports and all three components of the roof sagging displacement.

Regarding the solution of the plate theory boundary problem for domains of complex shape

Mathematical model construction of complicate physical phenomenon often leads to the setting and solving problems of parameters optimal control in differential equations in partial derivatives. Chosen equation with boundary and initial conditions is usually mathematical model basis of the object, which is under analysis. Optimal control of right-hand side function in non-linear problem for inhomogeneous biharmonic has been investigated. With the help of various gradient methods the problems of parameters control in such equations are solved successfully. Herewith linear problem is solved with the potential method on every step. The boundary value problem of plate theory, which is reduced to a system of Fredholm integral equations of the first kind and an algorithm of self-regularization of this system, is considered. The potential method is used to solve the linear problem for the harmonic equation. Examples of numerical implementation are shown that demonstrate high computational efficiency in the case of complex form regions. Algorithm for linear boundary value problem solution with boundary integral equations overcomes this problem successfully. Physical examples of numerical implementation have been presented, analysis of obtained solutions have been conducted. Their accuracy, algorithm simplicity and time spent evidence about this approach promising for practical results obtaining in plate theory and mathematical physics problems successful numerical solving.

An Extended Dielectric Crack Model for Fracture Analysis of a Thermopiezoelectric Strip

An extended dielectric crack model is proposed to capture the effects of the physical properties of crack interior on crack-tip thermoelectroelastic fields. The typical crack-face boundary conditions can be retrieved by considering the limiting cases of electrical permeability and thermal conductivity inside a crack. Making use of the Fourier transform technique, the problem of a thermopiezoelectric strip weakened by a Griffith crack is investigated and transformed to solve the system of the second kind Fredholm integral equations with Cauchy kernel. The Lobatto–Chebyshev collocation method is used to form a nonlinear system of algebraic equations, which is solved by elaborating on an algorithm. The crack-tip thermoelectroelastic fields are determined by using the approximate solutions. Numerical simulations are carried out to show the variations of the fracture parameters of concern under applied thermoelectromechanical loads, the physical properties of the dielectric medium inside the crack and the geometry of the cracked thermopiezoelectric strip. Some comparisons with the experimental results are reported to reveal the effectiveness of the extended dielectric crack model.

TRANSFER OF LOADS FROM A FINITE NUMBER OF ELASTIC OVERLAYS WITH FINITE LENGTHS TO AN ELASTIC STRIP THROUGH ADHESIVE SHEAR LAYERS

This article deals with the problem of an elastic infinite strip, which is strengthened along its free boundary by a finite number of finite overlays with different elastic characteristics and small constant thicknesses. The interaction between the strip and the overlays is mediated by adhesive shear layers. The overlays are deformed under the action of horizontal forces. The problem of determination of unknown stresses acting between the strip and overlays are reduced to a system of Fredholm integral equations of the second kind for a finite number of unknown functions defined on different finite intervals. It is shown that in the certain domain of variation of the characteristic parameter of the problem this system of integral equations in Banach space may be solved by the method of successive approximations. Particular cases are discussed and the character and behaviour of unknown shear stresses are investigated.

The Nyström Method and Convergence Analysis for System of Fredholm Integral Equations

In this paper, the efficient numerical solutions of a class of system of Fredholm integral equations are solved by the Nyström method, which discretizes the system of integral equations into solving a linear system. The existence and uniqueness of the exact solutions are proved by the Banach fixed point theorem. The format of the Nyström solutions is given, especially with the composite Trapezoidal and Simpson rules. The results of error estimation and convergence analysis are obtained in the infinite norm sense. The validity and reliability of the theoretical analysis are verified by numerical experiments.

Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind

This paper presents a new numerical approximation method to solve a system of nonlinear Fredholm integral equations of second kind. Spectral collocation method and their properties are applied to determine the general solution procedure for nonlinear Fredholm integral equations (FIEs). The convergence and error analysis of spectral collocation method are incorporated for the given nonlinear model. Legendre–Gauss–Lobatto (LGL) points are used as collocation points with various Legendre–Gauss quadrature with weight functions. The use of Legendre polynomials, together with the Gauss quadrature collocation points is well known for the accurate approximations that converge exponentially. Finally, we validate our theoretical results with a number of numerical examples, which further enhance the efficiency of our proposed scheme.

Fixed Point Results in Partial Symmetric Spaces with an Application

In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of Fredholm integral equations. Furthermore, we introduce an analogue of the Hausdorff metric in the context of partial symmetric spaces and utilize the same to prove an analogue of the Nadler contraction principle in such spaces. Our results extend and improve many results in the existing literature. We also give some examples exhibiting the utility of our newly established results.

Fracture of a Three-Layer Material with an Opening Mode Central Crack

This paper describes a problem of the theory of elasticity for a three-layer material in which two layers located symmetrically relative to the central layer have identical properties. The central layer has an opening mode crack whose tops are located in the neighboring layers. The solution of this problem is reduced to solving a system of Fredholm integral equations of the second kind. A stress intensity factor for opening mode cracks is determined, and its dependence on the problem parameters is investigated.

An approximate solution for solving the system of fredholm integral equations of the second kind

An approximate solution for solving the system of fredholm integral equations of the second kind

Идентификация неоднородных характеристик преднапряженных пироматериалов

Функционально-градиентные пироматериалы находятширокое применение при создании различных диагностических приборов. Дляправильного расчета устройств, использующих пироэффект, необходимо знаниематериальных характеристик. В случае неоднородных предварительно-напряженныхтел прямые измерения материальных характеристик невозможны, поскольку онипредставляют собой некоторые функции координат. Нахождение характеристикнеоднородных пироматериалов возможно только на основе аппаратакоэффициентных обратных задач термоэлектроупругости (КОЗТ), которыйпрактически не разработан. В работе приведена постановка обратной задачитермоэлектроупругости для предварительно-напряженногофункционально-градиентного стержня. Для этого на основе подхода,предложенного Гузем А.Н. для упругих тел, были получены уравнения термоэлектроупругости дляпредварительно-напряженного стержня. Проведено обезразмеривание задачи.Получена слабая постановка прямой задачи термоэлектроупругости. На основеслабой постановки и метода линеаризации получены операторные уравнения длярешения обратной задачи на основе итерационного процесса. В ходеитерационного процесса поправки к восстанавливаемым характеристикамтермоэлектроупругого стержня определялись из решения интегральных уравненийФредгольма 1-го рода. Прямая задача решалась на основе метода сведения ксистеме интегральных уравнений Фредгольма 2-го рода в трансформантах поЛапласу и использовании процедуры обращении, реализуемой в соответствии стеорией вычетов Проведена серия вычислительных экспериментов повосстановлению характеристик, изменение которых оказывает существенноевлияние на дополнительную информацию. В вычислительных экспериментахвосстанавливалась одна из характеристик термоэлектроупругого стержня приизвестных остальных. Даны практические рекомендации по выбору наиболееинформативных временных интервалов для измерения входной информации.Выяснено, что появление начальных напряжений существенно влияет нарезультаты реконструкции характеристик стержня.

Meshfree approach for solving multi-dimensional systems of Fredholm integral equations via barycentric Lagrange interpolation

A highly accurate meshfree approach based on the barycentric Lagrange basis functions is reported for solving the linear and nonlinear multi-dimensional systems of Fredholm integral equations (FIEs) of the second kind. The method is an improved Lagrange interpolation technique which possesses high precision and inexpensive procedure. The systems of FIEs are handled with the barycentric formula and transformed into the corresponding linear and nonlinear systems of algebraic equations. The stability of the solved integral equation systems is guaranteed by the numerical stability of the barycentric Lagrange formula, which is dominated by barycentric weights. The convergence analysis and error estimation are included. Some computational results indicate that the method is simple and efficient.

On the eigenfrequencies of preloaded rotationally restrained extensible circular beams by Green’s functions

The article is devoted to the vibrations of heterogeneous curved beams with centerline extensibility accounted for. The end supports are rotationally restrained pins, and the effect of a central, either compressive or tensile, force (preload) is incorporated into the model. The coupled differential equations that govern the problem are derived from the principle of virtual work. These are replaced by a system of homogeneous Fredholm integral equations. The kernel of the integral equation system is the Green’s function matrix, which is given in closed form. The eigenvalue problem determined by the homogeneous Fredholm integral equation system is solved numerically, and the results obtained are presented graphically. If the spring stiffness tends to (zero) [infinity], the beam behaves as if it were (pinned–pinned) [fixed–fixed]. When the external force tends to zero, the model returns the eigenfrequencies of the free vibrations.

A fast-converging iterative scheme for solving a system of Lane–Emden equations arising in catalytic diffusion reactions

In this paper, we present a fast-converging iterative scheme to approximate the solution of a system of Lane–Emden equations arising in catalytic diffusion reactions. In this method, the original system of differential equations subject to a Neumann boundary condition at $$X = 0$$ and a Dirichlet boundary condition at $$X = 1$$ , is transformed into an equivalent system of Fredholm integral equations. The resulting system of integral equations is then efficiently treated by the optimized homotopy analysis method. A numerical example is provided to verify the effectiveness and accuracy of the method. Results have been compared with those obtained by other existing iterative schemes to show the advantage of the proposed method. It is shown that the residual error in the present method solution is seven orders of magnitude smaller than in Adomian decomposition method and five orders of magnitude smaller than in the modified Adomian decomposition method. The proposed method is very simple and accurate and it converges quickly to the solution of a given problem.