Homogeneous Integral Equations Research Articles

Green functions for four-point boundary value problems with applications to heterogeneous beams

The main objective of this study is to define the Green functions for four-point boundary value problems. It is a further aim to clarify what properties the Green functions have and to present a method for calculating the elements of these Green functions. The examples are related to two heterogeneous beams with four supports: the (first) [second] beam is (fixed)[pinned] at the endpoints while the intermediate supports are two rollers. Determination of the eigenfrequencies leads to four-point eigenvalue problems associated with homogeneous boundary conditions. Utilizing the Green functions that belong to these eigenvalue problems we can transform those into eigenvalue problems governed by homogeneous Fredholm integral equations. Then a numerical solution is computed by reducing the homogeneous Fredholm integral equations into algebraic eigenvalue problems.

On Data Dependency and Solutions of Nonlinear Fredholm Integral Equations with the Three-Step Iteration Method

In this study, the solution of the second type of homogeneous nonlinear Fredholm integral equations is investigated using a three-step iteration algorithm. In other words, it has been shown that the sequences obtained from this algorithm converge to the solution of the mentioned equations. Also, data dependency is obtained for the second type of homogeneous nonlinear Fredholm integral equations and this result is supported by an example.

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

It is our main objective to find the critical load for three beams with cross sectional heterogeneity. Each beam has three supports, of which the intermediate one is a spring support. Determination of the critical load for these beams leads to three point boundary value problems (BVPs) associated with homogeneous boundary conditions—the mentioned BVPs constitute three eigenvalue problems. They are solved by using a novel solution strategy based on the Green functions that belong to these BVPs: the eigenvalue problems established for the critical load are transformed into eigenvalue problems governed by homogeneous Fredholm integral equations with kernels that can be given in closed forms provided that the Green function of each BVP is known. Then the eigenvalue problems governed by the Fredholm integral equations can be manipulated into algebraic eigenvalue problems solved numerically by using effective algorithms. It is an advantage of the way we attack these problems that the formalism established and the results obtained remain valid for homogeneous beams as well. The numerical results for the critical forces can be applied to solve some stability problems in the engineering practice.

A tripled coincidence point technique for solving integral equations via an upper class of type II

<abstract><p>The goal of this paper is to obtain some tripled coincidence point results for generalized contraction mappings in the setting of $ JS $-metric spaces endowed with a partial order. Furthermore, illustrative examples to support the theoretical results and the application are obtained. Finally, some theoretical results are applied to discuss the existence of a solution for a system of non-homogeneous and homogeneous integral equations as applications.</p></abstract>

On homogeneous integral equations of Fredholm type in the spaces of Bohr almost periodic functions

On homogeneous integral equations of Fredholm type in the spaces of Bohr almost periodic functions

Vibration of an axially loaded heterogeneous fixed–fixed beam with an intermediate roller support

The present paper investigates how an axial load can change the natural frequencies of heterogeneous fixed–fixed beams with an intermediate roller support. The problem is treated as a three-point boundary value problem (eigenvalue problem) that is paired with homogeneous boundary conditions. The Green functions are determined for the unloaded and pre-loaded beams as well—in the later case, both for compression and tension. With these, the eigenvalue problems can be transformed into eigenvalue problems governed by a homogeneous Fredholm integral equations. It is then replaced by an algebraic eigenvalue problem, that is solved numerically with an effective solution algorithm which is based on the boundary element method.

BUCKLING OF BEAMS WITH A BOUNDARY ELEMENT TECHNIQUE

The present work is devoted to the buckling study of non-homogeneous fixed- fixed beams with intermediate spring support. The stability issue of these beams leads to three-point boundary value problems. If the Green functions of these boundary value problems are known, the differential equations of the stability problems that contain the critical load sought can be turned into eigenvalue problems given by homogeneous Fredholm integral equations. The kernel function of these equations can be calculated from the associated Green functions. The eigenvalue issues can be reduced to algebraic eigenvalue problems, which are subsequently solvable numerically with the use of an efficient algorithm from the boundary element method. Within this article, the critical load findings of these beams are compared to those obtained using commercial finite element software, and the results are in excellent correlation.

Stability of heterogeneous beams with three supports through Green functions

The present paper is devoted to the issue how the critical load of some heterogeneous beams with three supports can be determined by using Green functions. The stability problems of these beams are equivalent to three-point boundary value problems, paired with homogeneous boundary conditions. If the Green functions of these boundary value problems are known, the eigenvalue problems that provide the critical load can be transformed into eigenvalue problems governed by homogeneous Fredholm integral equations. The later eigenvalue problems can be reduced to algebraic eigenvalue problems which then can be solved numerically with effective algorithms.

Solutions for the vibration and stability problems of heterogenous beams with three supports using Green functions

The goal of this study is to calculate the eigenvalues that provide the eigenfre- quencies and the critical loads for two heterogeneous beams with three supports: the (first) [second] beam is (fixed)[pinned] at the left end, the intermediate support is a roller while the right end of the beams can move vertically but the rotation is prevented there. The beams are referred to as FrsRp and PrsRp beams. Determination of the (eigenfrequencies) [critical loads] leads to three point eigenvalue problems associated with homogeneous boundary con- ditions. With the Green functions that belong to these eigenvalue problems we can transform them into eigenvalue problems governed by homogeneous Fredholm integral equations. The eigenvalue problems can then be reduced to algebraic eigenvalue problems that are solvable numerically by utilizing effective solution algorithms.

SOLUTION OF A TWO-DIMENSIONAL BOUNDARY VALUE PROBLEM OF HEAT CONDUCTION IN A DEGENERATING DOMAIN

In the paper we consider the boundary value problem of heat conduction outside the cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary condition contain a derivative with respect to the time variable. The peculiarity of the problem under consideration consists precisely in the presence of a moving boundary and the degeneration of the solution domain into a point at the initial moment of time. The well-known classical methods are generally not applicable to this type of problems. By the method of heat potentials, such boundary value problems of heat conduction are reduced to the solution of singular Volterra type integral equations of the second kind A singular Volterra type equation is understood as an equation whose kernel has the following property: the integral of the kernel of the equation does not tend to zero as the upper limit tends to the lower one. Such integral equations cannot be solved by the method of successive approximations, and in most cases the corresponding homogeneous integral equations have nonzero solutions. We prove a theorem on the solvability of the considered boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We found a nonzero solution of this singular integral equation.

Fixed points of $$(\varphi ,F)$$-weak contractions on metric-like spaces with applications to integral equations on time scales

In this paper, the notion of $$(\varphi , F)$$ -weak contractions in the framework of metric-like spaces is introduced and the corresponding fixed point theorems are established. As an attempt to answer the common question regarding the uniqueness of trivial solutions of homogeneous integral equations on time scales, one of our main results is applied to investigate sufficient criteria for existence and uniqueness of solution of nonlinear Fredholm integral equations of the second kind on time scales. The latter result is obtained through a new Lipschitz condition on the kernel different from the usual ones. Examples are constructed to show that our ideas herein are new and generalize some recently announced results in the literature.

Differential and integral equations for Legendre-Laguerre based hybrid polynomials

UDC 517.9 In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial differential equations are established. The analogous results for the three-variable Hermite – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, –E uler and – Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by deriving homogeneous Volterra integral equations for these polynomials and for their relatives.

The Neumann problem in plane deformations of a micropolar elastic solid with micropolar surface effects

We consider the Neumann problem in a theory of plane micropolar elasticity incorporating micropolar surface effects. The incorporation of surface elasticity utilizes the Eremeyev–Lebedev–Altenbach shell model, leading to a set of second-order boundary conditions describing the separate micropolar elasticity of the surface. The Neumann problem is of particular interest, since the question of solvability is complicated by the fact that the corresponding systems of homogeneous singular integral equations admit nontrivial solutions that affect the solvability of both the interior and exterior Neumann boundary value problems. We overcome this difficulty by constructing integral representations of the solutions based on specifically constructed auxiliary matrix functions leading to uniqueness and existence theorems in appropriate classes of smooth matrix functions.

Bell Polynomials in the Mathematica System and Asymptotic Solutions of Integral Equations

We consider the possibility of solving functional equations that arise when integrating homogeneous integral Fredholm equations of the second kind with a highly oscillatory kernel by using Bell polynomials. We review different types and properties of Bell polynomials. The focus of this paper is to promote using tools in the Bell polynomial package in the Mathematica system to solve certain problems in electrodynamics.

Quasi-Static Surface-PEEC Modeling of Electromagnetic Problem With Finite Dielectrics

This paper presents a quasi-static surface-based partial element equivalent circuit (PEEC) model for electromagnetic problems consisting of coupled conductors and heterogeneous dielectrics of finite size. The PEEC model is based on the surface equivalent principle. Unlike the traditional surface-based PEEC models, in which the integral equations are set up by enforcing the field continuity in the true field region, the homogeneous integral equations in this model are obtained by enforcing the tangential null field condition in the null field regions. Simplified integral equations are obtained under the quasi-static assumption, in which the circuit elements carrying electric currents and magnetic charges on the dielectric surfaces are vanished. Consequently, the size of the subcircuit describing dielectrics is reduced significantly compared to the full-wave surface-PEEC (S-PEEC) model. Since the quasi-static S-PEEC model only contains frequency-independent circuit elements, it can be conveniently used for time-domain simulation. Three numerical examples are presented to validate the new PEEC model for typical embedded RF passive components, packaging and interconnection problems in both the frequency domain and the time domain. Excellent agreement is observed between the results of the proposed PEEC model, S-PEEC model, and those of commercial software within the quasi-static frequency range.

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.

Differential and integral equations associated with some hybrid families of Legendre polynomials

The article aims to explore some new classes of differential and integral equations for some hybrid families of Legendre polynomials. Beginning with the recurrence relations and shift operators, the authors derived the differential, integro-differential and partial differential equations for the hybrid Legendre-Appell polynomials. Certain examples are framed for the hybrid Legendre-Bernoulli, Legendre-Euler and Legendre-Genocchi polynomials to show the applications of main results. Further, the homogeneous Volterra integral equations for the hybrid Legendre-Appell and other hybrid families of special polynomials are derived. The inclusion of integral equations is a bonus to this article.

Source-function synthesis in the self-consistent problem of radiation

A new approach to a rigorous theory of aperture antennas is proposed, which is based on solution of the self-consistent problem of radiation. It is ascertained that this formulation of the problem leads to a system of homogeneous Fredholm integral equations of the second kind—i.e., a problem for eigenfunctions and eigenvalues that provides a basis for solving the problem of field generation in the open space. The passage to Fredholm integral equations of the first kind allows one to solve the task of synthesis of the source function for a given radiation field.

Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra–Hammerstein integral equations. To do this, we introduce the concept of “blow-up collocation solution” and analyze numerically some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finally, we discuss the relationships between necessary conditions for blow-up of collocation solutions and exact solutions.

A Robust and Accurate Quasi-Monte Carlo Algorithm for Estimating Eigenvalue of Homogeneous Integral Equations

We present an efficient numerical algorithm for computing the eigenvalue of the linear homogeneous integral equations. The proposed algorithm is based on antithetic Monte Carlo algorithm and a low-discrepancy sequence, namely, Faure sequence. To reduce the computational time we reduce the variance by using the antithetic variance reduction procedure. Numerical results show that our scheme is robust and accurate.