Degenerate Kernel Research Articles

A Boundary Element Method Based on the Hierarchical Matrices and Multipole Expansion Theory for Acoustic Problems

The coefficient matrices of conventional boundary element method (CBEM) are dense and fully populated. Special techniques such as hierarchical matrices (H-matrices) format are required to extent its ability of handling large-scale problems. Adaptive cross approximation (ACA) algorithm is a widely adopted algorithm to obtain the H-matrices. However, the accuracy of the ACA boundary element method (ACABEM) cannot be adjusted by changing the tolerance [Formula: see text] when it exceeds a certain value. In this paper, the degenerate kernel approximation idea for the low-rank matrices is developed to build a fast BEM for acoustic problems by exploring the multipole expansion of the kernel, which is referred as the multipole expansion H-matrices boundary element method (ME-H-BEM). The newly developed algorithm compresses the far-field submatrices into low rank submatrices with the expansion terms of Green’s function. The obtained H-matrices are applied in conjunction with the generalized minimal residual method (GMRES) to solve acoustic problems. Numerical examples are carefully set up to compare the accuracy, efficiency as well as memory consumption of the CBEM, ACABEM, fast multipole boundary element method (FMBEM) and ME-H-BEM. The results of a pulsating sphere indicate that the ME-H-BEM keeps both storage and operation logarithmic-linear complexity of the H-matrices format as the ACABEM does. Moreover, the ME-H-BEM can achieve better convergence and higher accuracy than the ACABEM. For the analyzed complicated large-scale model, the ME-H-BEM with appropriate number of expansion terms has an advantage in terms of efficiency as compared with the ACABEM. Compared with the FMBEM, the ME-H-BEM is easier to be implemented.

ON AN INTEGRO-DIFFERENTIAL EQUATION OF PSEUDOPARABOLIC-PSEUDOHYPERBOLIC TYPE WITH DEGENERATE KERNELS

In the article the questions of solvability of boundary value problem for a homogeneous pseudoparabolic-pseudohyperbolic type integro-differential equation with degenerate kernels are considered. The Fourier method based on separation of variables is used. A criterion for the one-valued solvability of the considering problem is found. Under this criterion the one-valued solvability of the problem is proved.

Weakly Perturbed Fredholm Integral Equations with Degenerate Kernels in Banach Spaces

We consider weakly perturbed Fredholm equations with degenerate kernels in Banach spaces and establish conditions for e = 0 to be a bifurcation point for the solutions of weakly perturbed operator equations X in Banach spaces. A convergent iterative scheme for finding solutions in the form of series $$ {\Sigma}_{i=-1}^{+\infty }{\varepsilon}^i{z}_i(t) $$ in powers of e is proposed.

Bending of an Infinite beam on a base with two parameters in the absence of a part of the base

Currently, in connection with the rapid development of high-rise construction and the improvement of joint operation of high-rise structures and bases models, the questions connected with the use of various calculation methods become topical. The rigor of analytical methods is capable of more detailed and accurate characterization of the structures behavior, which will affect the reliability of objects and can lead to a reduction in their cost. In the article, a model with two parameters is used as a computational model of the base that can effectively take into account the distributive properties of the base by varying the coefficient reflecting the shift parameter. The paper constructs the effective analytical solution of the problem of a beam of infinite length interacting with a two-parameter voided base. Using the Fourier integral equations, the original differential equation is reduced to the Fredholm integral equation of the second kind with a degenerate kernel, and all the integrals are solved analytically and explicitly, which leads to an increase in the accuracy of the computations in comparison with the approximate methods. The paper consider the solution of the problem of a beam loaded with a concentrated force applied at the point of origin with a fixed value of the length of the dip section. The paper gives the analysis of the obtained results values for various parameters of coefficient taking into account cohesion of the ground.

Lyapunov operator L with degenerate kernel and Gibbs measures

Lyapunov operator L with degenerate kernel and Gibbs measures

Analytical derivation and numerical experiment of degenerate scale by using the degenerate kernel of the bipolar coordinates

Degenerate scales of an eccentric annulus and an infinite plane with two identical circular holes in the boundary integral equation method (BIEM) are analytically derived and numerically implemented in this paper. To analytically study the degenerate scale of the BIE, the closed-form fundamental solution of the two-dimensional Laplace equation, ln r, is expanded by a degenerate (separate) kernel in terms of the bipolar coordinates. It is proved that unit radius of the outer circle dominates the degenerate scale of eccentric annulus. An analytical formula of degenerate scale for the infinite plane with two identical circular boundaries was also derived at the first time. In addition, null fields of the domain and complementary domain for the ordinary and degenerate scales are both shown, respectively. Finally, comparison with available results and the BEM data are well done.

Scattering problems of the SH Wave by Using the Null-Field Boundary Integral Equation Method

The scattering problem of seismic waves is an important issue for studying earthquake engineering. In this paper, the null-field boundary integral equation approach was used in conjunction with degenerate kernels and eigenfunction expansion to solve the SH-wave scattering problem of a circular or an elliptical-arc hill. The original problem is divided into subdomains by taking a free-body diagram. One region is an interior boundary value problem. The other is a canyon scattering problem. For the boundary value problem, not only a simply connected domain (elliptical-arc hill problem) but also a doubly connected domain (a circular-arc hill problem containing a circular tunnel or a circular inclusion) is considered. The canyon scattering problem may be addressed in an infinite domain with an artificial boundary of a full plane such that the degenerate kernel can be fully utilized. The null-field integral equation method is used to match boundary conditions. Numerical results are compared favorably with the available data.

Discrete septic spline quasi-interpolants for solving generalized Fredholm integral equation of the second kind via three degenerate kernel methods

Three main contributions are presented in this paper. First, the septic quasi-interpolants are calculated with all their coefficients. Second, we explore the results to solve a generalized and broad class of Fredholm integral equations of the second kind. Finally, we present three degenerate kernel methods; the latter is a combination of the two previously established methods in the literature. Moreover, we provide a convergence analysis and we give new error bounds. Finally, we exhibit some numerical examples and compare them with previous results in the literature.

Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps

We consider piecewise deterministic Markov processes with degenerate transition kernels of the house-of-cards- type. We use a splitting scheme based on jump times to prove the absolute continuity, as well as some regularity, of the invariant measure of the process. Finally, we obtain finer results on the regularity of the one-dimensional marginals of the invariant measure, using integration by parts with respect to the jump times.

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

PurposeThe purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically.Design/methodology/approachTo avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered.FindingsResults using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function.Originality/valueThe analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.

Разрешимость и определение коэффициента в одной краевой задаче для интегро-дифференциального уравнения Фредгольма с вырожденным ядром

Разрешимость и определение коэффициента в одной краевой задаче для интегро-дифференциального уравнения Фредгольма с вырожденным ядром

Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel

This article examines questions of unique solvability for an inverse boundary value problem to recover the coefficient and boundary regime of a nonlinear integro-differential equation with degenerate kernel. We propose a novel method of degenerate kernel for the case of inverse boundary value problem for the considered ordinary integro-differential equation of second order. By the aid of denotation, the integro-differential equation is reduced to a system of algebraic equations. Solving this system and using additional conditions, we obtained a system of two nonlinear equations with respect to the first two unknown quantities and a formula for determining the third unknown quantity. We proved the single-value solvability of this system using the method of successive approximations.

Estimation of the maximum correlation coefficient using Bernstein copula

In this paper, the estimation of a nonlinear stochastic dependence between signals in information-measuring and control systems is considered. The maximum correlation coefficient and corresponding optimal transformations of the signals are estimated using the Bernstein approximation. Some basic concepts of the copula theory are discussed. Main calculation formulas for the maximum correlation coefficient in the case of a degenerate stochastic kernel are presented. An estimation algorithm for the maximum correlation coefficient and corresponding optimal (nonlinear) transformations of random signals is proposed. An example of correlation estimation for random signals is given.

Low cost numerical solution for three-dimensional linear and nonlinear integral equations via three-dimensional Jacobi polynomials

In recent years, numerical methods have been introduced to solve two-dimensional Volterra and Fredholm integral equations. In this study, a numerical scheme is constructed to solve classes of linear and nonlinear three-dimensional integral equations (Volterra, Fredholm, and mixed Volterra–Fredholm). This operational approach is proposed to easily and directly solve these equations at low computational costs. The scheme is based on the Jacobi polynomials on the interval [0, 1] where three-variable Jacobi polynomials are introduced and their operational matrices of integration and product are derived. Compared to other existing methods for multidimensional problems, the Jacobi operational method eliminates the time-consuming computations and solely employs the one-dimensional operational matrix to construct corresponding multidimensional operational matrices. The absolute error of the proposed method is almost constant on the studied interval even at higher dimensions, confirming the stability of the proposed operational Jacobi method. Required theorems on the convergence of the method are proved in Jacobi-weighted Sobolev space. It is established that the error function vanishes as N increases. The method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the Block–Pulse functions, differential transform, and degenerate kernel methods.

The relationship of degenerate kernel and projection methods on Fredholm integral equations of the second kind

In this paper, we show that the degenerate kernel method for some cases, on the condition that the source function is approximated by the same way of producing degenerate kernel, becomes as a projection method. We consider two ways, including Lagrange interpolation and best approximation methods, of producing degenerate kernel approximations of more general Fredholm integral equation of the second kind. For these two ways, we show that the degenerate kernel method becomes as a Lagrange-collocation method and Galerkin method respectively.

Nonlocal Mixed-Value Problem for a Boussinesq-Type Integrodifferential Equation with Degenerate Kernel

We consider the problem of unique solvability of a mixed problem for a nonlinear Boussinesq-type fourthorder integrodifferential equation with degenerate kernel and integral conditions. A method of degenerate kernel is developed for the case of nonlinear Boussinesq-type fourth-order partial integrodifferential equation. The Fourier method of separation of variables is used. After redenoting, the integrodifferential equation is reduced to a system of countable systems of algebraic equations with nonlinear and complex right-hand sides. As a result of the solution of this system of countable systems of algebraic equations and substitution of the obtained solution in the previous formula, we get a countable system of nonlinear integral equations. To prove the theorem on unique solvability of the countable system of nonlinear integral equations, we use the method of successive approximations. Further, we establish the convergence of the Fourier series to the required function of the mixed problem. Our results can be regarded as a subsequent development of the theory of partial integrodifferential equations with degenerate kernels.

Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel

A mixed problem for a certain nonlinear third-order intregro-differential equation of the pseudoparabolic type with a degenerate kernel is considered. The method of degenerate kernel is essentially used and developed and the Fourier method of variable separation is employed for this equation. A system of countable systems of algebraic equations is first obtained; after it is solved, a countable system of nonlinear integral equations is derived. The method of sequential approximations is used to prove the theorem on the unique solvability of the mixed problem.

Нелокальная краевая задача для неоднородного псевдопараболического интегро-дифференциального уравнения с вырожденным ядром

Нелокальная краевая задача для неоднородного псевдопараболического интегро-дифференциального уравнения с вырожденным ядром

О решении интегральных уравнений одного класса смешанных и контактных задач методом вырожденных ядер.

А fairly wide class of mixed and contact problems of mechanics of deformable solids is described by Fredholm integral equations of the second kind with symmetric kernels. For solving such equations, a well-known method of degenerate kernels is developed in the paper. The stated methodology is illustrated on the example of an integral equation of the E.Ja. Shtaerman generalized contact problem on indentation of a punch into an elastic half-plane taking into account the surface structure of the base.

On the Unique Solvability of the Boundary-Value Problems for Fredholm Integrodifferential Equations with Degenerate Kernel

We study a linear boundary-value problem for systems of Fredholm integrodifferential equations with degenerate kernels and present the definition of ν-regular partition of the interval. The coefficient necessary and sufficient conditions for the unique solvability of the considered problem are established.