Numerical Solution Of Integral Equations Research Articles

Transient in-plane analysis of cracked general orthotropic planes

An edge dislocation with a time-dependent Burgers vector in an orthotropic plane is analyzed. The stress field caused by a dynamic point force in the intact plane is determined. The displacement discontinuity of dislocation and direction of point load are at arbitrary angles with the Material Principal Axes (MPA), the general orthotropic plane. The density of dislocations is employed to construct integral equations for orthotropic planes weakened by several interacting cracks, making arbitrary angles with the MPA. The numerical solution of integral equations results in the density of dislocations on a crack surface. The results are employed to determine time-dependent Stress Intensity Factors (SIFs) at the tips of a crack and Crack Opening Displacement (COD). In the problem of general orthotropic planes containing cracks subjected to dynamic loads, crack orientations may cause their partial/total closing. Under the hypothesis of frictionless contact, we devise an iterative procedure to handle crack closing, as well. By contrast to cracks in the directions of MPA, expressions for the two modes of SIFs contain the density functions of both the climb and glide of the edge dislocation. Therefore, mode I SIF at an open tip of a crack may not be positive. Conversely, positive mode I SIF may occur at a closed crack tip. Numerical results of the SIFs and COD for several examples of orthotropic planes under dynamic point loads, while weakened by single and double parallel cracks, are presented. Moreover, in double cracks cases, the interaction between cracks is studied.

Non-uniform UE-spline quasi-interpolants and their application to the numerical solution of integral equations

A construction of Marsden's identity for UE-splines is developed and a complete proof is given. With the help of this identity, a new non-uniform quasi-interpolant that reproduces the spaces of polynomial, trigonometric and hyperbolic functions are defined. Efficient quadrature rules based on integrating these quasi-interpolation schemes are derived and analyzed. Then, a quadrature formula associated with non-uniform quasi-interpolation along with Nyström's method is used to numerically solve Hammerstein and Fredholm integral equations. Numerical results that illustrate the effectiveness of these rules are presented.

Higher order Haar wavelet method for numerical solution of integral equations

Higher order Haar wavelet method for numerical solution of integral equations

COMPUTATIONAL FEATURES OF CALCULAT-ING EXTENDED CONDUCTOR ANTENNAS BASED ON THE INTEGRAL EQUATIONS METHOD

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator

<abstract><p>In this paper, under some conditions in the Banach space $ C ([0, \beta], \mathbb{R}) $, we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space $ C([0, \beta], \mathbb{R}) $. Also, we propose an effective and efficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.</p></abstract>

Complex of programs for studying magnetic fields of magnetostriction level converters

They provided the description of a complex of programs developed by the authors that implements an effective numerical method to calculate the magnetic fields of magnetostrictive level converters (MLC) of the applied type. Mathematical modeling of the MLC was carried out with its help. For this purpose, it is proposed to obtain a mathematical model of magnetic fields MLG of an overhead or submerged type on a type on ultrasonic waves of torsion by means of a numerical solution of integral equations. This generalizes the solution of the problem for any computational domain and provides a minimal approximation error. A method is also given to increase the efficiency of the calculation by using more complex schemes of difference approximation. The results can be used to optimize the designs of existing applied MLCs. We can conclude that the continuous dependences of the strength of the magnetic field of a permanent magnet on the width of the non-magnetic wall of the reservoir obtained with the help of the developed software package "Applied MLC" provide an opportunity to choose its minimum required value.

Solution of Integral Equations Using Local Splines of the Second Order

Splines are an important mathematical tool in Applied and Theoretical Mechanics. Several Problems in Mechanics are modeled with Differential Equations the solution of which demands Finite Elements and Splines. In this paper, we consider the construction of computational schemes for the numerical solution of integral equations of the second kind with a weak singularity. To construct the numerical schemes, local polynomial quadratic spline approximations and second-order nonpolynomial spline approximations are used. The results of the numerical experiments are given. This methodology has many applications in problems in Applied and Theoretical Mechanics

ON A WAY FOR SOLVING VOLTERRA INTEGRAL EQUATION OF THE SECOND KIND

There are many classes’ methods for finding of the approximately solution of Volterra integral equations of the second kind. Recently, the numerical methods have been developed for solving the integral equations of Volterra type, which is associated with the using of computers. Volterra himself suggested quadrature formula for finding the numerical solution of integral equation with the variable bounders. By using some disadvantages of mentioned methods here proposed to use some modifications of the quadrature formula which have called as the multistep methods with the fractional step-size. This method has comprised with the known methods and found some relation between constructed here methods with the hybrid methods. And also, the advantages of these methods are shown. Constructed some simple methods with the fractional step-size, which have the degree p≤4 of the receiving results. Here is applied one of suggested methods to solve some model problem and receive results, which are corresponding to theoretical results

Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only 10 steps.

Splines of the Fourth Order Approximation and the Volterra Integral Equations

This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.

Numerical Solution of Integral Equation by using New Modified Adomian Decomposition Method and Newton Raphson Methods

In this paper, we discuss the numerical solution of Adomian decomposition method and Taylor’s expansion method in Volterra linear integral equation. And we apply modified Adomian decomposition method and Newton Raphson method in Volterra nonlinear integral equation with the help of example and estimated an error in MATLAB 13 versions.

Quintic B-spline collocation method for numerical solution of free vibration of tapered Euler-Bernoulli beam on variable Winkler foundation

The collocation method is the method for the numerical solution of integral equations and partial and ordinary differential equations. The main idea of this method is to choose a number of points in the domain and a finite-dimensional space of candidate solutions. So, that solution satisfies the governing equation at the collocation points. The current paper involves developing, and a comprehensive, step-by step procedure for applying the collocation method to the numerical solution of free vibration of tapered Euler-Bernoulli beam. In this stusy, it is assumed the beam rested on variable Winkler foundation. The simplicity of this approximation method makes it an ideal candidate for computer implementation. Finally, the numerical examples are introduced to show efficiency and applicability of quintic B-spline collocation method. Numerical results are demonstrated that quintic B-spline collocation method is very competitive for numerical solution of frequency analysis of tapered beam on variable elastic foundation.

Numerical solution of integral equations with Monte Carlo-collocation (MC-Coll) method and its application in neutron transport

In this study, we introduce a new procedure for the numerical solution of linear and nonlinear integral equations by combining the Monte Carlo simulation and collocation method. One of the major advantages of this method is reducing calculations by converting the integral equation to a system of algebraic equations. The numerical examples of various types of integral equations indicate the efficiency and accuracy of the proposed method. The results show this method is able to solve linear and nonlinear integral equations and also nonlinear weakly singular equations, with remarkable accuracy. Finally, the method is generalized for solving integral-differential equation of the neutron transport for the stationary case.

Numerical Treatment for Solving Fuzzy Volterra Integral Equation by Sixth Order Runge-Kutta Method

There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge–Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.

A Numerical Solution of Integral Equations with a Logarithmic Kernel by Product and Gauss-Legendre Methods. (Dept. P)

A Numerical Solution of Integral Equations with a Logarithmic Kernel by Product and Gauss-Legendre Methods. (Dept. P)

On the numerical solution of integral equations of the second kind over infinite intervals

In this paper, we discuss the numerical solution of a class of linear integral equations of the second kind over an infinite interval. The method of solution is based on the reduction of the problem to a finite interval by means of a suitable family of mappings so that the resulting singular equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. Several selected numerical examples are presented and discussed to illustrate the application and effectiveness of the proposed approach.

A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels

Abstract As is well known, piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC) are used to approximate the weakly singular integrals $$\begin{equation*}I(a,b,x) =\int^b_a \frac{u(y)}{|x-y|^\gamma}\textrm{d}y, \quad x \in (a,b),\quad 0< \gamma <1,\end{equation*}$$which have local truncation errors $\mathcal{O} (h^2 )$ and $\mathcal{O} (h^{4-\gamma } )$, respectively. Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., $\lambda u(x)- I(a,b,x) =f(x)$, $\lambda \neq 0$, the global convergence rates are also $\mathcal{O} (h^2 )$ and $\mathcal{O} (h^{4-\gamma } )$ by PLC and PQC in Atkinson (2009, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press). In this work we study the following nonlocal problems, which are similar to the above Fredholm integral equations: $$\begin{equation*}\int^b_a \frac{u(x)-u(y)}{|x-y|^\gamma}\textrm{d}y =f(x), \quad x \in (a,b),\quad 0< \gamma <1. \end{equation*}$$In the first part of this paper we prove that the weakly singular integrals $I(a,b,x)$ have optimal local truncation error $\mathcal{O}(h^4\eta _i^{-\gamma } )$ by PQC, where $\eta _i=\min \left \{x_i-a,b-x_i\right \}$ and $x_i$ coincides with an element junction point. Then the sharp global convergence orders $\mathcal{O}\left (h\right )$ and $\mathcal{O} (h^3)$ by PLC and PQC, respectively, are established for nonlocal problems. Finally, numerical experiments are shown to illustrate the effectiveness of the presented methods.

Numerical solution of integral equations using Bernoulli wavelets

Numerical solution of integral equations using Bernoulli wavelets

Numerical solution of an integral equation arising in the problem of cruciform crack using Daubechies scale function

This paper is concerned with obtaining approximate numerical solution of a classical integral equation of some special type arising in the problem of cruciform crack. This integral equation has been solved earlier by various methods in the literature. Here, approximation in terms of Daubechies scale function is employed. The numerical results for stress intensity factor obtained by this method for a specific forcing term are compared to those obtained by various methods available in the literature, and the present method appears to be quite accurate.

Algorithms for numerical solution of integral equations of three-dimensional scalar diffraction problem

The three-dimensional diffraction problem of stationary acoustic waves on a homogeneous inclusion is considered. It is reduced to the weakly singular boundary Fredholm integral equations of the first kind with one unknown function, each of which is conditionally equivalent to the original problem. By using the original method of averaging the integral operators kernels, these equations are approximated by systems of linear algebraic equations. The resulting systems are solved numerically by the generalized minimal residual method (GMRES). Then the solution of the initial problem is calculated. To find the solution on the spectrum of integral operators, where the condition of equivalence of integral equations to the original problem is violated, the interpolation solution method is proposed. It does not require knowledge of the spectrum and allows us to find the approximated solutions with high accuracy. The proposed algorithms have been implemented in the computing cluster of Computing Center FEB RAS. The results of the calculations that allow us to assess the possibilities of this approach are presented.