Cauchy-type singular integral equation with constant coefficients on the real line

In this paper, exact solutions of the characteristic and dominant equation with Cauchy kernel on the real line are presented. Next, trigonometric polynomials are used to derive approximate solutions of these equations. Moreover, estimations of errors of the approximated solutions are presented and proved.     

Numerical solution of a singular integral equation arising in a cruciform crack problem

A singular integral equation arising in a cruciform crack problem is investigated in the present paper. Based on the convex technique, the piecewise Taylor-series expansion method is extended by introducing a weight parameter. An approximate solution of the singular integral equation is constructed and its convergence and error estimate are made. The variations of the approximate solutions associating with stress intensity factors are analyzed by considering internal pressures of power and sine functions, respectively. By comparing with the known methods, the observations reveal that a good approximation can be achieved using less derivative times, less discretization points, and a suitable weight parameter. The obtained results show that the crack growth is dependent on applied mechanical loadings.     

Reproducing kernel method of solving singular integral equation with cosecant kernel

In the paper, a reproducing kernel method of solving singular integral equations (SIE) with cosecant kernel is proposed. For solving SIE, difficulties lie in its singular term. In order to remove singular term of SIE, an equivalent transformation is made. Compared with known investigations, its advantages are that the representation of exact solution is obtained in a reproducing kernel Hilbert space and accuracy in numerical computation is higher. On the other hand, the representation of reproducing kernel becomes simple by improving the definition of traditional inner product and requirements for image space of operators are weakened comparing with traditional reproducing kernel method. The final numerical experiments illustrate the method is efficient.     

A wavelet method for solving singular integral equation of MHD

This paper presents Haar wavelet approximation to solve a singular integral equation which has singularities on a diagonal of the domain R=[-1,1]×[-1,1]. The singularities arise basically due to modified Bessel function K₀ which appears as a part of the kernel. Thus the integral equation is weakly (logarithmically) singular only. The problem is solved considering all the singularities of the kernel and results are examined for approximations of different orders. Our interest to solve the problem using Haar wavelet basis is due to its simplicity and efficiency in numerical approximation. The results show convergence trend as mesh is refined. Comparison is made with some available results obtained earlier by partial consideration of the singularities.     

A multiscale Galerkin method for the hypersingular integral equation reduced by the harmonic equation

The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot(θ/2). Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j+1 × 2j+1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

Legendre spectral Galerkin method for second-kind Volterra integral equations

The Legendre spectral Galerkin method for the Volterra integral equations of the second kind is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ₂ norm) will decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical examples are given to illustrate the theoretical results.      

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.     

Existence of solution for a singular critical elliptic equation

In this paper, a singular semilinear elliptic problem involving the critical Sobolev exponent is studied by variational method, the existence of a solution is proved under certain conditions. The Hardy inequality is used and plays an important role in the discussion.     

The exact solution of a class of Volterra integral equation with weakly singular kernel

In this paper, the weakly singular Volterra integral equations with an infinite set of solutions are investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation. The aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory. Applying weighted integral, we provide a new definition for reproducing kernel space and obtain reproducing kernel function. Using the good properties of reproducing kernel function, the only smooth solution is exactly expressed in the form of series. The n-term approximate solution is obtained by truncating the series. Meanwhile, we prove that the derivative of approximation converges to the derivative of exact solution uniformly. The final numerical examples compared with other methods show that the method is efficient.     

Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. For hypersingular integral equations in 2D with a positive-order Sobolev space, we analyse the mathematical relation between the (h???h/2)-error estimator from [S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008), pp. 135–162], the two-level error estimator from [M. Maischak, P. Mund, and E. Stephan, Adaptive multilevel BEM for acoustic scattering, 585 Comput. Methods Appl. Mech. Eng. 150 (1997), pp. 351–367], and the averaging error estimator from [C. Carstensen and D. Praetorius, Averaging techniques for the a posteriori bem error control for a hypersingular integral equation in two dimensions, SIAM J. Sci. Comput. 29 (2007), pp. 782–810]. All of these a posteriori error estimators are simple in the following sense: first, the numerical analysis can be done within the same mathematical framework, namely localization techniques for the energy norm. Second, there is almost no implementational overhead for the realization.     

A semi-analytic solution for multiple curved cracks emanating from circular hole using singular integral equation

This paper provides an elastic solution for an infinite plate containing multiple curved edge cracks emanating from a circular hole. A fundamental solution is suggested, which represents a particular solution for a concentrated dislocation in an infinite plate with the traction free hole. The generalized image method and the concept of the modified complex potentials are used in the derivation of the fundamental solution. After using the fundamental solution and placing the distributed dislocations at the prospective sites of cracks, a singular integral equation is formulated. The singular integral equation is solved by using the curve length method in conjunction with the semi-opening quadrature rule. By taking an additional point dislocation at the hole center, the number of the unknowns is equal to the number of the resulting algebraic equations. This is a particular advantage of the suggested method. Finally, several numerical examples are given to illustrate the efficiency of the method presented. Numerical examinations are carried out and sufficient accurate results have been found.     

Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels

We propose iterated fast multiscale Galerkin methods for the second kind Fredholm integral equations with mildly weakly singular kernel by combining the advantages of fast methods and iteration post-processing methods. To study the super-convergence of these methods, we develop a theoretical framework for iterated fast multiscale schemes, and apply the scheme to integral equations with weakly singular kernels. We show theoretically that even the computational complexity is almost optimal, our schemes improve the accuracy of numerical solutions greatly, and exhibit the global super-convergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the methods.     

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.     

On solution sequences of the singular Liouville equations with an integral constraint

In this paper,we analyze the blow-up behavior of sequences{uk}satisfying the following conditionsΔuk=|x|2αk V k eukinΩ,(0.1)whereΩR2,V k→V in C1,|Vk|≤A,0a≤Vk≤b,0≤αk→α∈(0,∞),andΩ|x|2αkeuk dx≤Λ1.(0.2)Furthermore,we assume that there exists someq∈(1,2)such that rq 2Br(p)|uk|qdx≤Λ2(0.3)for anyB r(p)Ω.As a result,we give a new proof of the concentration-compactness theorem for the mean feld equation.     

On the approximate solution of some two‐dimensional singular integral equations

An approximation method for a wide class of two‐dimensional integral equations is proposed. The method is based on using a special function system. Orthonormality and good interaction with fundamental integral operators arising in partial differential equations are remarkable properties of this system. In addition, all the basis elements can easily be calculated by recurrence relations. Taking into account these properties we construct a numerical algorithm which does not require additional effort (such as quadrature) to compute the values of the fundamental operators on the basis elements. Copyright © 2001 John Wiley & Sons, Ltd.     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

Numerical solution of some classes of integral equations using Bernstein polynomials

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.     

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time‐step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. L²‐norm error estimates are derived for these procedures. Compared with the work of Dawson and Dupont [Math Comp 58 (1992), 21–35], these L²‐norm error estimates avoid the loss of H^?1/2 factor. Experimental results are presented to confirm the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009