On kalandiya’ method for the numerical solution of singular integral equations

For the numerical solution of one-dimensional singular integral equations with Cauchy type kernels, one can use an appropriate quadrature rule and an appropriate set of collocation points for the reduction of this equation to a system of linear equations. In this short paper, we use as collocation points the nodes of the quadrature rule and we rederive, in a more direct manner, Kalandiya’ method for the numerical solution of the aforementioned class of equations, which was originally based on a trigonometric interpolation formula. Furthermore, we test this method in numerical applications. Finally, a discussion on the accuracy of the same method is made.     

Chebyshev collocation method for solving singular integral equation with cosecant kernel

A collocation method based on Chebyshev polynomials is proposed for solving cosecant-type singular integral equations (SIE). For solving SIE, difficulties lie in its singular term. In order to remove singular term, we introduce Gauss–Legendre integration and integral properties of the cosecant kernel. An advantage of this method is to approximate the best uniform approximation by the best square approximation to obtain the unknown coefficients in the method. On the other hand, the convergence is fast and the accuracy is high, which is verified by the final numerical experiments compared with the existing references.     

CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel

In this paper, a computational method for numerical solution of a class of integro-differential equations with a weakly singular kernel of fractional order which is based on Cos and Sin (CAS) wavelets and block pulse functions is introduced. Approximation of the arbitrary order weakly singular integral is also obtained. The fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations by using the operational matrix of fractional integration of CAS wavelets. The error analysis of CAS wavelets is given. Finally, the results of some numerical examples support the validity and applicability of the approach.     

A natural quadrature formula for the numerical evaluation of the Macgregor-Westergaard complex potentials es crack problems

The method of singular integral equations can be used for the numerical solution of crack problems in plane and antiplane elasticity. Here we consider the problem of the subsequent numerical evaluation of the stress components in the whole cracked medium by using the MacGregor-Westergaard complex potentials. To this end we use a modified quadrature formula for Cauchy type (but not principal value) integrals and their derivatives, where the poles of the integrands are properly taken into consideration. This is achieved by using a natural interpolation-extrapolation formula for singular integral equations and, for this reason, the new term ‘natural quadrature formula’ is proposed. Two simple applications to specific crack problems, based on the Gauss- and Lobatto-Chebyshev quadrature formulas, show the efficiency of the suggested quadrature formula.     

On the error analysis of the numerical solution of linear Volterra integral equations of the second kind

In the numerical solution of linear Volterra integral equations, two kinds of errors occur. If we use the collocation method, these errors are the collocation and numerical quadrature errors. Each error has its own effect in the accuracy of the obtained numerical solution. In this study we obtain an error bound that is sum of these two errors and using this error bound the relation between the smoothness of the kernel in the equation and also the length of the integration interval and each of these two errors are considered. Concluded results also are observed during the solution of some numerical examples.     

A Galerkin collocation method for some integral equations of the first kind

Here we present a certain modified collocation method which is a fully discretized numerical method for the solution of Fredholm integral equations of the first kind with logarithmic kernel as principal part. The scheme combines high accuracy from Galerkin's method with the high speed of collocation methods. The corresponding asymptotic error analysis shows optimal order of convergence in the sense of finite element approximation. The whole method is an improved boundary integral method for a wide class of plane boundary value problems involving finite element approximations on the boundary curve. The numerical experiments reveal both, high speed and high accuracy.     

Efficient methods for highly oscillatory integrals with weak and Cauchy singularities

In this paper, we present two fast and accurate numerical schemes for the approximation of highly oscillatory integrals with weak and Cauchy singularities. For analytical kernel functions, by using the Cauchy theorem in complex analysis, we transform the integral into two line integrals in complex plane, which can be calculated by some proper Gauss quadrature rules. For general kernel functions, the non-oscillatory and nonsingular part of the integrand is replaced by a polynomial interpolation in Chebyshev points, and the integral is then evaluated by using recurrence relations. Furthermore, several numerical experiments are shown to verify the validity of such methods.     

Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations

Direct methods for solving Cauchy-type singular integral equations (S.I.E.) are based on Gauss numerical integration rule [1] where the S.I.E. is reduced to a linear system of equations by applying the resulting functional equation at properly selected collocation points. The equivalence of this formulation with the one based on the Lagrange interpolatory approximation of the unknown function was shown in the paper. Indirect methods for the solution of S. I. E. may be obtained after a reduction of it to an equivalent Fredholm integral equation and an application of the same numerical technique to the latter. It was shown in this paper that both methods are equivalent in the sense that they give the same numerical results. Using these results the error estimate and the convergence of the methods was established.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

A vortex-lattice method in the linear theory on a two-dimensional supercavitating flat plate foil

The vortex-lattice method has been found very satisfactory in the case of steady subsonic wing theory, however, the discrete numerical methods, such as the vortex-lattice method, have not been studied in detail for supercavitating flows. One of the discrete numerical method, a vortex-lattice method, is developed in the present paper for cavitating flows around a two dimensional flat plate foil. The governing equations in the linear theory are represented as a set of coupled integral equations with Cauchy kernel, and there are unknown functions which are not under integral signs. For solving them, they are exchanged to an alternative set of coupled integral equations by a new variable, and the present vortex-lattice method is schemed for equal spacing of the vortices and sources in this new variable. The position of the collocation points is determined, and it is sufficient to treat unknown functions which are not integral signs as step functions. Moreover, the proof of the convergence of this method is shown and the accuracy is estimated.     

Numerische Behandlung einer Volterraschen Integralgleichung

This paper deals with a generalized nonlinear Volterra integral equation, whose kernel contains the unknown function at two different arguments. The equation is solved by collocation with piecewise Hermite-polynomials. The method has order 2m,m ∈ ?, if polynomials of degree 2m?1 and appropriate integration formulas are used. The collocation points must be chosen in accordance with a certain stability condition.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

A numerical approach for solving the high-order linear singular differential-difference equations

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential-difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential-difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).     

The axisymmetric problem for a thick-walled spherical shell with an external edge crack

This paper considers the elasto-static axisymmetric problem for a thick-walled spherical shell containing a circumferential edge crack on the outer surface. Using the standard integral transform technique, the problem is formulated in terms of a singular integral equation of the first kind which has a generalized Cauchy kernel as the dominant part. As an example the axisymmetric load problem has been solved. The integral equation is solved numerically and the influences of the geometrical configurations on the stress intensity factors and crack opening displacements are shown graphically in detail.     

A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions

The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.

     

Generalized solutions to integral equations in the problem of identification of nonlinear dynamic models

Classes of nonlinear integral Volterra equations occurring in identifying dynamic systems are studied. A solution to a nonlinear system of integral Volterra equations of the first kind is constructed in the class of generalized functions with a point support in the form of a sum of singular and regular parts. In obtaining a singular part of the solution, a determined system of linear algebraic equations is used. The method of sequential approximations together with the method of undetermined coefficients allow constructing a regular part. Theorems of existence and uniqueness of generalized solutions are proved.     

Some remarks on the numerical solution of cauchy-type singular integral equations with index equal to —1

Several remarks concerning the Gauss-, Radau- and Lobatto-Jacobi direct quadrature methods of numerical solution of Cauchy-type singular integral equations of the first or the second kind with index equal to −1 are made. These remarks concern mainly the construction of the system of linear algebraic equations to which the above equations are reduced by using an appropriate number of collocation points. The same remarks apply also to the special cases where the Gauss-, Radau- and Lobatto-Chebyshev quadrature methods are used. These remarks are of particular interest for the numerical solution of smooth contact problems in the theory of elasticity.     

求解Cauchy型奇异积分方程的数值方法

1.引 言 断裂力学中许多裂纹问题的数学模型都可归结为奇异积分方程(SIE)[1,2].由于这些奇异积分方程的封闭解一般情况下都难以得到,因而数值方法受到广泛的注意.Muskhelishvili[3]对奇异积分方程的一般理论进行了深入的研究.这些研究成果为奇异积分方程的求解,不论     

Normalization techniques for the SVE of the Green function of Helmholtz operator

Electromagnetic and acoustic scattering problems can be usually formulated by suitable integral equations, where the kernel is given in terms of the fundamental solution of the Helmholtz operator. We can consider a special analytic method for the singular value expansion (SVE) of this integral kernel. Note that this is an important tool for the numerical solution of scattering problems, in fact, from the knowledge of the SVE of the integral kernel, we can easily solve the corresponding integral equation. In this paper, we study the numerical approximation of the SVE of this integral kernel, where we have to consider the asymptotic behavior of the Bessel functions.     

On the natural interpolation formula for cauchy type singular integral equations of the first kind

A Cauchy type singular integral equation of the first kind can be numerically solved either directly, through the use of a Gaussian numerical integration rule, or by reduction to an equivalent Fredholm integral equation of the second kind, where the Nyström method is applicable. In this note it is proved that under appropriate but reasonable conditions the expressions of the unknown function of the integral equation, resulting from the natural interpolation formulae of the direct method, as well as of the Nyström method, are identical along the whole integration interval.