Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

This paper is concerned with a trigonometric Hermite wavelet Galerkin method for the Fredholm integral equations with weakly singular kernel. The kernel function of this integral equation considered here includes two parts, a weakly singular kernel part and a smooth kernel part. The approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric Hermite wavelet constructed by E. Quak [Trigonometric wavelets for Hermite interpolation, Math. Comp. 65 (1996) 683–722] are developed. The use of trigonometric Hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix. It is shown that we only need to compute and store O(N)$\mathrm{O}(N)$ entries for the weakly singular kernel representation matrix with dimensions N²$N^2$ which can reduce the whole computational cost and storage expense. The computational schemes of the resulting matrix elements are provided for the weakly singular kernel function. Furthermore, the convergence analysis is developed for the trigonometric wavelet method in this paper.     

Superconvergence of collocation methods for a class of weakly singular Volterra integral equations

We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters, superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical examples are given which illustrate the theoretical results.     

A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind

In this paper an iterative approach for obtaining approximate solutions for a class of nonlinear Fredholm integral equations of the second kind is proposed. The approach contains two steps: at the first one, we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation, solution of the discretized form converges to the exact solution of the problem. Following that, in the next step, solution of the discretized form is approximated by an iterative approach. We finally on some examples show the efficiency of the proposed approach.     

Asymptotic error expansion for the Nyström method for nonlinear Fredholm integral equations of the second kind

In this paper the asymptotic error expansion for the Nyström method for one-dimensional nonlinear Fredholm integral equations of the second kind is considered. We show that the Nyström solution admits an error expansion in powers of the step-sizeh. Thus Richardson's extrapolation can be performed on the solution, and this will greatly increase the accuracy of the numerical solution.The project has been supported by the National Natural Science Foundation of China.     

Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations

We discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular Volterra integral equation associated with certain heat conduction problems. This work completes the previous studies of numerical methods for this type of equations with noncompact kernel. In particular, a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces. Some numerical examples illustrate the theoretical results.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented.     

A Nyström interpolant for some weakly singular nonlinear Volterra integral equations

We consider a second kind weakly singular nonlinear Volterra–Hammerstein integral equation defined by a compact operator and derive a Nyström type interpolant of the solution based on Gauss–Radau nodes. We prove the convergence of the interpolant and derive convergence estimates. For equations with nonlinearity of algebraic kind, we improve the rate of convergence by using a smoothing transformation. Some numerical examples are given.     

A Nyström interpolant for some weakly singular linear Volterra integral equations

We consider a second kind weakly singular Volterra integral equation defined by a non-compact operator and derive a Nyström type interpolant of the solution based on Gauss-Radau nodes. Assuming the stability of the interpolant, which is confirmed by the numerical tests, we derive convergence estimates.     

Fast solution methods for fredholm integral equations of the second kind

Summary The main purpose of this paper is to describe a fast solution method for one-dimensional Fredholm integral equations of the second kind with a smooth kernel and a non-smooth right-hand side function. Let the integral equation be defined on the interval [–1, 1]. We discretize by a Nyström method with nodes {cos(j/N)} _j ^=0/N . This yields a linear system of algebraic equations with an (N+1)×(N+1) matrixA. GenerallyN has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation. We show by Fourier analysis thatA can be approximated well by , a low-rank modification of the identity matrix. ReplacingA by in the linear system of algebraic equations yields a new linear system of equations, whose elements, and whose solution , can be computed inO (N logN) arithmetic operations. If the kernel has two more derivatives than the right-hand side function, then is shown to converge optimally to the solution of the integral equation asN increases.We also consider iterative solution of the linear system of algebraic equations. The iterative schemes use bothA andÃ. They yield the solution inO (N ²) arithmetic operations under mild restrictions on the kernel and the right-hand side function.Finally, we discuss discretization by the Chebyshev-Galerkin method. The techniques developed for the Nyström method carry over to this discretization method, and we develop solution schemes that are faster than those previously presented in the literature. The schemes presented carry over in a straightforward manner to Fredholm integral equations of the second kind defined on a hypercube.     

On projective methods of approximate solution of singular integral equations

The estimate for the rate of convergence of approximate projective methods with one iteration is established for one class of singular integral equations. The Bubnov-Galerkin and collocation methods are investigated.This author is also known as A. V. Dzhishkariani.     

On multilevel iterative methods for integral equations of the second kind and related problems

Summary We describe a unifying framework for multigrid methods and projection-iterative methods for integral equations of the second kind, and for the iterative aggregation method for solving input-output relations. The methods are formulated as iterations combined with a defect correction in a subspace. Convergence proofs use contraction arguments and thus involve the nonlinear case automatically. Some new results are presented.     

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.     

Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind

Summary This paper deals with the question of the attainable order of convergence in the numerical solution of Volterra and Abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals. The use of a (nonlinear) variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels.     

Nyström type methods for Fredholm integral equations with weak singularities

Nyström type methods are constructed and justified for a class of Fredholm integral equations of the second kind with kernels which may have weak diagonal and boundary singularities. The proposed approach is based on a suitable smoothing change of variables and product integration techniques. Global convergence estimates are derived and a collection of numerical results is given.     

Quadrature method for singular integral equations on closed curves

Summary A method which combines quadrature with trigonometric interpolation is proposed for singular integral equations on closed curves. For the case of the circle, the present method is shown to be equivalent to the trigonometric -collocation method together with numerical quadrature for the compact term, and is shown to be stable inL ² provided the operatorA is invertible inL ². The results are extended to arbitraryC curves, to give a complete error analysis in the scale of Sobolev spacesH ^s . In the final section the case of a non-invertible operatorA is considered.     

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

In this work, we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels. Particularly, we consider the case when the underlying solutions are sufficiently smooth. In this case, the proposed method leads to a fully discrete linear system. We show that the fully discrete integral operator is stable in both infinite and weighted square norms. Furthermore, we establish that the approximate solution arrives at an optimal convergence order under the two norms. Finally, we give some numerical examples, which confirm the theoretical prediction of the exponential rate of convergence.     

Backward differentiation type formulas for Volterra integral equations of the second kind

Summary Numerical integration formulas are discussed which are obtained by differentiation of the Volterra integral equation and by applying backward differentiation formulas to the resulting integro-differential equation. In particular, the stability of the method is investigated for a class of convolution kernels. The accuracy and stability behaviour of the method proposed in this paper is compared with that of (i) a block-implicit Runge-Kutta scheme, and (ii) the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas. The present method is particularly advantageous in the case of stiff Volterra integral equations.     

Superconvergence of piecewise polynomial Galerkin approximations,for Fredholm integral equations of the second kind

Summary Piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind are shown to posses superconvergence properties in some circumstances.     

Midpoint collocation for Cauchy singular integral equations

Summary A Cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid. Even if the grid is non-uniform, suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations. The same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential.This work was supported by the Australian Research Council through the program grant Numerical analysis for integrals, integral equations and boundary value problems