Regularized collocation method for Fredholm integral equations of the first kind

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An “unregularized” use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.     

Product integration methods for Volterra integral equations of the first kind

The numerical solution of Volterra integral equations of the first kind can be accomplished if the integral is replaced by certain simple quadrature rules, such as the midpoint or the trapezoidal methods. When the kernel of the integral equation oscillates more rapidly than the solution one can use product integration techniques to increase the accuracy. Such an approach is investigated in this paper.     

Reducible quadrature methods for Volterra integral equations of the first kind

Quadrature rules, generated by linear multistep methods for ordinary differential equations, are employed to construct a wide class of direct quadrature methods for the numerical solution of first kind Volterra integral equations. Our class covers several methods previously considered in the literature. The methods are convergent provided that both the first and second characteristic polynomial of the linear multistep method satisfy the root condition. Furthermore, the stability behaviour for fixed positive values of the stepsizeh is analyzed, and it turns out that convergence implies (fixedh) stability. The subclass formed by the backward differentiation methods up to order six is discussed and illustrated with numerical examples.     

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.     

Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ϕ(t, s) = (t − s)^−μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938–950], the error analysis for this approach is carried out for 0 < μ < 1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution.     

Stability of quadrature rule methods for first kind volterra integral equations

Gladwin [4] proved that Newton-Gregory formulas of order larger than 2 produce unstable algorithms when applied to nonlinear Volterra integral equations of the first kind. It is shown that similar results are true for all interpolatory quadrature rules using equidistant nodes. Upper bounds for the error order of quadrature rules, which lead to stable methods are given. Some higher order stable methods are indicated.     

On the asymptotic convergence of a collocation method for some boundary integral equations of the first kind

In this paper we present a certain collocation method for the numerical solution of a class of boundary integral equations of the first kind with logarithmic kernel as principle part. The transformation of the boundary value problem into boundary singular integral equation of the first kind via single-layer potential is discussed. A discretization and error representation for the numerical solution of boundary integral equations has been given. Quadrature formulae have been proposed and the error arising due to the quadrature formulae used has been estimated. The convergence of the solution with respect to the proposed numerical algorithm is shown and finally some numerical results have been presented.     

第二类Volterra型积分方程的Chebyshev-Legendre谱配置方法

提出了一种新的求解第二类线性Volterra型积分方程的Chebyshev谱配置方法.该方法分别对方程中积分部分的核函数和未知函数在Chebyshev-Gauss-Lobatto点上进行插值,通过Chebyshev-Legendre变换,把插值多项式表示成Legendre级数形式,从而将积分转换为内积的形式,再利用Legendre多项式的正交性进行计算.利用Chebyshev插值算子在不带权范数意义下的逼近结果,对该方法在理论上给出了L∞范数意义下的误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.     

Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations

Summary An existence and uniqueness result is given for nonlinear Volterra integral equations of the first kind. This permits, by means of analogous discrete manipulations, a general convergence analysis for a wide class of discretization methods for nonlinear first kind Volterra integral equations to be presented. A concept of optimal consistency allows twosided error bounds to be derived.     

An algorithm based on the regularization and integral mean value methods for the Fredholm integral equations of the first kind

In this paper, an algorithm based on the regularization and integral mean value methods, to handle the ill-posed multi-dimensional Fredholm equations, is introduced. The application of this algorithm is based on the transforming the first kind equation to a second kind equation by the regularization method. Then, by converting the first kind to a second kind, the integral mean value method is employed to handle the resulting Fredholm integral equations of the second kind. The efficiency of the approach will be shown by applying the procedure on some examples.     

Continuous time collocation methods for Volterra-Fredholm integral equations

Summary Integral equations of mixed Volterra-Fredholm type arise in various physical and biological problems. In the present paper we study continuous time collocation, time discretization and their global and discrete convergence properties.     

Full collocation methods for some boundary integral equations

In this paper we propose a fully discretized version of the collocation method applied to integral equations of the first kind with logarithmic kernel. After a stability and convergence analysis is given, we prove the existence of an asymptotic expansion of the error, which justifies the use of Richardson extrapolation. We further show how these expansions can be translated to a new expansion of potentials calculated with the numerical solution of a boundary integral equation such as those treated before. Some numerical experiments, confirming our theoretical results, are given. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the stability of the spline collocation method for a class of integral equations of the first kind

This paper is concerned with the stability of the spline collocation method for a class of integral equations of the first kind with logarithmic kernels. It is shown that a proper choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence for the approximate solutions.     

Asymptotic regularization for integral equations of the first kind

The paper concerns solving a certain class of i11-posed problems including integral equationsof the first kind. The proposed regularization consists in replacing the considered i11-posedproblem by an ass ociated dynamical system. Well posedness of introduced system and an asymptotic connection of its solution with the solution we look for are proved.     

Polynomially based multi-projection methods for Fredholm integral equations of the second kind

In this paper, we propose a multi-projection and iterated multi-projection methods for Fredholm integral equations of the second kind with a smooth kernel using polynomial bases. We obtain super-convergence rates for the approximate solutions, more precisely, we prove that in M-Galerkin and M-collocation methods not only iterative solution approximates the exact solution u in the supremum norm with the order of convergence n^-4k, but also the derivatives of approximate the corresponding derivatives of u in the supremum norm with the same order of convergence, n being the degree of polynomial approximation and k being the smoothness of the kernel.     

New variants of the collocation method for integral equations of the third kind

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 47–53, August, 1991.     

Quadrature methods for logarithmic-kernel integral equations on closed curves

The recently proposed method of Sloan and Burn for the logarithmic-kernelfirst-kind integral equation on closed curves is modified, soas to permit high-order convergence in appropriate negativenorm even when the solution is not smooth. In addition, theresult of Sloan and Burn for the original method is shown tobe exrtendable from circular to general smooth curves withoutloss.