Collocation Method For Integral Equations Research Articles

On the approximate solution of the boundary value problem for the Helmholtz equation with impedance condition

In this work, we present a justification of collocation method for integral equation of the impedance boundary value problem for the Helmholtz equation. We also build a sequence which converges to the exact solution of our problem and we obtain an error estimate.

Special Version of Collocation Method for Integral Equations of the Third Kind With Fixed Singularities in a Kernel

We consider a linear integral equation of the third kind with fixed singularities in its kernel. We offer and prove a special version of collocation method for its approximate solving.

A discrete regularization method for Ill-posed operator equations

Discrete regularization methods are often applied for obtaining stable approximate solutions for ill-posed operator equations $$Tx=y$$ , where $$T: X\rightarrow Y$$ is a bounded operator between Hilbert spaces with non-closed range R(T) and $$y\in R(T)$$ . Most of the existing such methods involve finite rank bounded projection operators on either the domain space X or on codomain space Y or on both. In this paper, we propose a discrete regularization based on finite rank projection-like operators on some subspace of the codomain space such that their ranges need not be subspaces of the codomain space. This method not only incudes some of the exisiting projection based methods but also a quadrature based collocation method considered by the author Nair (Quadrature based collocation method for integral equations of the first kind, Advances in Computational Mathematics, 2011) for integral equations of the first kind.

Special variants of collocation method for integral equations in a singular case

The present paper deals with a linear integral equation of the third kind with fixed singularities in its kernel. We propose and substantiate special generalized methods for its approximate solving in a space of generalized funtions.

Iterated Fast Collocation Methods for Integral Equations of the Second Kind

In this paper a new iteration technique is proposed based on fast multiscale collocation methods of Chen et al. (SIAM J Numer Anal 40:344---375, 2002) for Fredholm integral equations of the second kind. It is shown that an additional order of convergence is obtained for each iteration even if the exact solution of the integral equation is non-smooth, the kernel of the integral operator is weakly singular and the matrix compression is implemented. When the solution is smooth, this leads to superconvergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the method.

Quadrature based collocation methods for integral equations of the first kind

Problem of solving integral equations of the first kind, $\int_a^b k(s, t) x(t)\, dt = y(s),\, s\in [a, b]$ arises in many of the inverse problems that appears in applications. The above problem is a prototype of an ill-posed problem. Thus, for obtaining stable approximate solutions based on noisy data, one has to rely on regularization methods. In practice, the noisy data may be based on a finite number of observations of y, say y(? 1), y(? 2), ..., y(? n ) for some ? 1, ..., ? n in [a, b]. In this paper, we consider approximations based on a collocation method when the nodes ? 1, ..., ? n are associated with a convergent quadrature rule. We shall also consider further regularization of the procedure and show that the derived error estimates are of the same order as in the case of Tikhonov regularization when there is no approximation of the integral operator is involved.

A special version of the collocation method for integral equations of the first kind

A special version of the collocation method for integral equations of the first kind

A Special Version of the Collocation Method for Integral Equations of the Third Kind

A Special Version of the Collocation Method for Integral Equations of the Third Kind

A Fast Wavelet Collocation Method for Integral Equations on Polygons

We develop a fast wavelet collocation method for solving Fredholm integral equations of the second kind with weakly singular kernels on polygons, following the general setting introduced in a recent paper [11]. Specifically, we construct wavelet functions and multiscale collocation functionals having vanishing moments on polygons. Critical issues for numerical implementation of this method are considered, such as a practical matrix compression scheme, numerical integration of weakly singular integrals, error controls of numerical quadratures and numerical solutions of the resulting compressed linear systems. The estimate of the computational complexity ensures the proposed method is a fast algorithm. Numerical experiments are presented to demonstrate the proposed methods and confirm the theoretical estimates.

Fast Collocation Methods for Second Kind Integral Equations

In this paper we develop fast collocation methods for integral equations of the second kind with weakly singular kernels. For this purpose, we construct multiscale interpolating functions and collocation functionals having vanishing moments. Moreover, we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms. An optimal order of convergence of the approximate solutions obtained from the fast algorithms is proved and the computational complexity of the algorithms is estimated. The stability of the numerical method and the condition number of the truncated coefficient matrix are analyzed.

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

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

Piecewise Continuous Collocation for Integral Equations

The collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous. Difficulties with the usual function space setting of $L_\infty (D)$ are discussed, and a satisfactory sense of point evaluation is given for elements of $L_\infty (D)$. Other approaches which are discussed include (i) reformulation as a degenerate kernel method, (ii) the prolongation-restriction framework of Noble, (iii) other function space settings, and (iv) reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation.