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We propose an adaptive finite element method for the solution of a linear Fredholm integral equation of the first kind. We derive a posteriori error estimates in the functional to be minimized and in the regularized solution to this functional, and formulate corresponding adaptive algorithms. To do this we specify nonlinear results obtained earlier for the case of a linear bounded operator. Numerical experiments justify the efficiency of our a posteriori estimates applied both to the computationally...

An Adaptive Numerical Method for Solving Linear Fredholm Integral Equations of the First Kind.

R.J. Hanson, J.L. Phillips (1975)

Numerische Mathematik

An operational Haar wavelet method for solving fractional Volterra integral equations

Habibollah Saeedi, Nasibeh Mollahasani, Mahmoud Mohseni Moghadam, Gennady N. Chuev (2011)

International Journal of Applied Mathematics and Computer Science

A Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the validity and applicability of the developed method.

Boundary integral equations of plane elasticity in domains with peaks.

Maz'ya, V.G., Soloviev, A.A. (2001)

Georgian Mathematical Journal

Boundary value problems and controllability of linear systems with right invertible operators [Book]

Nguyen Van Mau (1992)

Continuous dependence of inverse fundamental matrices of generalized linear ordinary differential equations on a parameter

Zdeněk Halas (2005)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The problem of continuous dependence for inverses of fundamental matrices in the case when uniform convergence is violated is presented here.

Eine Verschärfung der Poincaré-Ungleichung

Leo Boček (1983)

Časopis pro pěstování matematiky

Entropy of $C (X)$ -valued operators and diverse applications.

Carl, Bernd, Edmunds, David E. (2001)

Journal of Inequalities and Applications [electronic only]

Exponential convergence of hp quadrature for integral operators with Gevrey kernels

Alexey Chernov, Tobias von Petersdorff, Christoph Schwab (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

Galerkin discretizations of integral equations in $ℝ^{d}$ require the evaluation of integrals $I = \int_{S^{(1)}} \int_{S^{(2)}} g (x, y) d y d x$ where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for x $\neq$ y and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules $𝒬_{N}$ using N function evaluations of g which achieves exponential convergence |I – $𝒬_{N}$ | ≤C exp(–rNγ) with...