### A certain inversion problem for the ray transform with incomplete data.

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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...

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.

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

Galerkin discretizations of integral equations in ${\mathbb{R}}^{d}$ require the evaluation of integrals $I={\int}_{{S}^{\left(1\right)}}{\int}_{{S}^{\left(2\right)}}g(x,y)\mathrm{d}y\mathrm{d}x$ 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$\ne $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 ${\mathcal{Q}}_{N}$ using N function evaluations of g which achieves exponential convergence |I – ${\mathcal{Q}}_{N}$| ≤C exp(–rNγ) with...