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Nyström type methods for Fredholm integral equations with weak singularities. (English) Zbl 1193.65226
Nyström type methods are constructed and justified for a class of Fredholm integral equations of the second kind
\[ u(x)=\int_0^1[a(x,y)|x-y|^{-v}+b(x,y)] u(y)dy=f(x), \;\;0\leq x\leq 1, \]
where \(0<v<1,\) functions \(a(x,y)\) and \(b(x,y)\) may have boundary singularities with respect to \(y:\)
\[ a(x,y),b(x,y)\in C^m([0,1]\times (0,1)), m\in N_0, \]
\[ \left|\left(\frac{\partial}{\partial x} \right)^k \left( \frac{\partial}{\partial y} \right)^l a(x,y) \right|\leq cy^{-\lambda_0-l}(1-y)^{-\lambda_1-l} \]
\[ \left|\left(\frac{\partial}{\partial x} \right)^k \left( \frac{\partial}{\partial y} \right)^l b(x,y)\right|\leq c y ^{-\mu_0-l}(1-y)^{-\mu_1-l} \]
where \((x,y)\in [0,1]\times (0,1),\;\;k,l\in N_0,\;\;k+l\leq m,\) \(\lambda_0,\lambda_1,\mu_0,\mu_1\in R,\) \(\mu_0,\mu_1<1,\) \(N_0={0}\cup N,\) \(N={1,2,\dots}, R\in (-\infty,\infty)\).
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.

MSC:
65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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