Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets (Q5948572)

The authors investigate wavelet-Galerkin methods for the solution of second kind integral equations as NEWLINE\[NEWLINE x(t)+\int\limits^1_0 k(t,\tau)x(\tau) d\tau=f(t), NEWLINE\]NEWLINE where \(k(t,\tau)\) is a smooth or weakly singular kernel, which satisfies the conditions NEWLINE\[NEWLINE \left|\frac{\partial^r k(t,\tau)}{\partial t^r}\right|\leq M|t- \tau|^{\alpha-r}, \quad \left|\frac{\partial^r k(t,\tau)}{\partial \tau^r}\right|\leq M|t- \tau|^{\alpha-r} NEWLINE\]NEWLINE for \(t \neq \tau.\) Here \(-1 < \alpha <0,\;\) \(r\) is an integer. They use continuous and smooth wavelet functions in the wavelet-Galerkin method and construct quadrature rules for calculating inner products of any functions.