Multiscale approximation of the solution of weakly singular second kind Fredholm integral equation in Legendre multiwavelet basis

Numerical solution of Fredholm integral equation of second kind with weakly singular kernel is obtained in this paper by employing Legendre multi-wavelet basis. The low- and high-pass filters for two-scale relations involving Legendre multiwavelets having four or five vanishing moments of their wavelets have been derived and are used in the evaluation of integrals for the multiscale representation of the integral operator. Explicit expressions for the elements of the matrix associated with the multiscale representation are given. An estimate for the Hölder exponent of the solution of the integral equation at any point in its domain is obtained. A number of examples is provided to illustrate the efficiency of the method developed here.