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In solving integral equations with logarithmic kel which arises from thernboundary integral equation reformulation of some boundary value problems forrnthe two dimensional Helmholtz equation, we combine the Galerkin method withrnBeylkin’s ([2]) approach, series of dense and nonsymmetric matrices may appearrnif we use traditional method. By appealing the so-called periodic quasi-waveletrn(PQW in abbr.) ([5]), some of these matrices become diagonal, therefore we canrnfind a algorithm with only O(K(m)2) arithmetic operations where m is the highestrnlevel. The Galerkin approximation has a polynomial rate of convergence.rn