Abstract
In this paper a Multistep Collocation (MC) method for the second kind Fredholm integral equations (FIEs) is proposed and analyzed. The multistep collocation method is applied to FIE with smooth kernels under uniform mesh and weakly singular kernels |s−t|−α(0<α<1) using a graded mesh then the same convergence rate as collocation method but with a lower degree of freedom is obtained. Moreover, in order to avoid the round-off errors caused by graded mesh, a Hybrid Multistep Collocation (HMC) method by combining multistep collocation and hybrid collocation method is proposed. The HMC method converges faster with lower degrees of freedom and more efficiently captures the weakly singular properties by nonpolynomial interpolation at the first subinterval. The L∞-norm convergence results are analysed and proved. Numerical examples are presented to demonstrate the efficiency of the proposed methods.
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