A product integration method for a class of singular first kind Volterra equations

Summary

Convergence of a midpoint product integration method for singular first kind Volterra equations with kernels of the formk(t, s)(t−s) ^−α, 0<α<1, wherek(t, s) is continuous, is examined. It is shown that convergence of order one holds if the solution of the Volterra equation has a Lipschitz continuous first derivative andk(t, s) is suitably smooth. In addition, convergence is shown to hold when the solution has only Lipschitz continuity and the same conditions onk(t, s) apply. An existence theorem of Kowalewski is used to relate these conditions on the solution to conditions on the data andk(t, s).