Fractional linear multistep methods for Abel-Volterra integral equations of the second kind 论文

摘要

Fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular Volterra integral equations. The proposed methods are convergent of the order of the underlying multistep method, also in the generic case of solutions which are not smooth at the origin. The stability properties (stability region, <italic>A</italic> -stability, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis alpha right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> α </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A(\alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -stability) are closely related to those of the underlying linear multistep method.