Stability of reducible quadrature methods for Volterra integral equations of the second kind

Abstract

Summary. Stability analysis of reducible quadrature methods for Volterra integral equations based on the test equationis presented. The concept of absolute stability is defined and necessary and sufficient conditions for the method to be absolutely stable for given 2, #, and v are derived. These conditions are illustrated for the class of 0-methods for integral equations. The main tool in our stability analysis is the theory of difference equations of Poincar6 type.

Stability analysis of reducible quadrature methods for Volterra integro-differential equations

“…It was observed that this equation is of Poincare type, and stability analysis was performed with the use of an extension of the classical theorem of Perron. A similar analysis is presented here for VIDEs based on the test equation •t (2) y'(t) = y y(t)…”

“…The method (6) -(7) with w nJ subject to (8) is called a ((Q, a); (Q, CT)) method. The investigation of stability properties of these methods will follow the approach of Wolkenfelt [12] and the authors [2], [3], in which application of these methods to the test equation 9leads to a difference equation of fixed order which characterizes the solution of (9). Thus, the method (6) -(7) applied to (9) takes the form…”

“…Note added in proof. The authors recently shown that for 2\i + v < 0 and fi + v < 0 the solutions of the test equation (2) are bounded. The approach is to consider the associated third order differential equation.…”

“…The analysis is facilitated by taking advantage of the fact that application of reducible multistep methods to this test equation leads to a difference equation of finite order with constant coefficients. Applying the approach of Wolkenfelt to a more general test equation the authors [2] obtained a difference equation of finite order but with variable coefficients. It was observed that this equation is of Poincare type, and stability analysis was performed with the use of an extension of the classical theorem of Perron.…”