Sense from sensitivity and variation of parameters

Abstract

It is known that Alekseev's variation of parameters formula for ordinary differential equations can be generalized to other types of causal equations (including delay differential equations and Volterra integral equations), and corresponding discrete forms. Such variation of parameters formulae can be employed, together with appropriate inequalities, in discussing the behaviour of solutions of continuous and discretized problems, the significance of parameters in mathematical models, sensitivity and stability issues, bifurcation, and (in classical numerical analysis) error control, convergence and super-convergence of densely defined approximations and error analysis in general. However, attempts to extend Alekseev's formula to nonlinear Volterra integral equations are not straightforward, and difficulties can recur in attempts to analyze the sensitivity of functionally-dependent or structurally-dependent solutions. In analyzing sensitivity we discuss behaviour for infinitesimally small perturbations. In discussions of stability we need to establish the existence of bounds on changes to solutions (or their decay in the limit as t → ∞ ) that ensue from perturbations in the problem. Yet the two topics are related, not least through variation of parameters formulae, and (motivated by some of our recent results) we discuss this and related issues within a general framework.