9 - Epilogue

Summary

Our voyage through the preceding eight chapters has shown that we have certainly not yet reached the end of the story on collocation methods for Volterra functional integral and integro-differential equations. Many important questions remain unanswered. It is my belief that we have to find new mathematical approaches and tools (likely from very unexpected areas) if we are to make substantial progress towards finding complete solutions to these open problems.

It is the purpose of this brief final chapter to point to some possible, and seemingly very promising, new approaches for the numerical analysis of collocation solutions to Volterra functional equations.

Semigroups and abstract resolvent theory

The long-time integration of Volterra integral and integro-differential equations by collocation methods, in particular the asymptotic behaviour of collocation solutions, is not yet understood. As a number of papers and books have shown (see, e.g. Ito and Kappel (1989, 1991, 2002), Ito and Turi (1991), Brunner, Kauthen and Ostermann (1995), Bellen and Maset (1999), Maset (1999, 2003), and Bellen and Zennaro (2003, pp. 56–60)) the appropriate reformulation of the given equation as an abstract Cauchy problem and the exploitation of the underlying semigroup or abstract resolvent framework (integrability and asymptotic behaviour of resolvents) will often lead to deep insight into the qualitative properties of approximate solutions.