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  • Cited by 6
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    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Brunner, Hermann and Ou, Chunhua 2014. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure and Applied Analysis, Vol. 14, Issue. 2, p. 397.

    Adivar, Murat Koyuncuoğlu, H. Can and Raffoul, Youssef N. 2013. Periodic and asymptotically periodic solutions of systems of nonlinear difference equations with infinite delay. Journal of Difference Equations and Applications, Vol. 19, Issue. 12, p. 1927.

    Song, Yihong Ni, Jingsong and Baker, Christopher T.H. 2011. Non-negative convergent solutions of discrete Volterra equations. Journal of Difference Equations and Applications, Vol. 17, Issue. 3, p. 423.

    Brunner, Hermann Huang, Qiumei and Xie, Hehu 2010. Discontinuous Galerkin Methods for Delay Differential Equations of Pantograph Type. SIAM Journal on Numerical Analysis, Vol. 48, Issue. 5, p. 1944.

    Kirk, C.M. 2009. Numerical and asymptotic analysis of a localized heat source undergoing periodic motion. Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, Issue. 12, p. e2168.

    Brunner, Hermann and Hu, Qiya 2007. Optimal Superconvergence Results for Delay Integro‐Differential Equations of Pantograph Type. SIAM Journal on Numerical Analysis, Vol. 45, Issue. 3, p. 986.

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  • Print publication year: 2004
  • Online publication date: August 2010

The numerical analysis of functional integral and integro-differential equations of Volterra type

Summary

The qualitative and quantitative analysis of numerical methods for delay differential equations is now quite well understood, as reflected in the recent monograph by Bellen and Zennaro (2003). This is in remarkable contrast to the situation in the numerical analysis of functional equations, in which delays occur in connection with memory terms described by Volterra integral operators. The complexity of the convergence and asymptotic stability analyses has its roots in new ‘dimensions’ not present in DDEs: the problems have distributed delays; kernels in the Volterra operators may be weakly singular; a second discretization step (approximation of the memory term by feasible quadrature processes) will in general be necessary before solution approximations can be computed.

The purpose of this review is to introduce the reader to functional integral and integro-differential equations of Volterra type and their discretization, focusing on collocation techniques; to describe the ‘state of the art’ in the numerical analysis of such problems; and to show that - especially for many ‘classical’ equations whose analysis dates back more than 100 years - we still have a long way to go before we reach a level of insight into their discretized versions to compare with that achieved for DDEs.

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Acta Numerica 2004
  • Online ISBN: 9780511569975
  • Book DOI: https://doi.org/10.1017/CBO9780511569975
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