TY - JOUR
T1 - Recent advances in the numerical analysis of Volterra functional differential equations with variable delays
AU - BRUNNER, Hermann
N1 - Funding Information:
This paper is based on two invited lectures presented at the Workshop on Innovative Methods for Solving Evolution Problems with Memory (Capri, June 19–21, 2006). The research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The author also gratefully acknowledges the financial support extended to him by the Università degli Studi di Napoli “Federico II” during a visit in June 2006.
PY - 2009/6/15
Y1 - 2009/6/15
N2 - The numerical analysis of Volterra functional integro-differential equations with vanishing delays has to overcome a number of challenges that are not encountered when solving 'classical' delay differential equations with non-vanishing delays. In this paper I shall describe recent results in the analysis of optimal (global and local) superconvergence orders in collocation methods for such evolutionary problems. Following a brief survey of results for equations containing Volterra integral operators with non-vanishing delays, the discussion will focus on pantograph-type Volterra integro-differential equations with (linear and nonlinear) vanishing delays. The paper concludes with a section on open problems; these include the asymptotic stability of collocation solutions uh on uniform meshes for pantograph-type functional equations, and the analysis of collocation methods for pantograph-type functional equations with advanced arguments.
AB - The numerical analysis of Volterra functional integro-differential equations with vanishing delays has to overcome a number of challenges that are not encountered when solving 'classical' delay differential equations with non-vanishing delays. In this paper I shall describe recent results in the analysis of optimal (global and local) superconvergence orders in collocation methods for such evolutionary problems. Following a brief survey of results for equations containing Volterra integral operators with non-vanishing delays, the discussion will focus on pantograph-type Volterra integro-differential equations with (linear and nonlinear) vanishing delays. The paper concludes with a section on open problems; these include the asymptotic stability of collocation solutions uh on uniform meshes for pantograph-type functional equations, and the analysis of collocation methods for pantograph-type functional equations with advanced arguments.
KW - Collocation solutions
KW - Optimal order of superconvergence
KW - Pantograph equation
KW - Proportional delays
KW - Vanishing delays
KW - Volterra functional integro-differential equations
UR - http://www.scopus.com/inward/record.url?scp=64549093775&partnerID=8YFLogxK
U2 - 10.1016/j.cam.2008.03.024
DO - 10.1016/j.cam.2008.03.024
M3 - Article
AN - SCOPUS:64549093775
SN - 0377-0427
VL - 228
SP - 524
EP - 537
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 2
ER -