TY - JOUR
T1 - Collocation methods for third-kind VIEs
AU - Allaei, Sonia Seyed
AU - Yang, Zhan Wen
AU - Brunner, Hermann
N1 - This research was supported by the Hong Kong Research Grants Council (GRF Grant HKBU 200212) and the Natural Sciences and Engineering Research Council of Canada (DG Grant 9406).
PY - 2017/7/1
Y1 - 2017/7/1
N2 - The spline collocation method in piecewise polynomial spaces is applied to a class of third-kind Volterra integral equations (VIEs). Under certain conditions the operator associated with the equivalent secondkind VIE is compact, and this guarantees that the resulting algebraical system is uniquely solvable for all sufficiently small mesh diameters. However, the solvability of this system is not ensured, both on uniform and classical graded meshes, when the operator is noncompact (which typically is the case). Hence, we introduce a modified graded mesh to overcome the solvability problem. For such meshes we establish results on the optimal order of global convergence of the collocation solutions for third-kind VIEs. Numerical tests confirm the validity of these results.
AB - The spline collocation method in piecewise polynomial spaces is applied to a class of third-kind Volterra integral equations (VIEs). Under certain conditions the operator associated with the equivalent secondkind VIE is compact, and this guarantees that the resulting algebraical system is uniquely solvable for all sufficiently small mesh diameters. However, the solvability of this system is not ensured, both on uniform and classical graded meshes, when the operator is noncompact (which typically is the case). Hence, we introduce a modified graded mesh to overcome the solvability problem. For such meshes we establish results on the optimal order of global convergence of the collocation solutions for third-kind VIEs. Numerical tests confirm the validity of these results.
KW - collocation methods
KW - cordial Volterra integral equations
KW - global convergence order
KW - noncompact integral operators
KW - Volterra integral equations of the third kind
KW - weakly singular kernel
UR - http://www.scopus.com/inward/record.url?scp=85027080456&partnerID=8YFLogxK
U2 - 10.1093/imanum/drw033
DO - 10.1093/imanum/drw033
M3 - Journal article
AN - SCOPUS:85027080456
SN - 0272-4979
VL - 37
SP - 1104
EP - 1124
JO - IMA Journal of Numerical Analysis
JF - IMA Journal of Numerical Analysis
IS - 3
ER -