Abstract
This paper concerns iterated collocation methods for a class of nonlinear Volterra delay integral equations (VDIEs) with noncompact operators. Some properties of corresponding operators as well as existence, uniqueness and regularity of exact solution are discussed. Under a special graded meshes, the existence and uniqueness of collocation solutions are proved. Moreover, we present the global supper-convergence order. Finally, some numerical examples are given to verify the validity of the theoretical orders of convergence.
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References
Brauer F (1976) Constant rate harvesting of populations governed by Volterra integral equations. J Math Anal Appl 56:18–27. https://doi.org/10.1016/0022-247X(76)90004-4
Brunnera H, Yatsenko Y (1996) Spline collocation methods for nonlinear Volterra integral equations with unknown delay. J Comput Appl Math 71:67–81. https://doi.org/10.1016/0377-0427(95)00228-6
Brunner H (2004) Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge
Brunner H, Hu Q (2005) Optimal superconvergence of iterated collocation solutions for Volterra integral equations with vanishing delays. SIAM J Numer Anal 43:1934–1949. https://doi.org/10.1137/040615705
Brunner H (2017) Volterra integral equations, an introduction to theory and applications. Cambridge University Press, Cambridge
Cooke KL (1976) An epidemic equation with immigration. Math Biosci 29:135–158. https://doi.org/10.1016/0025-5564(76)90033-X
Krantz SG, Parks HR (2002) The implicit function theorem, history, theory and applications. Birknause, Boston
Ming WY, Huang CM, Li M (2016) Superconvergence in collocation methods for Volterra integral equations with vanishing delays. J Comput Appl Math 308:361–378. https://doi.org/10.1016/j.cam.2016.06.010
Ming WY, Huang CM (2017) Collocation methods for Volterra functional integral equations with non-vanishing delays. Appl Math Comput 296:198–214. https://doi.org/10.1016/j.amc.2016.10.021
Ming WY, Huang CM, Zhao LB (2018) Optimal superconvergence results for Volterra functional integral equations with proportional vanishing delays. Appl Math Comput 320:292–301. https://doi.org/10.1016/j.amc.2017.09.045
Seyed Allaei S, Yang ZW, Brunner H (2017) Collocation methods for third-kind VIEs. IMA J Numer Anal 37:1104–1124. https://doi.org/10.1093/imanum/drw033
Song HM, Yang ZW, Brunner H (2019) Analysis of collocation methods for nonlinear Volterra integral equations of the third kind. Calcolo. https://doi.org/10.1007/s10092-019-0304-9
Song HM, Yang ZW, Diogo T (2021) Collocation methods for cordial Volterra integro-differential equations. J Comput Appl Math 393:113321
Song HM, Yang ZW, Xiao Y (2021) Analysis of blow-up behavior of solutions to CVIEs. Comput Appl Math 40. https://doi.org/10.1007/s40314-021-01520-8
Song HM, Xiao Y, Chen Minghao (2021) Collocation methods for third-kind Volterra integral equations with proportional delays. Appl Math Comput 388:125509
Vainikko G (2009) Cordial Volterra integral equations I. Numer Funct Anal Optim 30:1145–1172. https://doi.org/10.1080/01630560903393188
Vainikko G (2011) Spline collocation-interpolation method for linear and nonlinear cordial Volterra integral equations. Numer Funct Anal Optim 32:83–109. https://doi.org/10.1080/01630563.2010.526412
Wang JL, Zhang R, Kuniya T (2015) Global dynamics for a class of age-infection HIV models with nonlinear infection rate. J Math Anal Appl 432:289–313. https://doi.org/10.1016/j.jmaa.2015.06.040
Xie HH, Zhang R, Brunner H (2011) Collocation methods for general Volterra functional integral equations with vanishing delays. SIAM J Sci Comput 33:3303–3332. https://doi.org/10.1137/100818595
Zhang K, Li J, Song HM (2012) Collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays. Appl Math Comput 218:10848–10860. https://doi.org/10.1016/j.amc.2012.04.045
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Communicated by Justin Wan.
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Song, H., Yang, Z. & Xiao, Y. Iterated collocation methods for nonlinear third-kind Volterra integral equations with proportional delays. Comp. Appl. Math. 41, 191 (2022). https://doi.org/10.1007/s40314-022-01775-9
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DOI: https://doi.org/10.1007/s40314-022-01775-9