Abstract
Within the theoretical framework of the numerical stability analysis for the Volterra integral equations, we consider a new class of test problems and we study the long-time behavior of the numerical solution obtained by direct quadrature methods as a function of the stepsize. Furthermore, we analyze how the numerical solution responds to certain perturbations in the kernel.
Similar content being viewed by others
References
Baker, C.: A perspective on the numerical treatment of Volterra equations. Numerical analysis 2000, vol. vi, ordinary differential equations and integral equations. J. Comput. Appl. Math. 125(1–2), 217–249 (2000). doi:10.1016/S0377-0427(00)00470-2
Brunner, H., van der Houwen, P.: The numerical solution of Volterra equations. North-Holland (1986)
Burton, T.: Volterra integral and differential equations. Elsevier B. V., Amsterdam (2005)
Cardone, A., Messina, E., Vecchio, A.: An adaptive method for Volterra-Fredholm integral equations on the half line. J. Comput. Appl. Math. 228 (2), 538–547 (2009). doi:10.1016/j.cam.2008.03.036
Eggermont, P., Lubich, C.: Uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line. Math. Comp 56(193), 149–176 (1991). doi:10.2307/2008535
Feng, Z., Xu, D., H.Z.: Epidemiological models with non-exponentially distributed disease stages and applications to disease control. Bull. Math. Biol. 69 (5), 1511–1536 (2007). doi:10.1007/s11538-006-9174-9
Ford, N., Baker, C.: Qualitative behaviour and stability of solutions of discretised nonlinear Volterra integral equations of convolution type. J. Comput. Appl. Math. 66(1-2), 213–225 (1996). doi:10.1016/0377-0427(95)00180-8
Gripenberg, G.: On the resolvents of nonconvolution Volterra kernels. Funkcial. Ekvac. 23(1), 83–95 (1980)
Gripenberg, G., Londen, S.O., Staffans, O.: Volterra integral and functional equations. Cambridge University Press, Cambridge (1990)
Győri, I., Reynolds, D.: On admissibility of the resolvent of discrete Volterra equations. J. Difference Equ. Appl. 16(12), 1393–1412 (2010)
Hartung, N.: Efficient resolution of metastatic tumor growth models by reformulation into integral equations. Discret. Contin. Dyn. Syst. - Ser. B 20(2), 445–467 (2015). doi:10.3934/dcdsb.2015.20.445
de Hoog, F., Anderssen, R.: Kernel perturbations for a class of second-kind convolution Volterra equations with non-negative kernels. Appl. Math. Lett. 25(9), 1222–1225 (2012). doi:10.1016/j.aml.2012.02.058
Linz, P.: Analytical and numerical methods for Volterra equations. SIAM, Philadelphia (1985)
Lubich, C.: On the stability of linear multistep methods for Volterra convolution equations. IMA J. Numer. Anal. 3(4), 439–465 (1983). doi:10.1093/imanum/3.4.439
Messina, E., Vecchio, A.: Nonlinear stability of direct quadrature methods for Volterra integral equations. Math. Comput. Simul. 110, 155–164 (2015). doi:10.1016/j.matcom.2013.04.007
Messina, E., Vecchio, A.: Stability and boundedness of numerical approximations to Volterra integral equations. Submitted (2016)
Reynolds, D.W.: Parameter-dependent Volterra summation equations. In: The International Conference on Difference Equations and Applications. Private communication. Bialystok, Poland (2015)
Strauss, A.: On a perturbed Volterra integral equation. J. Math. Anal. Appl. 30, 564–575 (1970)
Vecchio, A.: Boundedness of the global error of some linear and nonlinear methods for Volterra integral equations. J. Integr. Equ. Appl. 12(4), 449–465 (2000). doi:10.1216/jiea/1020282238
Wolkenfelt, P.H.M.: Linear multistep methods and the construction of quadrature formulae for Volterra integral and integro-differential equations. Afdeling Numerieke Wiskunde [Department of Numerical Mathematics] 76 (1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Messina, E., Vecchio, A. A sufficient condition for the stability of direct quadrature methods for Volterra integral equations. Numer Algor 74, 1223–1236 (2017). https://doi.org/10.1007/s11075-016-0193-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0193-9