Abstract
We analyze the stability of convolution quadrature methods for weakly singular Volterra integral equations with respect to a linear test equation. We prove that the asymptotic behavior of the numerical solution replicates the one of the continuous problem under some restriction on the stepsize. Numerical examples illustrate the theoretical results.
Similar content being viewed by others
References
Aceto, L., Magherini, C., Novati, P.: Fractional convolution quadrature based on generalized Adams methods. Calcolo 51(3), 441–463 (2014)
Becker, L.C.: Resolvents and solutions of weakly singular linear Volterra integral equations. Nonlinear Anal. 74(5), 1892–1912 (2011)
Brunner, H.: Volterra Integral Equations. An Introduction to Theory and Applications. Cambridge University Press, Cambridge (2017)
Burns, J.A., Cliff, E.M., Herdman, T.L.: A state-space model for an aeroelastic system. In: Proc. 22nd IEEE Conference on Decision and Control, pp. 1074–1077 (1983)
Carlone, R., Figari, R., Negulescu, C.: The quantum beating and its numerical simulation. J. Math. Anal. Appl. 450, 1294–1316 (2017)
Choi, U.Jin, MacCamy, R.C.: Fractional order Volterra equations with applications to elasticity. J. Math. Anal. Appl. 139(2), 448–464 (1989)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)
Diogo, T., Edwards, J.T., Ford, N.J., Thomas, S.M.: Numerical analysis of a singular integral equation. Appl. Math. Comput. 167, 372–382 (2005)
Fedotov, S., Iomin, A., Ryashko, L.: Non-markovian models for migration-proliferation dichotomy of cancer cells: anomalous switching and spreading rate. Phys. Rev. E 84(6), 061131 (2011)
Garrappa, R.: Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math. Comput. Simul. 110, 96–112 (2015)
Garrappa, R., Galeone, L.: On multistep methods for differential equations of fractional order. Mediterr. J. Math. 3(3–4), 565–580 (2006)
Garrappa, R., Messina, E., Vecchio, A.: Effect of perturbation in the numerical solution of fractional differential equations. Discrete Contin. Dyn. Syst. Ser. B (2017). https://doi.org/10.3934/dcdsb.2017188
Garrappa, R., Popolizio, M.: On accurate product integration rules for linear fractional differential equations. J. Comput. Appl. Math. 235(5), 1085–1097 (2011)
Gorenflo, R., Vessella, S.: Abel Integral Equations. Analysis and Applications, Volume 1461 of Lecture Notes in Mathematics. Springer, Berlin (1991)
Győri, I., Reynolds, D.W.: On admissibility of the resolvent of discrete Volterra equations. J. Differ. Equ. Appl. 16(12), 1393–1412 (2010)
Linz, P.: Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia (1985)
Lubich, Ch.: Fractional linear multistep methods for Abel–Volterra integral equations of the second kind. Math. Comput. 45(172), 463–469 (1985)
Lubich, Ch.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)
Lubich, Ch.: A stability analysis of convolution quadratures for Abel–Volterra integral equations. IMA J. Numer. Anal. 6(1), 87–101 (1986)
Messina, E., Vecchio, A.: Stability and convergence of solutions to Volterra integral equations on time scales. Discrete Dyn. Nat. Soc. 2015(ID612156), 6 (2015)
Messina, E., Vecchio, A.: Stability and boundedness of numerical approximations to Volterra integral equations. Appl. Numer. Math. 116, 230–237 (2017)
Messina, E., Vecchio, A.: A sufficient condition for the stability of direct quadrature methods for Volterra integral equations. Numer. Algorithms 74(4), 1223–1236 (2017)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Surguladze, T.A.: On certain applications of fractional calculus to viscoelasticity. J. Math. Sci. 112(5), 4517–4557 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
G. Izzo, E. Messina and A. Vecchio are members of the INdAM Research group GNCS.
Rights and permissions
About this article
Cite this article
Izzo, G., Messina, E. & Vecchio, A. Stability of Numerical Solutions for Abel–Volterra Integral Equations of the Second Kind. Mediterr. J. Math. 15, 113 (2018). https://doi.org/10.1007/s00009-018-1149-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-018-1149-1