nach oben

BIT Numerical Mathematics

Erschienen in:

09.03.2021

Block boundary value methods for linear weakly singular Volterra integro-differential equations

verfasst von: Yongtao Zhou, Martin Stynes

Erschienen in: BIT Numerical Mathematics | Ausgabe 2/2021

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A class of block boundary value methods (BBVMs) is constructed for linear weakly singular Volterra integro-differential equations (VIDEs). The convergence and stability of these methods is analysed. It is shown that optimal convergence rates can be obtained by using special graded meshes. Numerical examples are given to illustrate the sharpness of our theoretical results and the computational effectiveness of the methods. Moreover, a numerical comparison with piecewise polynomial collocation methods for VIDEs is given, which shows that the BBVMs are comparable in numerical precision.

Vorheriger Artikel Optimal quadratic element on rectangular grids for problems

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

Brugnano, L.: Essentially symplectic boundary value methods for linear Hamiltonian systems. J. Comput. Math. 15(3), 233–252 (1997)MathSciNetMATH

Brugnano, L., Trigiante, D.: High-order multistep methods for boundary value problems. Appl. Numer. Math. 18(1–3), 79–94 (1995)MathSciNetCrossRef

Brugnano, L., Trigiante, D.: Convergence and stability of boundary value methods for ordinary differential equations. J. Comput. Appl. Math. 66(1–2), 97–109 (1996)MathSciNetCrossRef

Brugnano, L., Trigiante, D.: Block boundary value methods for linear Hamiltonian systems. Appl. Math. Comput. 81(1), 49–68 (1997)MathSciNetMATH

Brugnano, L., Trigiante, D.: Boundary value methods: the third way between linear multistep and Runge-Kutta methods. Comput. Math. Appl. 36(10–12), 269–284 (1998)MathSciNetCrossRef

Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods, volume 6 of Stability and Control: Theory, Methods and Applications. Gordon and Breach Science Publishers, Amsterdam (1998)

Brunner, H., Pedas, A., Vainikko, G.: A spline collocation method for linear Volterra integro-differential equations with weakly singular kernels. BIT 41(5, suppl.), 891–900 (2001). BIT 40th Anniversary Meeting

Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J. Numer. Anal. 20(6), 1106–1119 (1983)MathSciNetCrossRef

Brunner, H.: Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels. IMA J. Numer. Anal. 6(2), 221–239 (1986)MathSciNetCrossRef

10.

Brunner, H.: Collocation methods for Volterra integral and related functional differential equations, volume 15 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2004)

11.

Brunner, H., Pedas, A., Vainikko, G.: Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J. Numer. Anal. 39(3), 957–982 (2001)MathSciNetCrossRef

12.

Chen, H., Zhang, C.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218(6), 2619–2630 (2011)MathSciNetMATH

13.

Chen, H., Zhang, C.: Block boundary value methods for solving Volterra integral and integro-differential equations. J. Comput. Appl. Math. 236(11), 2822–2837 (2012)MathSciNetCrossRef

14.

Chen, H., Zhang, C.: Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations. Appl. Numer. Math. 62(2), 141–154 (2012)MathSciNetCrossRef

15.

Diethelm, K., Garrappa, R., Stynes, M.: Good (and not so good) practices in computational methods for fractional calculus. Mathematics 8(324), (2020)

16.

Ghiat, M., Guebbai, H.: Analytical and numerical study for an integro-differential nonlinear Volterra equation with weakly singular kernel. Comput. Appl. Math. 37(4), 4661–4674 (2018)MathSciNetCrossRef

17.

Iavernaro, F., Mazzia, F.: Block-boundary value methods for the solution of ordinary differential equations. SIAM J. Sci. Comput. 21(1), 323–339 (1999)MathSciNetCrossRef

18.

Kangro, R., Parts, I.: Superconvergence in the maximum norm of a class of piecewise polynomial collocation methods for solving linear weakly singular Volterra integro-differential equations. J. Integ. Equ. Appl. 15(4), 403–427 (2003)MathSciNetCrossRef

19.

Parts, I.: Optimality of theoretical error estimates for spline collocation methods for linear weakly singular Volterra integro-differential equations. Proc. Estonian Acad. Sci. Phys. Math. 54(3), 162–180 (2005)MathSciNetMATH

20.

Tang, T.: A note on collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J. Numer. Anal. 13(1), 93–99 (1993)MathSciNetCrossRef

21.

Vainikko, G.: Multidimensional Weakly Singular Integral Equations. Lecture Notes in Math. 1549. Springer, Berlin, Heidelberg, New York (1993)

22.

Xu, X., Xu, D.: A semi-discrete scheme for solving fourth-order partial integro-differential equation with a weakly singular kernel using Legendre wavelets method. Comput. Appl. Math. 37(4), 4145–4168 (2018)MathSciNetCrossRef

23.

Zhou, Y., Zhang, C.: Convergence and stability of block boundary value methods applied to nonlinear fractional differential equations with Caputo derivatives. Appl. Numer. Math. 135, 367–380 (2019)MathSciNetCrossRef

Titel: Block boundary value methods for linear weakly singular Volterra integro-differential equations
verfasst von: Yongtao Zhou
Martin Stynes
Publikationsdatum: 09.03.2021
Verlag: Springer Netherlands
Erschienen in: BIT Numerical Mathematics / Ausgabe 2/2021
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-020-00840-1

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Weitere Artikel der Ausgabe 2/2021

Runge–Kutta Lawson schemes for stochastic differential equations

Inexact rational Krylov method for evolution equations

Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

Efficient exponential Runge–Kutta methods of high order: construction and implementation

A three-term recurrence relation for accurate evaluation of transition probabilities of the simple birth-and-death process

Numerical conservative solutions of the Hunter–Saxton equation

Premium Partner