Top

Journal of Scientific Computing

Published in:

04-07-2017

Generalized Störmer–Cowell Methods for Nonlinear BVPs of Second-Order Delay-Integro-Differential Equations

Authors: Chengjian Zhang, Cui Li

Published in: Journal of Scientific Computing | Issue 3/2018

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper deals with the numerical solutions of nonlinear boundary value problems (BVPs) of second-order delay-integro-differential equations. The generalized Störmer–Cowell methods (GSCMs), combined with the compound quadrature rules, are extended to solve this class of BVPs. It is proved under some suitable conditions that the extended GSCMs are uniquely solvable, stable and convergent of order \(\min \{p,q\}\), where p, q are consistent order of the GSCMs and convergent order of the compound quadrature rules, respectively. Several numerical examples are presented to illustrate the proposed methods and their theoretical results. Moreover, a numerical comparison with the existed methods is also given, which shows that the extended GSCMs are comparable in numerical precision and computational cost.

previous article On Solving an Acoustic Wave Problem Via Frequency-Domain Approach and Tensorial Spline Galerkin Method

next article Manifold Based Low-Rank Regularization for Image Restoration and Semi-Supervised Learning

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

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!

inform now

Aceto, L., Ghelardoni, P., Magherini, C.: PGSCM: a family of \(p\)-stable boundary value methods for second order initial value problems. J. Comput. Appl. Math. 236, 3857–3868 (2012)MathSciNetCrossRefMATH

Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)CrossRefMATH

Al-Mdallal, Q.M.: Monotone iterative sequences for nonlinear integro-differential equations of second order. Nonlinear Anal. RWA 12, 3665–3673 (2011)MathSciNetCrossRefMATH

Atkinson, K.E.: An Introduction to Numerical Analysis. John Wiley & Sons, Canada (1989)MATH

Bakke, V.L., Jackiewicz, Z.: The numerical solution of boundary-value problems for differential equations with state dependent deviating arguments. Apl. Mat. 34, 1–17 (1989)MathSciNetMATH

Brunner, H., van der Houwen, P.J.: The Numerical Solution of Volterra Equations (CWI Monographs 3). North-Holland, Amsterdam (1986)

Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004)CrossRefMATH

Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach science publishers, Amsterdam (1998)MATH

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

10.

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

11.

Cravero, I., Pittaluga, G., Sacripante, L.: An algorithm for solving linear Volterra integro-differential equations. Numer. Algor. 60, 101–114 (2012)MathSciNetCrossRefMATH

12.

Garey, L., Shaw, R.: Efficient algorithms for solving nonlinear Volterra integro-differential equations with two point boundary conditions. Int. J. Math. Math. Sci. 21, 755–760 (1998)MathSciNetCrossRefMATH

13.

Hrusa, W., Nohel, J.: Global existence and asymptotics in one-dimensional nonlinear viscoelasticity. In: Ciarlet, P.G., Roseau, M. (eds.), Trends and Application of Pure Mathematics to Mechanics, Lect. Notes in Phys., p. 195. Springer (1984)

14.

Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Springer Science & Business Media, New York (2013)MATH

15.

Linz, P.: Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia (1985)CrossRefMATH

16.

Li, C., Zhang, C.: The extended generalized Störmer–Cowell methods for second order delay boundary value problems. Appl. Math. Comput. 294, 87–95 (2017)MathSciNetCrossRef

17.

Makroglou, A.: Numerical solution of some second order integro-differential equations arising in ruin theory. In: Proceedings of the 3rd Conference in Actuarial Science and Finance, pp. 2–5 (2004)

18.

Shaw, R., Garey, L.: A parallel shooting method for second order Volterra integro-differential equations with two point boundary conditions. Int. J. Comput. Math. 49, 61–66 (1993)CrossRefMATH

19.

Shaw, R., Garey, L.: A fast method for solving second order boundary value Volterra integro-differential equations. Int. J. Comput. Math. 65, 121–129 (1997)MathSciNetCrossRefMATH

20.

Wazwaz, A.M.: A reliable algorithm for solving boundary value problems for higher-order integro-differential equations. Appl. Math. Comput. 118, 327–342 (2001)MathSciNetCrossRefMATH

21.

Zhang, C., Vandewalle, S.: Stability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J. Numer. Anal. 24, 193–214 (2004)MathSciNetCrossRefMATH

22.

Zhang, C., Vandewalle, S.: Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization. J. Comput. Appl. Math. 164, 797–814 (2004)MathSciNetCrossRefMATH

23.

Zhang, C., Vandewalle, S.: General linear methods for Volterra integro-differential equations with memory. SIAM J. Sci. Comput. 27, 2010–2031 (2006)MathSciNetCrossRefMATH

24.

Zhang, C., He, Y.: The extended one-leg methods for nonlinear neutral delay-integro-differential equations. Appl. Numer. Math. 59, 1409–1418 (2009)MathSciNetCrossRefMATH

25.

Zhao, J., Cao, Y., Xu, Y.: Legendre spectral collocation methods for Volterra delay-integro-differential equations. J. Sci. Comput. 67, 1110–1133 (2016)MathSciNetCrossRefMATH

Title: Generalized Störmer–Cowell Methods for Nonlinear BVPs of Second-Order Delay-Integro-Differential Equations
Authors: Chengjian Zhang
Cui Li
Publication date: 04-07-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0491-y

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 3/2018

On Solving an Acoustic Wave Problem Via Frequency-Domain Approach and Tensorial Spline Galerkin Method

Simulating Compressible Two-Medium Flows with Sharp-Interface Adaptive Runge–Kutta Discontinuous Galerkin Methods

Spectral Methods for Substantial Fractional Differential Equations

A Hybrid High-Order Method for the Steady Incompressible Navier–Stokes Problem

High Resolution Finite Volume Scheme Based on the Quintic Spline Reconstruction on Non-uniform Grids

Numerical Approximations for the Cahn–Hilliard Phase Field Model of the Binary Fluid-Surfactant System

Premium Partner