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

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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.

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References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. 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)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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  Google Scholar 

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

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Chengjian Zhang & Cui Li

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Chengjian Zhang

  3. School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450046, China

    Cui Li

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  1. Chengjian Zhang
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  2. Cui Li
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Correspondence to Chengjian Zhang.

Additional information

This work is supported by NSFC (Grant No. 11571128).

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Zhang, C., Li, C. Generalized Störmer–Cowell Methods for Nonlinear BVPs of Second-Order Delay-Integro-Differential Equations. J Sci Comput 74, 1221–1240 (2018). https://doi.org/10.1007/s10915-017-0491-y

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  • DOI: https://doi.org/10.1007/s10915-017-0491-y

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