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The adapted block boundary value methods for singular initial value problems

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Abstract

This paper deals with the numerical methods for solving singular initial value problems. By adapting the block boundary value methods (BBVMs) for regular initial value problems, a class of adapted BBVMs are constructed for singular initial value problems. It is proved under some suitable conditions that the adapted BBVMs are uniquely solvable, stable and convergent of order p, where p is the consistence order of the methods. Several numerical examples are performed to verify the stability, efficiency and accuracy of the adapted methods. Moreover, a comparison between the adapted BBVMs and the IEM-based iterated defect correction methods is given. The numerical results show that the adapted BBVMs are comparable.

Keywords

Block boundary value methods Singular initial value problems Unique solvability Stability Convergence 

Mathematics Subject Classifiacation

65L05 65L20 

References

  1. 1.
    Amodio, P., Budd, C.J., Koch, O., Settanni, G., Weinmüller, E.: Asymptotical computations for a model of flow in saturated porous media. Appl. Math. Comput. 237, 155–167 (2014)MathSciNetMATHGoogle Scholar
  2. 2.
    Amodio, P., Iavernaro, F.: Symmetric boundary value methods for second order initial and boundary value problems. Mediterr. J. Math. 3, 383–398 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Amodio, P., Settanni, G.: Variable step/order generalized upwind methods for the numerical solution of second order singular perturbation problems. J. Numer. Anal. Ind. Appl. Math. 4, 65–76 (2009)MathSciNetMATHGoogle Scholar
  4. 4.
    Auzinger, W., Koch, O., Weinmüller, E.: Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31, 5–25 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Auzinger, W., Koch, O., Weinmüller, E.: Analysis of a new error estimate for collocation methods applied to singular boundary value problems. SIAM J. Numer. Anal. 42, 2366–2386 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boreskov, G.K., Slin’ko, M.G.: Modelling of chemical reactors. Pure Appl. Chem. 10, 611–624 (1965)CrossRefGoogle Scholar
  7. 7.
    Brugnano, L., Trigiante, D.: Convergence and stability of boundary value methods for ordinary differential equations. J. Comput. Appl. Math. 66, 97–109 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam (1998)MATHGoogle Scholar
  9. 9.
    Brugnano, L., Zhang, C., Li, D.: A class of energy-conserving Hamiltonian boundary value methods for nonlinear \(\text{ Schr }\ddot{o}\text{ dinger }\) equation with wave operator. Commun. Nonlinear Sci. Numer. Simul. 60, 33–49 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chan, C.Y., Hon, Y.C.: A constructive solution for a generalized Thomas–Fermi theory of ionized atoms. Q. Appl. Math. 45, 591–599 (1987)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, H., Zhang, C.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218, 2619–2630 (2011)MathSciNetMATHGoogle Scholar
  12. 12.
    Chen, H., Zhang, C.: Block boundary value methods for solving Volterra integral and integro-differential equations. J. Comput. Appl. Math. 236, 2822–2837 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    de Hoog, F.R., Weiss, R.: The application of linear multistep methods to singular initial value problems. Math. Comput. 31, 676–690 (1977)CrossRefMATHGoogle Scholar
  15. 15.
    de Hoog, F.R., Weiss, R.: Collocation methods for singular boundary value problems. SIAM J. Numer. Anal. 15, 198–217 (1978)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    de Hoog, F.R., Weiss, R.: The application of Runge–Kutta schemes to singular initial value problems. Math. Comput. 44, 93–103 (1985)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Iavernaro, F., Mazzia, F.: Convergence and stability of multistep methods solving nonlinear initial value problems. SIAM J. Sci. Comput. 18, 270–285 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Iavernaro, F., Mazzia, F.: Block-boundary value methods for the solution of ordinary differential equations. SIAM J. Sci. Comput. 21, 323–339 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Keller, H.B., Wolfe, A.W.: On the nonunique equilibrium states and buckling mechanism of spherical shells. J. Soc. Ind. Appl. Math. 13, 674–705 (1965)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Koch, O.: Asymptotically correct error estimation for collocation methods applied to singular boundary value problems. Numer. Math. 101, 143–164 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Koch, O., Kofler, P., Weinmüller, E.: The implicit Euler method for the numerical solution of singular initial value problems. Appl. Numer. Math. 34, 231–252 (2000)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Koch, O., Kofler, P., Weinmüller, E.: Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind. Analysis 21, 373–389 (2001)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Koch, O., Weinmüller, E.: Iterated defect correction for the solution of singular initial value problems. SIAM J. Numer. Anal. 38, 1784–1799 (2001)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Koch, O., Weinmüller, E.: The convergence of shooting methods for singular boundary value problems. Math. Comput. 71, 289–305 (2003)MathSciNetMATHGoogle Scholar
  25. 25.
    Koch, O., Weinmüller, E.: Analytical and numerical treatment of a singular initial value problem in avalanche modeling. Appl. Math. Comput. 148, 561–570 (2004)MathSciNetMATHGoogle Scholar
  26. 26.
    Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications. Marcel Dekker, New York (2002)CrossRefMATHGoogle Scholar
  27. 27.
    Li, C., Zhang, C.: Block boundary value methods applied to functional differential equations with piecewise continuous arguments. Appl. Numer. Math. 115, 214–224 (2017)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Li, C., Zhang, C.: The extended generalized \(\text{ St }\ddot{o}\text{ rmer }\)–Cowell methods for second-order delay boundary value problems. Appl. Math. Comput. 294, 87–95 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Luke, Y.L.: The Special Functions and Their Approximations. Academic Press, New York (1969)MATHGoogle Scholar
  30. 30.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)MATHGoogle Scholar
  31. 31.
    Russell, D.L.: Numerical solution of singular initial value problems. SIAM J. Numer. Anal. 7, 399–417 (1970)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wang, H., Zhang, C., Zhou, Y.: A class of compact boundary value methods applied to semi-linear reaction–diffusion equations. Appl. Math. Comput. 325, 69–81 (2018)MathSciNetGoogle Scholar
  33. 33.
    Xu, Y., Zhao, J., Gao, Z.: Stability analysis of block boundary value methods for neutral pantograph equation. J. Differ. Equ. Appl. 19, 1227–1242 (2013)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Xu, Y., Zhao, J., Gao, Z.: Stability analysis of block boundary value methods for the neutral differential equation with many delays. Appl. Math. Model. 38, 325–335 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhang, C., Chen, H.: Asymptotic stability of block boundary value methods for delay differential-algebraic equations. Math. Comput. Simul. 81, 100–108 (2010)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Zhang, C., Chen, H.: Block boundary value methods for delay differential equations. Appl. Numer. Math. 60, 915–923 (2010)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zhang, C., Chen, H., Wang, L.: Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations. Numer. Linear Algebra Appl. 18, 843–855 (2011)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Zhang, C., Li, C.: Generalized \(\text{ St }\ddot{o}\text{ rmer }\)–Cowell methods for nonlinear BVPs of second-order delay-integro-differential equations. J. Sci. Comput. 74, 1221–1240 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina
  2. 2.Hubei Key Laboratory of Engineering Modeling and Scientific ComputingHuazhong University of Science and TechnologyWuhanChina

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