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On some boundary value methods for stiff IVPs in ODEs

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Abstract

A boundary value method (BVM) leads to a discrete boundary value problem with some conditions imposed at the initial points and the remaining ones at the final points. The interest in this class of methods is the fact that BVMs overcome the limitations of the well-known Dahlquist order and stability barrier for an A-stable linear multistep method (LMM). Yet, A-stability is fundamental to the integration of stiff ordinary differential equations (ODEs). This presents multiple families of linear multistep formulas (LMFs) with future points based on extended backward differentiation formula (BDF) which can be used as a main method in a BVM implementation for the solution of initial value problems (IVPs) in ODEs.

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Acknowledgements

We wish to acknowledge the useful comments of the anonymous referees which have greatly improved the quality of this work.

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Authors and Affiliations

  1. Advanced Research Laboratory, Department of Mathematics, University of Benin, Benin City, Nigeria

    G. C. Nwachukwu, M. N. O. Ikhile & J. Osaghae

Authors
  1. G. C. Nwachukwu
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  2. M. N. O. Ikhile
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  3. J. Osaghae
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Correspondence to G. C. Nwachukwu.

Additional information

J. Osaghae: Passed on before this work was concluded. His contribution especially with the preliminary investigation in this work is acknowledged.

Appendix

Appendix

See Tables 6, 7, 8, 9.

Table 6 Condition numbers of the matrices \(A-qB\) associated with the \(A_{k,s}\)-stable methods for \(s=1(1)5\); \(h=1\), \(\lambda =1\), \(q=h\lambda \)
Full size table
Table 7 Condition numbers of the matrices \(A-qB\) associated with the \(A_{k,s}\)-stable methods for \(s=1(1)5\); \(h=1\), \(\lambda =1\), \(q=h\lambda \) (continuation)
Full size table
Table 8 The coefficients, error constant (EC) and order p of (2.1) for \(s=1\), \(y_{n+k}=\sum _{j=0}^{k-1}\alpha _j y_{n+j}+h\sum _{j=k}^{k+s}\beta _j f_{n+j},p=k+1\)
Full size table
Table 9 The coefficients, error constant (EC) and order p of (2.1) for \(s=1\), \(y_{n+k}=\sum _{j=0}^{k-1}\alpha _j y_{n+j}+h\sum _{j=k}^{k+s}\beta _j f_{n+j},p=k+1\) (continuation)
Full size table

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Nwachukwu, G.C., Ikhile, M.N.O. & Osaghae, J. On some boundary value methods for stiff IVPs in ODEs. Afr. Mat. 29, 731–752 (2018). https://doi.org/10.1007/s13370-018-0574-4

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