High order boundary value linear multistep method for the numerical solution of IVPs in ODEs

Authors

  • S. E. Ogunfeyitimi
    Department of Computing,Wellspring University, Benin City, Nigeria
    https://orcid.org/0000-0001-5072-9009
  • M. N. O Ikhile
    Advanced Research Laboratory, Department of Mathematics, University of Benin, P.M.B 1154, Benin City, Nigeria
  • P. O. Olatunji
    Department of Mathematical Sciences, Adekunle Ajasin University, P.M.B 001, Akungba Akoko,Ondo State, Nigeria

Keywords:

Boundary value methods, Linear multistep formulas, Extended backward differentiation formulas, Stiff systems

Abstract

In this paper, we introduce High order boundary value linear multistep method (HOBVLMM) for the numerical solution of stiff systems of initial value problems (IVPs). The order, error constant, zero stability and the region of absolute stability for the HOBVLMM are discussed. The proposed scheme posses 0k,k-1 -stability and (Ak,k-1 )-stability, achieving a high order of p = 2k - 1, where k represents the step number of the LMM. The methods prove to be effective for stiff systems of IVPs in ordinary differential equations (ODEs), as evidenced by our numerical experiments, which shows superior performance compared to some existing methods.

Dimensions

[1] J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley and Sons Ltd, England, 2003. http://dx.doi.org/10.1002/0470868279.

[2] J. D. Lambert, Numerical methods for ordinary differential equations: the initial value problem, Wiley, New York, 1991. https://api.semanticscholar.org/CorpusID:122798404.

[3] C. W. Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, New Jersey, 1971. https://books.google.com.ng/books?id=2RyRwgEACAAJ.

[4] E. Hairer & G. Wanner, Solving ordinary differential equations II. Stiff and differential-algebraic problems, Springer-Verlag, 1996. https://doi.org/10.1007/978-3-642-05221-7.

[5] G. Dahlquist, “A special stability problem for linear multistep methods”, BIT Numerical Mathematics 3 (1963) 27. https://doi.org/10.1007/BF01963532.

[6] J. R. Cash, “On the integration of stiff systems of ODEs using extended backward differentiation formulae”, Numerische Mathematik 34 (1980) 234. https://doi.org/10.1007/BF01396701.

[7] J. R. Cash, “Second derivative extended backward differentiation formula for the numerical integration of stiff system”, SIAM Journal on Numerical Analysis 18 (1981) 21. https://doi.org/10.1137/0718003.

[8] P. O. Olatunji, M. N. O. Ikhile & R. I. Okunoghae, “Nested second derivative two-step Runge-Kutta methods”, International Journal of Applied and Computational Mathematics 6 (2021) 1. https://doi.org/10.1007/s40819-021-01169-1.

[9] J. R. Cash, “The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae”, Computers and Mathematics with Applications 9 (1983) 645. https://doi.org/10.1016/0898-1221(83)90122-0.

[10] J. R. Cash, “Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs”, Journal of Computational and Applied Mathematics 125 (2000) 117. https://doi.org/10.1016/S0377-0427(00)00463-5.

[11] J. R. Cash, “Review paper: efficient numerical methods for the solution of stiff initial value problems and differential algebraic equations”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 459 (2003) 797. https://doi.org/10.1098/rspa.2003.1130.

[12] X. Jia-Xiang & K. Jiao-Xun, “A class of DBDF methods with the derivative modifying term”, Journal of Computational Mathematics 6 (1998) 7. https://www.jstor.org/stable/43692367.

[13] P. O. Olatunji & M. N. O. Ikhile, “Modified second derivative backward differentiation formulas with recursively nested hybrid evaluation”, Journal of the Nigerian Association of Mathematical Physics 40 (2017) 83. https://www.ajol.info/index.php/jonamp/article/view/206644.

[14] P. O. Olatunji & M. N. O. Ikhile, “Second derivative multistep method with nested hybrid evaluation”, Asian Research Journal of Mathematics 11 (2018) 1. https://doi.org/10.9734/ARJOM/2018/41601.

[15] P. O. Olatunji & M. N. O. Ikhile, “Variable order nested hybrid multistep methods for stiff ODEs”, Journal of Mathematical and Computational Science 10 (2020) 78. https://scik.org/index.php/jmcs/article/view/4147.

[16] B. T. Olabode & P. O. Olatunji, “A class of generalized hybrid multistep methods for the solution of stiff differential equations”, World Scientific News 187 (2024) 129. https://worldscientificnews.com/wp-content/uploads/2023/12/WSN-187-2024-129-144.pdf.

[17] A. O. H. Axelsson & J. G. Verwer, “Boundary-value techniques for initial value problems in ordinary differential equations”, Mathematics of Computation 45 (1985) 153. https://ir.cwi.nl/pub/1819.

[18] L. Brugnano & D. Trigiante, “Convergence and stability of boundary-value methods for ordinary differential equations”, Journal of Computational and Applied Mathematics 66 (1996) 97. https://doi.org/10.1016/0377-0427(95)00166-2.

[19] L. Brugnano & D. Trigiante, Solving differential problems by multistep initial and boundary-value methods, Gordon and Breach Science Publishers, 1998. https://web.math.unifi.it/users/brugnano/Libri/SDP/index.html.

[20] J. Ehigie, S. Jator, A. B. Sofoluwe & S. A. Okunuga, “Boundary-value technique for initial value problems with continuous second derivative multistep method of Enright”, Computational and Applied Mathematics 33 (2014) 81. https://doi.org/10.1007/s40314-013-0044-4.

[21] S. N. Jator & R. K. Sahi, “Boundary-value technique for initial value problems based on Adams-type second derivative methods”, International Journal of Mathematical Education in Science and Technology 41 (2010) 819. https://doi.org/10.1080/00207391003675141.

[22] G. Dahlquist, “Convergence and stability in the numerical integration of ordinary differential equations”, Mathematica Scandinavica 4 (1956) 33. https://www.mscand.dk/article/view/10454.

[23] P. Amodio & F. Mazzia, “Boundary-value methods based on Adams-type methods”, Applied Numerical Mathematics 18 (1995) 23. https://doi.org/10.1016/0168-9274(95)00041-R.

[24] P. Amodio, W. L. Golik & F. Mazzia, “Variable-step boundary-value methods based on reverse Adams schemes and their grid distribution”, Applied Numerical Mathematics 18 (1995) 5. https://doi.org/10.1016/0168-9274(95)00044-U.

[25] L. Brugnano & D. Trigiante, “High order multistep methods for boundary-value problems”, Applied Numerical Mathematics 18 (1995) 79. https://doi.org/10.1016/0168-9274(95)00045-V.

[26] L. Brugnano & D. Trigiante, Block implicit methods for ODEs, in recent trends in numerical analysis, Nova Science, New York, 2000, pp. 105. https://dl.acm.org/doi/10.5555/380227.380239.

[27] F. Iavernaro & F. Mazzia, “Convergence and stability of multistep methods solving nonlinear initial value problems”, SIAM Journal on Numerical Analysis 18 (1997) 270. https://doi.org/10.1137/S1064827595287122.

[28] J. Oboyi, S. E. Ekoro & S. E. Ogunfeyitimi, “On some multi-block reverse Adams methods for stiff”, International Journal of Applied and Computational Mathematics 8 (2022) 166. https://doi.org/10.1007/s40819-022-01367-5.

[29] S. E. Ogunfeyitimi & M. N. O. Ikhile, “Second derivative generalized extended backward differentiation formulas for stiff problems”, Journal of the Korean Society for Industrial and Applied Mathematics 23 (2019) 179. https://doi.org/10.12941/jksiam.2019.23.179.

[30] S. E. Ogunfeyitimi & M. N. O. Ikhile, “Generalized second derivative linear multistep methods based on the methods of Enright”, International Journal of Applied and Computational Mathematics 6 (2020) 76. https://doi.org/10.1007/s40819-020-00827-0.

[31] T. Okor & G. C. Nwachukwu, “Modified generalized second derivative extended backward differentiation formulas for highly stiff and stiffly oscillatory systems of ODEs”, Computational and Applied Mathematics 43 (2024) 138. https://doi.org/10.1007/s40314-024-02624-7.

[32] T. Okor & G. C. Nwachukwu, “High order extended boundary value methods for the solution of stiff systems of ODEs”, Journal of Computational and Applied Mathematics 400 (2022) 113750. https://doi.org/10.1016/j.cam.2021.113750.

[33] P. Amodio & F. Mazzia, “Boundary-value methods for the solution of differential-algebraic equations”, Numerische Mathematik 66 (1993) 411. https://doi.org/10.1007/BF01385705.

[34] J. C. Zhang & H. Chen, “Block boundary-value methods for delay differential equations”, Applied Numerical Mathematics 60 (2010) 915. https://doi.org/10.1016/j.apnum.2010.05.001.

[35] J. Zhao, X. Jiang & Y. Xu, “Convergence of block boundary value methods for solving delay differential algebraic equation with index-1 and index-2”, Applied Mathematics and Computation 399 (2021) 1. https://doi.org/10.1016/j.amc.2021.126034.

[36] F. Mazzia & A. M. Nagy, “Solving Volterra integro-differential equations by variable stepsize block BS methods: properties and implementation techniques”, Applied Mathematics and Computation 239 (2014) 198. https://doi.org/10.1016/j.amc.2014.04.030.

[37] C. Zhang & H. Chen, “Convergence and stability of extended block boundary-value methods for Volterra delay integro-differential equations”, Applied Numerical Mathematics 62 (2012) 141. https://doi.org/10.1016/j.apnum.2011.11.001.

[38] C. Zhang & X. Yan, “Convergence and stability of extended BBVMs for nonlinear delay-differential-algebraic equations with piecewise continuous arguments”, Numerical Algorithms 87 (2021) 921. https://doi.org/10.1007/s11075-020-00993-8.

[39] Y. Xu, J. Gao & Z. Gao, “Stability analysis of block boundary-value methods for neutral pantograph equation”, Journal of Difference Equations and Applications 119 (2013) 1127. https://doi.org/10.1080/10236198.2012.733703.

[40] Y. Xu, J. Gao & Z. Gao, “Stability analysis of block boundary-value methods for neutral pantograph equation with many delays”, Applied Mathematical Modelling 38 (2014) 325. https://doi.org/10.1016/j.apm.2013.06.013.

[41] Y. Zhou, C. Zhang & H. Wang, “Boundary value methods for Caputo fractional differential equations”, Journal of Computational Mathematics 39 (2021) 108. https://api.semanticscholar.org/CorpusID:229411190.

[42] X. Jiang, Y. Xu & J. Zhao, “A new kind of block fractional generalized Adams method for solving fractional differential equations”, Numerical Algorithms (2025). https://doi.org/10.1007/s11075-025-02067-z.

[43] S. E. Ogunfeyitimi & M. N. O. Ikhile, “Multi-block boundary value methods for ordinary differential and differential algebraic equations”, Journal of the Korean Society for Industrial and Applied Mathematics 24 (2020) 243. https://doi.org/10.12941/jksiam.2020.24.243.

[44] S. E. Ogunfeyitimi & M. N. O. Ikhile, “Multi-block generalized Adams-type integration methods for differential algebraic equations”, International Journal of Applied and Computational Mathematics 7 (2021) 197. https://doi.org/10.1007/s40819-021-01135-x.

[45] X. Wu & J. Xia, “Two low accuracy methods for stiff systems”, Applied Mathematics and Computation 123 (2001) 141. https://doi.org/10.1016/S0096-3003(00)00010-2.

[46] O. A. Akinfenwa, “Seven step Adams type block method with continuous coefficient for periodic ordinary differential equation”, World Academy of Science, Engineering and Technology 5 (2011) 2072. https://doi.org/10.5281/zenodo.1058163.

[47] F. F. Ngwane & S. N. Jator, “Block hybrid-second derivative method for stiff systems”, International Journal of Pure and Applied Mathematics 80 (2012) 543. https://www.ijpam.eu/contents/2012-80-4/7/7.pdf.

[48] D. J. Higham & N. J. Higham, MATLAB guide, 3rd ed., SIAM, Philadelphia, 2005. https://epubs.siam.org/doi/10.1137/1.9781611974669.

[49] G. Ismail & I. Ibrahim, “New efficient second derivative multistep methods for stiff systems”, Applied Mathematical Modelling 23 (1999) 279. https://doi.org/10.1016/S0307-904X(98)10086-0.

[50] G. Hojjati, M. Rahimi Ardabili & S. Hosseini, “New second derivative multistep methods for stiff systems”, Applied Mathematical Modelling 30 (2006) 466. https://doi.org/10.1016/j.apm.2005.06.007.

Published

2025-11-01

How to Cite

High order boundary value linear multistep method for the numerical solution of IVPs in ODEs. (2025). Journal of the Nigerian Society of Physical Sciences, 7(4), 2718. https://doi.org/10.46481/jnsps.2025.2718

Issue

Volume 7, Issue 4, November 2025

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2025 S. E. Ogunfeyitimi, M. N. O Ikhile, P. O. Olatunji

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

High order boundary value linear multistep method for the numerical solution of IVPs in ODEs. (2025). Journal of the Nigerian Society of Physical Sciences, 7(4), 2718. https://doi.org/10.46481/jnsps.2025.2718

Similar Articles

121-130 of 184

You may also start an advanced similarity search for this article.