Falkner hybrid block methods for second-order IVPs: A novel approach to enhancing accuracy and stability properties

Robert I. Okuonghae; Joshua K. Ozobokeme

doi:10.33993/jnaat532-1450

Authors

Robert I. Okuonghae University of Benin, Nigeria https://orcid.org/0000-0003-3776-1172
Joshua K. Ozobokeme University of Benin, Nigeria https://orcid.org/0009-0007-8518-6949

DOI:

https://doi.org/10.33993/jnaat532-1450

Abstract views: 0

Abstract

Second-order initial value problems (IVPs) in ordinary differential equations (ODEs) are ubiquitous in various fields, including physics, engineering, and economics. However, their numerical integration poses significant challenges, particularly when dealing with oscillatory or stiff problems. This article introduces a novel Falkner hybrid block method for the numerical integration of second-order IVPs in ODEs. The newly developed method is of order six with a large interval of absolute stability and is implemented using a fixed step size technique. The numerical experiments show the accuracy of our methods when compared with Falkner linear multistep methods, block methods, and other hybrid codes proposed in the scientific literature. This innovative approach demonstrates improved accuracy and stability in solving second-order IVPs, making it a valuable tool for researchers and practitioners.

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References

A. Abdulsalam, N. Senu, Z. A. Majid, N.A. Nik-Long, Development of high-order adaptive multi-step Runge-Kutta-Nystrom method for special second-order ODEs, Comput. in Simulation, 216 (2024), pp. 104-125. https://doi.org/10.1016/j.matcom.2023.09.006 DOI: https://doi.org/10.1016/j.matcom.2023.09.006

R. B. Adeniyi, and M. O. Alabi, A collocation method for direct numerical integration of initial value problems in higher-order ordinary differential equations, Analele Stiintifice Ale Universitatii AL. I. Cuza din Iasi (SN), Matematica, 2 (2011), pp. 311-321. DOI: https://doi.org/10.2478/v10157-011-0028-x

E. O. Adeyefa, A Model for Solving First, Second, and Third Order IVPs Directly Int, J. Appl. Comput. Math., 7 (2021) no. 131, pp. 1-9.https://doi.org/10.1007/s40819-021-01075-6 DOI: https://doi.org/10.1007/s40819-021-01075-6

E. O. Adeyefa, and j. O. Kuboye, Derivation of New Numerical Model Capable of Solving Second and Third Order Ordinary Differential Equations Directly, IAENG Int. J. Appl. Math., 50 (2020) no. 2, pp. 1-9.

O. Adeyeye, and Z. Omar, Maximal order block method for the solution of second order ordinary differential equations, IAENG Int. J. Appl. Math., 46 (2016) no. 4.

R. Allogmany, and F. Ismail, Direct solution of u” = f (t, u, u’) using three point block method of order eight with applications, J. King Saud Univ. Sci., 33 (2021), 101337. https://doi.org/10.1016/j.jksus.2020.101337 DOI: https://doi.org/10.1016/j.jksus.2020.101337

E. A.,Areo, N. O. Adeyanju, and S. J. Kayode, Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods, FUOYE Journal of Engineering and Technology (FUOYEJET), 5 (2020) no. 2, pp. 2579-0617. https://doi.org/10.46792/fuoyejet.v5i2.537 DOI: https://doi.org/10.46792/fuoyejet.v5i2.537

D. O. Awoyemi, A class of continuous methods for general second order initial value problems in ordinary differential equations, Int. J. Comput. Math., 72 (1999), pp. 29-37. https://doi.org/10.1080/00207169908804832 DOI: https://doi.org/10.1080/00207169908804832

D.O. Awoyemi, A New Sixth Order Algorithms for General Second Order Ordinary Differential Equation, Int. J. Comput. Math., 77 (2001), pp. 117-124. https://doi.org/10.1080/00207160108805054 DOI: https://doi.org/10.1080/00207160108805054

D.O. Awoyemi, and S. J. Kayode, A Maximal Order Collocation Method for Direct Solution of Initial Value Problems of General Second Order Ordinary Differential Equations, Proceedings of the conference organized by the National Mathematical Centre, Abuja, (2005).

A. M.Badmus, A new eighth order implicit block algorithms for the direct solution of second order ordinary differential equations, Am. J. Comput. Math., 4(2014) no. 4, pp. 376-386. https://doi.org/10.4236/ajcm.2014.44032 DOI: https://doi.org/10.4236/ajcm.2014.44032

A. M. Badmus , An efficient seven-point hybrid block method for the direct solution of y” = f(x, y, y’), J. Adv. Math. Comput. Sci., 4 (2014), pp. 2840-2852. https://doi.org/10.9734/BJMCS/2014/6749 DOI: https://doi.org/10.9734/BJMCS/2014/6749

A. M. Badmus, and Y. A. Yahaya, An accurate uniform order 6 blocks method for direct solution general second order ordinary differential equations, Pacif. J. Sci. Technol., 10 (2009), pp. 248-254.

L. Collatz, The Numerical Treatment of Differential Equations, Springer, Berlin, 1966.

J. O. Ehigie, S. A. Okunuga, A. B. Sofoluwe, and M. A. Akanbi, On generalized 2-step continuous linear multistep method of hybrid type for the integration of second order ordinary differential equation, Arch. Appl. Sci. Res., 2 (2010) no 6, pp. 362-372.

M. V. Falkner, A method of numerical solution of differential equations, Phil. Mag. S., 7 (1936), pp. 621-640. https://doi.org/10.1080/14786443608561611 DOI: https://doi.org/10.1080/14786443608561611

T. M. Falkner, and S. W. Skan, A hybrid block method for solving second-order boundary value problems, J. Comput. Phys., 357 (2018), 109924.

S. O. Fatunla, Block Methods for Second-order ODEs, Int. J. Comput. Math., 41 (1991), pp. 55-63. https://doi.org/10.1080/00207169108804026. DOI: https://doi.org/10.1080/00207169108804026

I. C. Felix, and R. I. Okuonghae, On the generalization of Pad´e approximation approach for the construction of p-stable hybrid linear multistep methods, Int. J. Appl. Comput. Math., 5 (2019), pp. 1-20. DOI: https://doi.org/10.1007/s40819-019-0685-0

C. W. Gear, Argonne National Laboratory, Report no. ANL-7126, 1966.

S. N. Jator, A sixth-order linear multistep method for direct solution of Int, J. Pure Appl. Math., 40(2007) no. 1, pp. 407-472.

S. J. Kayode, A zero-stable optimal order method for direct solution of second order differential equations, J. Math. Stat., 6 (2010), pp. 367-371. https://doi.org/10.3844/jmssp.2010.367.371 DOI: https://doi.org/10.3844/jmssp.2010.367.371

S. J. Kayode, and O. Adeyeye, Two-step two-point hybrid methods for general second order differential equations, Afr. J. Math. Comput. Sci. Res., 6 (2013), pp. 191-196. https://doi.org/10.5897/AJMCSR2013.0502

J. O. Kuboye, Z. Omar, O. E. Abolarin, and R. Abdelrahim, Generalized hybrid block method for solving second-order ordinary differential equations directly, Res. Rep. Math., 2 (2018)(2), pp. 1-7.

J. D. Lambert, and A. Watson, Symmetric multistep methods for periodic IVPs, J. Inst. Math. Applics., 18 (1976), pp. 189-202. https://doi.org/10.1093/imamat/18.2.189 DOI: https://doi.org/10.1093/imamat/18.2.189

U. Mohammed, and R. B. Adeniyi, Derivation of block hybrid backward difference formula (HBDF) through multistep collocation for solving second-order differential equations, Asia Pac. J. Sci. Technol., 15 (2014), pp. 89-95.

R. I. Okuonghae, and M. N. O. Ikhile, Second derivative general linear methods, Numer. Algorithms, 67 (2014) no. 3, pp. 637-654.https://doi.org/10.1007/s11075-013-9814-8 DOI: https://doi.org/10.1007/s11075-013-9814-8

Z. Omar, and J. O. Kuboye, A New Implicit Block Method for Solving Second-Order Ordinary Differential Equations Directly, GU J Sci., 28(4) (2015), pp. 689-694. https://doi.org/10.19026/rjaset.11.1671 DOI: https://doi.org/10.19026/rjaset.11.1671

H. Ramos, S. Mehta, and J. Vigo-Aguia, A unified approach for the development of k-step block Falkner-type methods for solving general second-order initial-value problems in ODEs, J. Comput. Appl. Math., 318 (2017), pp. 550-564. DOI: https://doi.org/10.1016/j.cam.2015.12.018

N. Waeleh, and Z. A. Majid, Numerical algorithm of block method for general second order odes using variable step size, Sains Malays, 46 (2017), pp. 817-824. https://doi.org/10.17576/jsm-2017-4605-16 DOI: https://doi.org/10.17576/jsm-2017-4605-16