An Efficient New Scheme for Direct Simulation of Higher Order Third and Fourth Oscillatory Differential Equations
DOI:
https://doi.org/10.62054/ijdm/0101.06Schlagwörter:
Direct simulation; Convergence; Consistency; Error constant; Linear stability; Oscillatory differential equationsAbstract
In this research, we have examined the general block approach for solving higher-order oscillatory differential equations using the linear block approach (LBA). The basic properties of the new method, such as order, error constant, zero-stability, consistency, convergence, linear stability, and region of absolute stability, were also analyzed and satisfied. Some distinct fourth-order oscillatory problems were directly applied to the new method in order to overcome the setbacks of the reduction method. The results obtained were compared with those in the literature, and the new method takes away the burden of solving fourth-order oscillatory differential equations. Therefore, from the results, the new method has shown better accuracy and faster convergence graphically. One of the advantages of the new method is that it does not require much computational burden and is also self-starting.
Convergence, Consistency, Direct simulation, Linear stability, Oscillatory differential equations
Literaturhinweise
Abolarin, O. E., Akinola, L. S., Adeyefa, E. O. and Ogunware, B. G. (2022). Implicit hybrid block methods for
solving second, third and fourth orders ordinary differential equations directly. Italian Journal of Pure
and Applied Mathematics. 48, 1–21.
Adeyeye, O. and Omar, Z. (2019). Solving third order ordinary differential equation using one-step block
method with four equidistance generalized hybrid points. International Journal of Applied
Mathematics. 49(2), 1-9. https://www.iaeng.org/IJAM/issues_v49/issue_2/IJAM_49_2_15
Adeyeye, O. and Omar. Z. (2019). Implicit five-step block method with generalised equidistant points for
solving fourth order linear and non-linear initial value problems. Ain Shams Engineering Journal. 10.
https://doi.org/10.1016/j.asej.2017.11.011
Adoghe, L. O. and Omole, E. O. (2019). A two-step hybrid block method for the numerical integration of higher
order initial value problem of ordinary differential equations, World science news. An International
ScientificJournal.118,246. http://www.worldscientificnews.com/wp-content/uploads/2018/11/WSN 118-
-236-250
Adoghe, L. O., Ogunwari, B. G. and Omole, E. O. (2016). A family of symmetric implicit higher order methods
for the solution of third order initial value problems in ordinary differential equations. Theoretical
Mathematics and Application, 6(3), 67-84. http://www.scienpress.com/Upload/TMA/Vol%206_3_5
Agarwal, R. P., Bohner, M., Li, T., Zhang, C. A. (2013). New approach in the study of oscillatory behavior of
even-order neutral delay differential equations. Appl. Math. Comput., 225, 787-794.
https://doi.org/10.1016/j.amc.2013.09.037
Ahamad, N. and Charan, S. (2019). Study of numerical solution of fourth order ordinary differential equations
by fifth order Runge-Kutta method. Int. J. Sci. Res. Sci. Eng. Technol. 6, 235-236.
https://doi.org/10.32628/IJSRSET196142
Areo, E. A. and Rufai, M. A. (2016). An efficient one-eight step hybrid block method for solving Second order
initial value problems of ODEs. International Journal of Differential Equation and Application, 15(2),
-139. https://doi.org/10.12732/ijdea.v15i2.3352
Allogmany, R., Fudziah, I., Majid, Z. N. and Zarina, B. I. (2020). Implicit two-point block method for solving
fourth-order initial value problem directly with application. Mathematical Problems in Engineering. 1-
https://doi.org/10.1155/2020/6351279
Areo, E. A. and Omojola, M. T. (2017). One-twelveth step continuous block method for the solution of
y′′′ = f(x, y, y′, y′′). International Journal of Pure and Applied Mathematics. 114(2), 165-178.
https://doi.org/10.12732/ijpam.v114i2.1
Awoyemi, D. O. (2003). A P-stable linear multistep method for solving third order ordinary differential
equations. Inter. J. Computer Math. 80(8), 85-91. https://doi.org/10.1080/0020716031000079572
Atabo, V. O. and Adee, S. O. (2021). A new special 15-step block method for solving general fourth ordinary
differential equations. Journal of Nigerian Society of Physical Sciences, 3, 308-331.
https://doi:10.46481/jnsps.2021.337
Ayinde, A. M., Ibrahim, S., Sabo, J. and Silas, D. (2023). The physical application of motion using single step
block method. Journal of Material Science Research and Review. 6(4), 708-719.
https://journaljmsrr.com/index.php/JMSRR
Blanka B. (2019). Oscillation of second-order nonlinear non-canonical differential equations with deviating
argument. Applied Mathematics Letters. 91, 68–75. https://doi.org/10.1016/j.aml.2018.11.021
Donald, J. Z., Skwame, Y., Sabo, J. and Ayinde, A. M. (2021). The use of linear multistep method on implicit
one-step second derivative block method for direct solution of higher order initial value problems.
Abacus (Mathematics Science Series). 48(2), 224-237.
Duromola M. K. (2022). Single-step block method of p-stable for solving third-order differential equations
(IVPs): Ninth order of accuracy. American Journal of Applied Mathematics and Statistics, 10(1), 4-13.
https://doi.org/10.12691/ajams-10-1-2
Fasasi, K. M. Adesanya, A. O. and Adee, S. O. (2015). One step continuous hybrid block method for the
solution of y’’’= f(x, y, y’, y’’). Natural Science Research. 4, 55-62. www.iiste.org
Gear, C. W. (1966). The numerical integration of ordinary differential equations. Math. Comp., 21, 146-156.
https://doi.org/10.1090/S0025-5718-1967-0225494-5
Gear, C. W. (1971). Numerical initial value problems in ordinary differential equations. New Jersey: Prentice
Hall.
Gear, C. W. (1978). The stability of numerical methods for second order ordinary differential equations. SIAMJ.
Numer. Anal. 15(1), 118-197. https://www.jstor.org/stable/2156571
Hall, G. and Suleiman, M. B. (1981). Stability of Adams types formulae for second order ordinary differential
equations. IMA J. Numer. Anal., 1, 427-428.
Jator, S. N. and Li. (2009). A self-starting linear multistep method for the direct solution of a general second
order initial value problem. International Journal of Computer Math, 86(5), 817-836.
http://dx.doi.org/10.1080/00207160701708250
Kayode, S. J. (2008). An order six zero-stable method for direct solution of fourth order ordinary differential
equations. American Journal of Applied Sciences, 5(11), 1461-1466.
https://doi.org/10.3844/ajassp.2008.1461.1466
Kayode, S. J. and Obaruha, F. O. (2017). Symmetric 2-step 4-point hybrid method for the solution of general
third order differential equations. Journal of Applied and Computational Mathematics. 6(2). 1-4.
https://doi.org/10.4172/2168-9679.1000348
Kuboye, J. and Omar, Z. (2015). New zero-stable block method for direct solution of fourth order ordinary
differential equations. Indian Journal of Science and Technology, 8(12), 1-8.
https://doi.org/10.17485/ijst/2015/v8i12/70647
Kuboye, J. O. Elusakin, O. R. and Quadri. O. F. (2022). Numerical algorithm for the direct solution of fourth
Order ordinary differential equation, Nigerian Society of Physical Sciences 2, 218.
https://doi.org/10.46481/jnsps.2020.100
Kwari, L. J. Sunday, J. Ndam, J. N. Shokri, A. and Wang, Y. (2023). On the simulations of second-order
oscillatory problems with applications to physical systems. Axioms. 12 (2023) 9-10.
https://doi.org/10.3390/axioms12030282
Lambert, J. D. (1973). Computational methods in ordinary differential equations. Introductory Mathematics for
Scientists and Engineers. Wiley. https://doi.org/10.1002/zamm.19740540726
Modebeia, M. I., Olaiya, O. O. and Ngwongwo, I. P. (2021). Computational study of some three-step hybrid
integrators for solution of third order ordinary differential equations. J. Nig. Soc. Phys. Sci. 3, 459–468
https://doi.org/10.46481/jnsps.2021.323
Olabode, B. T., and Momoh, A. L. (2016). Continuous hybrid multistep methods with Legendre basic function for
treatment of second order stiff ODEs. American Journal of Computational and Applied Mathematics,
(2), 38-49. https://doi.org/10.5923/j.ajcam.20160602.03
Omar, Z. and Abdelrahim, R. (2016). Application of single step with three generalized hybrid points block
method for solving third order ordinary differential equations. J. Nonlinear Sci. Appl. 9, 2705-2717.
https://doi.org/10.22436/jnsa.009.05.67
Raymond, D., Pantuvu, T. P., Lydia, A., Sabo, J. and Ajia, R. (2023). An optimized half-step scheme third
derivative methods for testing higher order initial value problems. African Scientific Reports, 2(76), 1-
https://doi.org/10.46481/asr.2023.2.1.76
Sabo, J., Kyagya, T. Y. and Vashawa, W. J. (2021). Numerical simulation of one step block method for
treatment of second order forced motions in mass spring systems. Asian Journal of Research and
Reviews in Physic.s 5(2), 1-11 https://doi.org/10.9734/ajr2p.2021/v5i230157
Sabo, J., Kyagya, T. Y., Ayinde, A. M. and Otaide, I. J. (2022). Mathematical simulation of the linear block
algorithm for modeling third-order initial value problems. BRICS Journal of Educational Research, 12
(3), 88-96. https://scholar.google.com/citations?user=LqKrNl4AAAAJandhl=en
Suleiman, M. B. (1979). Generalized multistep Adams and backward differentiation methods for the solution of
stiff and non-stiff ordinary differential equations. Doctoral Dissertation. University of Manchester,
England. file:///C:/Users/OWNER/Downloads/511-Article%20Text-504-504-10-20140218
Suleiman, M. B. (1989). Solving higher order ODEs directly by the integration method. Applied Mathematics
and Computation. 33, 197-219. https://doi.org/10.1016/0096-3003(89)90051-9
Sunday J. (2018). On the oscillation criteria and computation of third order oscillatory differential equations.
Communication in Mathematics and Applications. 6(4), 615-625.
https://doi.org/10.26713/cma.v9i4.968
Sunday, J., Joshua, K. V. and Danladi, N. (2019). On the derivation and implementation of a one-step algorithm
for third order oscillatory problems. Adamawa State University, Journal of Scientific Research
(ADSUJSR). 7(1), 8-20. http://www.adsujsr.com
Ukpebor, L. A. Omole, E. O. and Adoghe, L.O. (2020). An order four numerical scheme for fourth-order initial
value problems using lucas polynomial with application in ship dynamics. International Journal of
Mathematical Research. 9(1), 28-41. https://doi.org/10.18488/journal.24.2020.91.28.41
Wend, D. V. (1969). Existence and uniqueness of solutions of ordinary differential equations. Proceedings of
the American Mathematical Society, 27-33. https://doi.org/10.1090/S0002-9939-1969-02458794
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Dear Editor-in-Chief,
I would like to send the International Journal of Development Mathematics (IJDM) a new work titled “An Efficient New Scheme for Direct Simulation of Higher Order Third and Fourth Oscillatory Differential Equations" for publication. Outstanding research in this area includes the title is succinct and captures the goals and conclusions of the study. The title does not contain any unnecessary equations or unusual abbreviations. The full first and last names of the author(s) have been written below the title.
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Aloko Macdonald Damilola-Writing original draft, Validation and Resources
Abdullahi Muhammed Ayinde- Writing review and editing, Visualization and checking plagiarism
John Sabo-Formal analysis, Methodology, Conceptualization, Analysis of the results
Usman Ahmed Danbaba-Software and Project administration
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