Robust Numerical Solutions of Higher-Order Stiff and Non-Stiff Third-Order Oscillatory Differential Equations Using a Direct Block Approach

Abstract

This work introduces a block algorithm formulated for directly solving third-order ordinary differential equations (ODEs), without the intermediate step of converting them into systems of first-order equations. The method is applied to three distinct types of problems non-stiff, oscillatory and stiff previously investigated by other researchers. Numerical results obtained using the proposed method are compared with those from existing methods in literature. The results reveal that the block algorithm consistently produces numerical solutions that closely align with the analytical solutions across all problem types. Graphical comparisons further demonstrate the superiority of the new method in terms of accuracy and stability. This confirms the method's effectiveness and versatility in solving higher-order ODEs directly, making it a valuable tool in computational mathematics.

Keywords

• Block Algorithm
• Third-order Differential Equations
• Direct Method
• Numerical Methods
• Numerical Solution
• Stiff ODEs
• Oscillatory Systems
• Non-stiff Problems

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References

1. J. Sunday, “On the oscillation criteria and computation of third order oscillatory differential equations,” Commun. Math. Appl., vol. 9, no.4, pp. 615-626, Jan. 2018, doi: 10.26713/cma.v9i4.968 .
2. J. O. Kuboye and Z. Omar, “Numerical solution of third order ordinary differential equations using a seven-step block method,” Int. J. Math. Anal. , vol. 9, no.15, pp.743-754, 2015, doi:10.12988/ijma.2015.5125.
3. M.K. Duromola and B. Bolarinwa, “Fractional step method for the numerical integration of initial value problems of third order ordinary differential equations,” NAMP Journal , vol. 36, pp. 23-30, 2016.
4. S.J. Kayode and F.O. Obarhua, “Symmetric 2-step 4-point hybrid method for the solution of general third order differential equations,” J. Comput. Appl. Math., vol. 6, no. 2, pp. 1-4, Jan. 2017, doi:10.4172/2168-9679.1000348.
5. N. S. Yakusak, S.T. Akinyemi and I.G. Usman, “Three off-steps hybrid method for the numerical solution of third order initial value problems,” IOSR Journal of Mathematics, vol. 12, no. 3, pp. 59-65, 2016.
6. H. Soomro, N. Zainuddin, H. Daud and J. Sunday, “ Optimized hybrid block Adams method for solving first order ordinary differential equations,” Comput. Mater. Contin., vol. 72, no. 2, pp. 2948–2950, Mar. 2022, doi:10.32604/cmc.2022.025933.
7. J.M. Althemai, J. Sabo and M. Yaska, “The use of implicit single-step linear block method on third order ordinary differential equations by interpolation and collocation procedure,” DUJOPAS , vol. 8, no. 1, pp. 106-116, May 2022, doi:10.4314/dujopas.v8i1b.13.
8. M. D., Aloko, M.A., Ayinde, A.D., Usman and J. Sabo, “An efficient scheme for direct simulation of third and fourth oscillatory differential equations,” IJDM, vol. 1, no.1, pp. 72–89, Mar. 2024, doi:10.62054/ijdm/0101.06.

9. A.K. Jimoh, “Two-step hybrid block method for the numerical solution of third order ordinary differential equations,” IJMSI, vol. 12, no. 3, pp. 29–39, May-Jun. 2024, doi:10.35629/4767-12032939.
10. F.L. Joseph, “A continuous hybrid scheme for initial value problem of third order ordinary differential equations,” IJMCER, vol. 4, no. 5, pp.55–65, 2022.
11. M.K. Duromola and A.L. Momoh, “Hybrid numerical method with block extension for direct solution of third order ordinary differential equations,” Am. J. Comput. Math., vol. 9, no. 2, pp. 68–80, Jun. 2019, doi:10.4236/ajcm.2019.92006.
12. R. Abdelrahim and Z. Omar, “Solving third order ordinary differential equations using hybrid block method of order five,” Int. J. Appl. Eng., vol. 10, no. 24, pp. 44307–44310, 2015.
13. O. Adeyeye and Z. Omar, “Solving third order ordinary differential equation using one-step block method with four equidistance generalized hybrid points,” Int. J. Appl. Math.", vol. 49, no. 2, pp. 1–9, 2019.
14. K.M. Fasasi, “New continue hybrid constant block method for the solution of third order initial value problem of ordinary differential equations,” AJAMS, vol. 4, no. 6, pp. 53–60, 2018.
15. Y. Skwame, P.I. Dalatu, J. Sabo and M. Mathew, “Numerical application of third derivative hybrid block methods on third order initial value problem of ordinary differential equations,” Int. J. Stat. Appl. Math, vol. 4, no. 6, pp. 90–100, 2019.
16. D. Raymond, T.P. Pantuvu, A. Lydia, J. Sabo and R.Y. Ajia, “An optimized half step scheme third derivative methods for testing higher order initial value problems,” African Scientific Reports, vol. 2, no. 76, pp. 1–8, Apr. 2023, doi:10.46481/asr.2023.2.1.76.