SELECTED SINGLE-STEP HYBRID BLOCK FORMULA FOR SOLVING THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATION IN THIN FILM FLOW

  • Oboni Atabo Ahmadu Ribadu College, Yola
  • Praveen ARGAWAL
  • Alvary Kefas KWALA
  • Niongon Reuben ANONGO
Keywords: Selected single-step, basis function, Hybrid block formula, interpolation and collocation, thin film flow problem

Abstract

This research paper examines the derivation of selected hybrid single-step block method for the numerical integration of third order ordinary differential equations. The method has the advantage of selecting only odd off-grid points within the interval of interest. The basis function for the formula is interpolated at three selected non-grid points within a single-step interval and collocated at all points. Further analysis of the basic numerical properties were established. The method was found to be A-stable, zero-stable and consistent. The small scale errors observed from numerical experiment indicate that the derived formula has better numerical approximations than some comparable methods in literature, while its application on practical thin film flow problem also showed improved solutions.

References

Arqub, O. A. & Maayah, B. (2019a). Modulation of producing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense. Chaos Solutions Fractals, 1(125): 163-170, https://doi.Org/10.1016/j.chaos.2019.05.025.

Arqub, A. O. & Maayah, B. (2019b). Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC-Fractional Volterra integro-differential equations, Chaos Solutions Fractals, 1(126): 394-402, https://doi.org/10.1016Zj.chaos.2019.07.023.

Arqub, O. A. & Maayah, B. (2018). Numerical solutions of integro-differential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator, Chaos Solutions Fractals,

(117):117-124, https://doi.org/10.1016/j.chaos.2018.10.007.

Henrici, P. (1962). Discrete variable methods in ordinary differential equations, John Wiley & Sons, New York, USA, (1962).

Gragg, W. B. & Stetter, H. J. (1964). Generalized multistep predictor-corrector methods, Journal of the ACM, 11(2): 188-209.

Dahlquist, G. (1956). Convergence and stability in the numerical integration of ordinary differential equations, Mathematica Scandinavica, 1(4): 33-53, http://doi.org/10.7146/math.scan.a-10454.

Althemai, J. M., Sabo, J. & Yaska, M. (2022). The Use of Implicit Single-Step Linear Block Method on Third Order Ordinary Differential Equations by Interpolation and Collocation Procedure, Dutse Journal of Pure and Applied Sciences (DUJOPAS), 8(1b): 106-116, https://dx.doi.org/10.4314/ dujopas.v8i1b.13.

Kuboye, J. O., Quadri, O. F. & Elusakin, O. R. (2020). Solving Third Order Ordinary Differential Equations Directly using Hybrid Block Method, Nigerian Society of Physical Sciences, 2(2): 69-76.

Duromola, M. K. (2022). Single-Step Block Method of P-Stable for Solving Third Order Ordinary Differential Equations (IVPs): Ninth Order Accuracy, American Journal of Applied Mathematics and Statistics, 10(1):4-13, https://dx.doi.org/10.12691/ajams-10-1-2.

Duromola, M. K. & Momoh, A. L. (2019). Hybrid numerical method with block extension for direct solution og third order ordinary differential equations, Amer. J. of Compt. Math., 9: 68-80.

Abdelrahim, R. & Omar, Z. (2016). One step block method for the direct solution of third order initial value problems of ordinary differential equations, Far East Journal of Mathematical Sciences (FEJMS), Pushpa Publishing House, Allahabad, India, 99(6): 945-958, https://dx.doi.org/10. 17654/MS099060945.

Adeyeye, O. & Omar, Z. (2017). Solving Third Order Ordinary Differential Equations Using One Step Block Method with Four Equidistant Generalized Hybrid Points, IAENG International Journal of Applied Mathematics (IJAM), 49(2).

Modebei, M. I., Olaya, O. O. & Nwongwo, I. P. (2021). Computational Study of Some Three-Step Hybrid Integrators for Solution of Third Order Ordinary Differential Equations, Journal of Nigerian Society of Physical Sciences, 3(4): 459-468, https://doi.org/10.46481/jnsps.2021.323.

Haweel, M. T., Zahran, O. & Abd El-Samie, F. E. (2021). Adaptive Polynomial Method for Solving Third Order ODE with Application in Thin Flow, Open Access Journal, 9, https://doi.org/10. 1109/ACCESS.2021.3072944.

Lawal, K. O., Yahaya, Y. A. & Yakubu, S. D. (2018). Four-Step Block Method for Solving Third Order Ordinary Differential Equations, International Journal of Mathematics Trends and Technology (IJMTT), 57 331, http:

//www.ijmttjournal.org.

Kashkari, B. S. H. & Alqarni, S. (2019). Optimization of two-step block method with three hybrid points for solving third order initial value problems, Journal of Nonlinear Sciences and Applications, 12(7): 450-469, http://doi.org/10.22436/jnsa.012.07.04

Obarhua, F. O. & Kayode, S. J. (2016). Symmetric Hybrid Linear Multistep Method for General Third Order Ordinary Differential Equations, Open Access Library Journal, 3, http://dx.doi.org/ 10.4236/oalib.1102583.

Abolarin, O. E., Kuboye, J. O., Adeyefa, E. O. & Ogunware, B. G. (2020). New Efficient Numerical Model for Solving Second, Third and Fourth Order Ordinary Differential Equations Directly, Journal of Science, 33(4): 821-833, https://doi.org/10.35378/gujs.627677.

Ogunware, B. G., Omole, E. O. & Olanegan, O. O. (2015). Hybrid and non-hybrid implicit schemes for solving third order ODEs using block method as predictors, Journal of Mathematical Theory and Modeling (IISTE), 5(3): 10-25, https://doi.org/10.35378/gujs.627677.

Lambert, J. D. (1991). Numerical methods for ordinary differential systems, John Wiley and Sons, New York.

Butcher, J. C. (2008). Numerical Methods for Ordinary Diffferential Equations, John Wiley & Sons, Ltd., The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, England.

Lambert, J. D. (1973). Computational methods in ODEs, John Wiley and Sons, New YorK.

Areo, E. A. & Omole, E. O. (2015). Half-step symmetric continuous hybrid block method for the numerical solution of fourth order ordinary differential equations, Archives of Applied Science Research, 7(10): 39-49.

Adeniran, O. & Omotoye, A. (2016). One step hybrid block method for the numerical solution of general third order ordinary differential equations, International Journal of Mathematical Sciences, 2(5): 1-12.

Kayode, S. J. & Obarhua, F. O. (2017). Symmetric 2-step 4-point hybrid block method for the solution of general third order ordinary differential equations. Journal of Applied Computational Mathematics, 6(2): 1-4, http://doi.org/10.4172/2168-9679.1000348.

Momoniat, E. & Mahomed, F. M. (2010). Symmetry reduction and numerical solution of a third-order ODE from thin film flow,Mathematical and Computational Applications, 4(15): 709-719.

Mechee, M. Senu, N., Ismail, F., Nikouravan, B. & Siri, Z. (2013). A three-stage fifth-order Runge- Kutta method for directly solving special third-order differential equation with application to thin film flow problem, Mathematical Problems in Engineering.

Tuck, E. O. & Schwartz, L. W. (1990). A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows, SIAM Review, 3(32): 453.

Yap, L. K., Ismail, F. & Senu, N. (2014). An accurate block hybrid collocation method for third order ordinary differential equations, Hindawi Publishing Corporation, Journal of Applied Mathematics, http://doi.org/10.1155/2014/549597.

Aigbiremhon, A. A., Familua, A. B. and Omole, E. O. (2021). A Three-Step Interpolation Technique with Perturbation Term for Direct Solution Third-Order Ordinary Differential Equations, FUDMA Journal of Sciences (FJS) , 5(2): 365-376, https://doi.org/10.33003/fjs-2021-0502-556

Published
2023-01-07
How to Cite
AtaboO., ARGAWAL P., KWALA A. K., & ANONGO N. R. (2023). SELECTED SINGLE-STEP HYBRID BLOCK FORMULA FOR SOLVING THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATION IN THIN FILM FLOW. FUDMA JOURNAL OF SCIENCES, 6(6), 150 - 168. https://doi.org/10.33003/fjs-2022-0606-1152