Olusola Ezekiel ABOLARIN, John Olusola KUBOYE, Emmanuel Oluseye ADEYEFA, Gbenga OGUNWARE

New Efficient Numerical Model for Solving Second, Third and Fourth Order Ordinary Differential Equations Directly

This article presents a two-step hybrid linear multistep block method for solving second, thirdand fourth order initial value problems of ordinary differential equations directly. The derivationof the method was done using collocation and interpolation techniques, while approximatedpower series was used as an interpolating polynomial. The fourth derivative of the power serieswas collocated at the entire grid and off-grid points, while the fifth and sixth derivatives of thepolynomial were collocated at the endpoint only. The basic properties of the developed method,that is, order, error constant, zero stability, region of absolute stability, convergence andconsistency of the method were properly investigated. The numerical results demonstrated thatthe scheme developed handles: second, third and fourth order ordinary differential equationsefficiently and accurately when compared with existing methods. The proposed method takesaway the burden of developing a separate method for the solution of second, third and fourthorder initial value problem of ordinary differential equations.

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[1] Waeleh, N., Majid, Z. A., Ismail, F. and Suleiman, M., “Numerical solution of higher order ordinary differential equations by direct block code”, Journal of Mathematics and Statistics, 8 (1): 77-81, (2012).
[2] Cakır, M. and Arslan, D., “Reduced differential transform method for singularly perturbed sixthorder boussinesq equation”, Math Stat ,2: 14, 1-8, (2016).
[3] Arslan, D., “A novel hybrid method for singularly perturbed delay differential equations”, Gazi University Journal of Science, 32(1): 217 – 223, (2019).
[4] Cakir, M., Masiha, Y. R. and Arslan, D., “A novel numerical approach for solving convectiondiffusion problem with boundary layer behavior”, Gazi University Journal of Science, 33 (1): 152 – 166, (2020).
[5] Lambert, J. D., “Computational Methods in Ordinary Differential Equation”, John Wiley & Sons Inc., New York, (1973).
[6] Awoyemi, D. O., “A class of continuous linear multistep methods for general second order initial value problems in ordinary differential equations”, International Journal of Computer Mathematics, 72: 29-37, (1999).
[7] Brugnano, L. and Trigiante, D., “Solving Differential Problems by Multistep IVPs and BVP Methods”, Gordon and Breach Science Publishers, (1998).
[8] Gear, C. W., “The numerical integration of ordinary differential equations”, Math. Comp., 21: 146 - 156, (1966).
[9] Gear, C. W., “Numerical Initial Value Problems in Ordinary Differential Equations”, New Jersey, Prentice Hall, (1971).
[10] Gear, C. W., “The stability of numerical methods for second order ordinary differential equations”, SIAM Journal of Numerical Analysis, 15(1): 1187 – 197, (1987).
[11] Hall, G. and Suleiman, M. B., “Stability of Adams – type formulae for second order ordinary differential equations”, SIAM Journal of Numerical Analysis, 1: 427 – 428, (1981).
[12] Mohammed, U., “A six step block method for solution of fourth order ordinary differential equations”, The Pacific Journal of Science and Technology, 11(1): 259-265, (2010).
[13] Badmus, A. M. and Yahaya, Y. A., “A class of collocation methods for general second order ordinary differential equations”, African Journal of Mathematics and Computer Science Research, 2(4): 69-71, (2009).
[14] Ogunware, B. G., Omole, E. O. and Olanegan, O. O., “Hybrid and non-hybrid ımplicit schemes for solving third order odes using block method as predictors”, Journal of Mathematical Theory and Modeling (IISTE), 5(3): 10-25, (2015).
[15] Omole, E. O. and Ogunware, B. G., “3- point single hybrid block method (3pshbm) for direct solution of general second order ınitial value problem of ordinary differential equations”, JSRR, 20(3): 1-11, (2018).
[16] Areo, E. A. and Omole, E. O., “Half-step symmetric continuous hybrid block method for the numerical solutions of fourth order ordinary differential equations”, Archives of Applied Science Research, 7 (10): 39-49, (2015).
[17] Kuboye, J. O, Omar, Z., Abolarin, O. E. and Abdelrahim, R., “Generalized hybrid block method for solving second order ordinary differential equations directly”, Res. Rep. Math. , 2(1): 1-7, (2018).
[18] Omar, Z. B. and Suleiman, M. B., “Solving higher order odes directly using parallel 2-point explicit block method”, Matematika Pengintegrasian Teknologi Dalam Sains Matematik, Universiti Sains Malaysia, 21(1): 15-23, (2005).
[19] Majid, Z. A., “Parallel Block Methods for Solving Ordinary Differential Equations”, PhD Thesis, University Putra Malaysia, 25-38 (2004).
[20] Fatunla, S.O., “Block methods for second order ivps”, Intern. J. Computer Math., 41(9): 55-63, (1991).
[21] Awoyemi, D. O., “A p-stable linear multistep method for solving third order ordinary differential equation”, Int. J. Compt, Math., 80(8): 85-99, (2003).
[22] Adeniran, A. O., Odejide, S. A. and Ogundare, S. B., “One step hybrid numerical scheme for the direct solution of general second order ordinary differential equations”, International Journal of Applied Mathematics”, 28(3): 197-212, (2015).
[23] Adoghe, L. O., Ogunware, B. G. and Omole, E. O., “A family of symmetric implicit higher order methods for the solution of third order initial value problems in ordinary differential equations”, Theoretical Mathematics & Applications, 6(3): 67-84, (2016).
[24] Duromola, M.K., “An accurate five off-step points ımplicit block method for direct solution of fourth order differential equations”, Open Access Library Journal, 3: 1-14, (2016).

Gazi University Journal of Science-Cover

Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 1988
Yayıncı: Gazi Üniversitesi, Fen Bilimleri Enstitüsü

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