Three-Step Block Method for Solving Second Order Differential Equations
Keywords:
Three-step Block method, Legendre Polynomials and absolutely stable.
Abstract
In this paper, we developed a three-step block Method for numerical solution of second order differential equations using Legendre polynomials as the basic function. Interpolation and collocation procedures are used by choosing interpolation points at s=2 steps points using power series, while collocation points at r=k step points, using a combination of power series and perturbation term gotten from the Legendre polynomials, giving rise to a polynomial of degree r+s-2 and r+s equations. All the analysis on the scheme derived shows that it is stable, convergent and has region of Absolute Stability. Numerical examples were provided to test the performance of the method. Results obtained when compared with existing methods in the literature, shows that the method is accurate and efficient.
References
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[3] Ehigie J.O., Okunuga, S.A, Sofoluwe, A.B. and Akanbi, M.A. On Generalized 2-step continuous linear multistep method of hybrid type for the integration of second ODEs, Journal of Scholars Research Library 2(6): 362-372 (2010).
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[5] Kayode and Adebeye Two step point hybrid methods for general second order Differential Equations. African Journal of Mathematics and Computer Sciences research 6(10), 191 – 196 (2013).
[6] Ademiluyi R.A., Duromola, M.K and Bolarinwa B. Modified Block Method for the direct solution of initial value problem of Fourth Order Ordinary Differential Equation. Austrialian Journal of Basic and Applied sciences, Journal of Basic and Applied 8(10); 389 – 394 (2014).
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[9] Abdulganiy, R.I, Akinfewa, O.A. and Okunuga, S.A. Maximal order Block Trigonometrially Fitted scheme for the Numerical Treatment of second order Initial Value Problem with ossillating solution. International Journal of Mathematical Analysis and Optimization Theory and Application. VOL. 2017, 168-186 (2017).
[10] Peter, O.J. and Ibrahim, M.O. Application of Differential Transformation Method in Solving a Typhoid Fever Model. International Journal of Mathematical Analysis and Optimization Theory and Application. Vol. 2017, 250-260 (2017).
[11] Zarina, B. I., Mohammed, S., Kharil, I. and Zanariah, M. Block method for generalized multistep Adams method and backward differentiation formula in solving first – order ODEs, Mathematika 25-33 (2005).
[12] Yahaya. Y.A., and Mohammed U. Fully implicit three point backward differentiation formulae for solution of first order initial value problems. International Journal of Numerical mathematics 5(3): 384 – 398 (2010).
[13] Odekunle, M.R.; Adesanya, A.O.; and Sunday, J. A New Block integrator for the solution of initial value problems of first order ordinary differential equation. International Journal of Pure and Applied Sciences and Technology 11(1):92-100 (2012).
[14] Sunday, J; James, A.A; Ibijola, E.A.; Ogunrinde, R.B.; and Ogunyebi, S.A., A computational Approach to Verhulst – Pearl Model 10SR Journal of Mathematics 4; 06 – 13 (2012).
[15] Awoyemi, D.O On some continuous linear multistep methods for initial value problems unpublished doctoral dissatisfaction, University of Ilorin (1992).
[16] Lambert J.D. Computational methods in ordinary differential equation. John Wiley, New York (1973).
[17] Abhulimen C. E. and Aigbiremhon, A.A. Continuous Implicit Block Methods Using Legendre Polynomials for Numerical Solution of Ordinary Differential Equations of Order One, Journal of Mathematical physics vol. 41. 465-474 (2017).
[18] Chollon, J.P., Ndam, J.N, and Kunleng, G. M On some properties of the block linear multistep methods, scienes world Journal, 2(3), 11-17 (2007).
[19] Jator, S.N., A sixeth order linear multistep method for the direct solution of ordinary differential equations. International Journal of Pure and Applied Mathematics 40(1), 457-472 (2007).
[20] Lambert J.D. Computational methods in ordinary differential equation. John Wiley, New York. (1991).
[21] Henrici, P. Discrete variable method in Ordinary Differential Equation. John Wiley and Sons. New York (1962).
[2] Anake T.A. Continuous Implicit one – step methods for the solution IVPs of General Second ODEs, using power series unpublished doctoral dissertation, Convenant University, Ota. (2011).
[3] Ehigie J.O., Okunuga, S.A, Sofoluwe, A.B. and Akanbi, M.A. On Generalized 2-step continuous linear multistep method of hybrid type for the integration of second ODEs, Journal of Scholars Research Library 2(6): 362-372 (2010).
[4] Adeniyi R.B. & Adeyefa, E.O. Chebyshev Collocation Approach for a Continuous formation of implicit hybrid methods for IVPs in second ODEs. Journal of Mathematics, 6(4):9 – 12 (2013).
[5] Kayode and Adebeye Two step point hybrid methods for general second order Differential Equations. African Journal of Mathematics and Computer Sciences research 6(10), 191 – 196 (2013).
[6] Ademiluyi R.A., Duromola, M.K and Bolarinwa B. Modified Block Method for the direct solution of initial value problem of Fourth Order Ordinary Differential Equation. Austrialian Journal of Basic and Applied sciences, Journal of Basic and Applied 8(10); 389 – 394 (2014).
[7] Bolarinwa B. Implicit hybrid block methods for the numerical solution of IVPs of third order ODEs. A Ph.D thesis in mathematical sciences Department of Federal University of Technology, Akure, Nigeria, (2012).
[8] Osilagun, J.A., Adesanya, A.O., Anake, T.A. and Oghoyon, G. J. Four steps implicit method for the solution of General second order Ordinary Differential Equations. Journal of Natural Sciences, Engineering and Technology 8(1), 52-61 (2009).
[9] Abdulganiy, R.I, Akinfewa, O.A. and Okunuga, S.A. Maximal order Block Trigonometrially Fitted scheme for the Numerical Treatment of second order Initial Value Problem with ossillating solution. International Journal of Mathematical Analysis and Optimization Theory and Application. VOL. 2017, 168-186 (2017).
[10] Peter, O.J. and Ibrahim, M.O. Application of Differential Transformation Method in Solving a Typhoid Fever Model. International Journal of Mathematical Analysis and Optimization Theory and Application. Vol. 2017, 250-260 (2017).
[11] Zarina, B. I., Mohammed, S., Kharil, I. and Zanariah, M. Block method for generalized multistep Adams method and backward differentiation formula in solving first – order ODEs, Mathematika 25-33 (2005).
[12] Yahaya. Y.A., and Mohammed U. Fully implicit three point backward differentiation formulae for solution of first order initial value problems. International Journal of Numerical mathematics 5(3): 384 – 398 (2010).
[13] Odekunle, M.R.; Adesanya, A.O.; and Sunday, J. A New Block integrator for the solution of initial value problems of first order ordinary differential equation. International Journal of Pure and Applied Sciences and Technology 11(1):92-100 (2012).
[14] Sunday, J; James, A.A; Ibijola, E.A.; Ogunrinde, R.B.; and Ogunyebi, S.A., A computational Approach to Verhulst – Pearl Model 10SR Journal of Mathematics 4; 06 – 13 (2012).
[15] Awoyemi, D.O On some continuous linear multistep methods for initial value problems unpublished doctoral dissatisfaction, University of Ilorin (1992).
[16] Lambert J.D. Computational methods in ordinary differential equation. John Wiley, New York (1973).
[17] Abhulimen C. E. and Aigbiremhon, A.A. Continuous Implicit Block Methods Using Legendre Polynomials for Numerical Solution of Ordinary Differential Equations of Order One, Journal of Mathematical physics vol. 41. 465-474 (2017).
[18] Chollon, J.P., Ndam, J.N, and Kunleng, G. M On some properties of the block linear multistep methods, scienes world Journal, 2(3), 11-17 (2007).
[19] Jator, S.N., A sixeth order linear multistep method for the direct solution of ordinary differential equations. International Journal of Pure and Applied Mathematics 40(1), 457-472 (2007).
[20] Lambert J.D. Computational methods in ordinary differential equation. John Wiley, New York. (1991).
[21] Henrici, P. Discrete variable method in Ordinary Differential Equation. John Wiley and Sons. New York (1962).
Published
2018-08-08
How to Cite
Abhulimen, C. E., & Aigbiremhon, A. (2018). Three-Step Block Method for Solving Second Order Differential Equations. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2018, 364 - 381. Retrieved from http://ijmso.unilag.edu.ng/article/view/53
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