Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative

Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative

A great number of physical phenomena can be expressed as initial or boundary value problems of higher order ordinary differential equations (ODEs) which may not have analytical solutions. Thus, there is a need to develop numerical methods for approximating the solution of higher order ODEs. One of t...

Full description

Saved in:
Bibliographic Details
Main Author: Alkassasbeh, Mohammad Farhan
Format: Thesis
Language:eng
eng
Published: 2018
Subjects:
QA273-280 Probabilities
Mathematical statistics
Online Access:https://etd.uum.edu.my/7523/1/Depositpermission_s900374.pdf
https://etd.uum.edu.my/7523/2/s900374_01.pdf
Tags: Add Tag
No Tags, Be the first to tag this record!
id my-uum-etd.7523
record_format uketd_dc
institution Universiti Utara Malaysia
collection UUM ETD
language eng
eng
advisor Omar, Zurni
topic QA273-280 Probabilities
Mathematical statistics
spellingShingle QA273-280 Probabilities
Mathematical statistics
Alkassasbeh, Mohammad Farhan
Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative
description A great number of physical phenomena can be expressed as initial or boundary value problems of higher order ordinary differential equations (ODEs) which may not have analytical solutions. Thus, there is a need to develop numerical methods for approximating the solution of higher order ODEs. One of the well-known direct methods which frequently employed is block method. Even though this method is capable of finding the approximate solutions at several points simultaneously, it fails to overcome the zero-stability barrier. Thus, a hybrid block method was introduced to tackle this drawback. The main benefit of this method is its ability of using data at off-step points which contribute to better accuracy. Most of the existing hybrid block methods, however, only focus on specific off-step point(s) in deriving the methods with the exception of the method proposed by Abdelrahim in 2016. Although he has successfully developed one-step hybrid block methods with generalised off-step point(s) for solving high order ODEs directly, nevertheless, the methods are only confined to initial value problems. Moreover, he did not consider higher derivative in developing those methods. Thus, this study introduced new one-step hybrid block methods with generalised off-step point(s) in the presence of higher derivative for directly solving higher order ODEs. In developing these methods, a power series was used as an approximate solution to the problems of ODEs of order m. The power series was interpolated at m points, while its mth and (m+1)th derivatives were collocated at all points in the given interval. Investigations on the properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also carried out. Several initial and boundary value problems of higher order ODEs considered in literature were then solved by using the newly developed methods in order to investigate the accuracy of the solution in terms of error. The numerical results revealed that, in general, the new methods were able to produce smaller errors compared to the existing methods in solving the same problems. In conclusion, this study has successfully developed viable methods for directly solving both initial and boundary value problems of higher order ODEs.
format Thesis
qualification_name Ph.D.
qualification_level Doctorate
author Alkassasbeh, Mohammad Farhan
author_facet Alkassasbeh, Mohammad Farhan
author_sort Alkassasbeh, Mohammad Farhan
title Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative
title_short Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative
title_full Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative
title_fullStr Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative
title_full_unstemmed Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative
title_sort direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points in the presence of higher derivative
granting_institution Universiti Utara Malaysia
granting_department Awang Had Salleh Graduate School of Arts & Sciences
publishDate 2018
url https://etd.uum.edu.my/7523/1/Depositpermission_s900374.pdf
https://etd.uum.edu.my/7523/2/s900374_01.pdf
_version_ 1747828230325272576
spelling my-uum-etd.75232021-08-09T03:59:43Z Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative 2018 Alkassasbeh, Mohammad Farhan Omar, Zurni Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts & Sciences QA273-280 Probabilities. Mathematical statistics A great number of physical phenomena can be expressed as initial or boundary value problems of higher order ordinary differential equations (ODEs) which may not have analytical solutions. Thus, there is a need to develop numerical methods for approximating the solution of higher order ODEs. One of the well-known direct methods which frequently employed is block method. Even though this method is capable of finding the approximate solutions at several points simultaneously, it fails to overcome the zero-stability barrier. Thus, a hybrid block method was introduced to tackle this drawback. The main benefit of this method is its ability of using data at off-step points which contribute to better accuracy. Most of the existing hybrid block methods, however, only focus on specific off-step point(s) in deriving the methods with the exception of the method proposed by Abdelrahim in 2016. Although he has successfully developed one-step hybrid block methods with generalised off-step point(s) for solving high order ODEs directly, nevertheless, the methods are only confined to initial value problems. Moreover, he did not consider higher derivative in developing those methods. Thus, this study introduced new one-step hybrid block methods with generalised off-step point(s) in the presence of higher derivative for directly solving higher order ODEs. In developing these methods, a power series was used as an approximate solution to the problems of ODEs of order m. The power series was interpolated at m points, while its mth and (m+1)th derivatives were collocated at all points in the given interval. Investigations on the properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also carried out. Several initial and boundary value problems of higher order ODEs considered in literature were then solved by using the newly developed methods in order to investigate the accuracy of the solution in terms of error. The numerical results revealed that, in general, the new methods were able to produce smaller errors compared to the existing methods in solving the same problems. In conclusion, this study has successfully developed viable methods for directly solving both initial and boundary value problems of higher order ODEs. 2018 Thesis https://etd.uum.edu.my/7523/ https://etd.uum.edu.my/7523/1/Depositpermission_s900374.pdf text eng public https://etd.uum.edu.my/7523/2/s900374_01.pdf text eng public http://sierra.uum.edu.my/record=b1697805~S1 Ph.D. doctoral Universiti Utara Malaysia Abdelrahim, R. (2016). One-step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations (Unpublished doctoral dissertation). Universiti Utara Malaysia, Malaysia. Abdelrahim, R., & Omar, Z. (2015). Uniform order one-step hybrid block method with two generalized off-step points for solving third order ordinary differential equations directly. Global Journal of Pure and Applied Mathematics, 11(6), 4809–4823. Abdullah, A. S., Majid, Z., & Senu, N. (2013). Solving third order boundary value problem using fourth order block method. Applied Mathematical Sciences, 7(53-56), 2629–2645. Ademiluyi, R. D. M., & Bolarinwa, B. (2014). Modified block method for the direct solution of initial value problems of fourth order ordinary differential equations. Australian Journal of Basic and Applied Sciences, 8(10), 89-394. Adeniran, O., & Ogundare, B. (2015). An efficient hybrid numerical scheme for solving general second order initial value problems. International Journal of Applied Mathematical Research, 4(2), 411–419. Adeniyi, R., & Adeyefa, E. (2013). Chebyshev collocation approach for a continuous formulation of implicit hybrid methods for VIPs in second order ODEs. Journal of Mathematics, 12, 9–12. Adesanya, A., Abdulqadri, B., & Ibrahimb, Y. (2014). Hybrid one-step block method for the solution of third order initial value problems of ordinary differential equations. International Journal of Applied Mathematics and Computation, 6(1), 10–16. Adesanya, A., Fasasi, M. K., & Anake, T. (2013). Three-steps hybrid block method for the solution of general second order ordinary differential equations. Journal of Applied Mathematics, 8(1 , 2), 1–11. Adesanya, A., Odekunle, M., & Adeyeye, O. (2012). Continuous block hybrid predictor corrector method for the solution of y00 = f (x;y;y0). International Journal of Mathematics and Soft Computing, 2(2), 35–42. Adesanya, A., Udoh, D. M., & Ajileye, A. M. (2013). A new hybrid block method for the solution of general third order initial value problems of ordinary differential equations. International Journal of Pure and Applied Mathematics, 86(2), 365–375. Adeyeye, O. (2013). Two-step two-point hybrid methods for general second order differential equations. African Journal of Mathematics and Computer Science Research, 6(10), 191–196. Adeyeye, O., & Omar, Z. (2016). New uniform order eight hybrid third derivative block method for solving second order initial value problems. Far East Journal of Mathematical Sciences, 100(9), 1515. Adeyeye, O., & Omar, Z. (2017). Linear block method derived from direct Taylor series expansions for solving linear third order boundary value problems. Advanced Science, Engineering and Medicine, 9(10), 895–900. Anake, T., Awoyemi, D. O., & Adesanya, A. (2012a). One-step implicit hybrid block method for the direct solution of general second order ordinary differential equations. International Journal of Applied Mathematics, 42(4), 1-5. Anake, T., Awoyemi, D. O., & Adesanya, A. (2012b). A one-step method for the solution of general second order ordinary differential equations. International Journal of Science and Technology, 2(4), 159–163. Atkinson, K. (1989). An introduction to numerical analysis. New York, NY: John Wiley and Sons. Atkinson, K., & Han, W. (2004). Elementary numerical analysis. New York, NY: John Wiley and Sons. Awoyemi, D. (2003). A p-stable linear multi-step method for solving general third order ordinary differential equations. International Journal of Computer Mathematics, 80(8), 985–991. Awoyemi, D., Adebile, E., Adesanya, A., & Anake, T. A. (2011). Modified block method for the direct solution of second order ordinary differential equations. International Journal of Applied Mathematics and Computation, 3(3), 181–188. Awoyemi, D., Kayode, S. J., & Adoghe, L. (2015). A six-step continuous multistep method for the solution of general fourth order initial value problems of ordinary differential equations. Journal of Natural Sciences Research, 5(5), 131–138. Badmus, A. (2014). An efficient seven-point hybrid block method for the direct solution of y00 = f (x;y;y0). British Journal of Mathematics & Computer Science, 4(19), 2840–2852. Burden, R., & Faires, J. (2011). Numerical analysis. Boston, Massachusetts: Brooks/Cole Cengage Learning. Butcher, J. C. (1965). A modified multi-step method for the numerical integration of ordinary differential equations. Journal of the ACM, 12(1), 124–135. Chen, J. (2011). Fast multilevel augmentation methods for nonlinear boundary value problems. Computers & Mathematics with Applications, 61(3), 612–619. Chen, J., & Li, C. (2012). High accuracy finite difference schemes for linear fourth order boundary value problem and derivatives. Journal of Information and Computational Science, 9(10), 2751–2759. Chu, M. T., & Hamilton, H. (1987). Parallel solution of ODEs by multiblock methods. SIAM Journal on Scientific and Statistical Computing, 8(3), 342–353. Enright, W. (1974). Second derivative multi-step methods for stiff ordinary differential equations. SIAM Journal on Numerical Analysis, 11(2), 321–331. Faires, J., & Burden, R. (2002). Numerical methods. Boston, Massachusetts: Brooks/Cole Cengage Learning. Fatunla, S. (1991). Block methods for second order ODEs. International journal of computer mathematics, 41(1-2), 55–63. Gbenga, O. B., Olaoluwa, O. E., & Olayemi, O. O. (2015). Hybrid and non-hybrid implicit schemes for solving third order ODEs using block method as predic-tors. Mathematical Theory and Modeling, 5(3), 10–25. Gear, C. W. (1965). Hybrid methods for initial value problems in ordinary differential equations. Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis, 2(1), 69–86. Gragg, W. B., & Stetter, H. J. (1964). Generalized multi-step predictor-corrector methods. Journal of the ACM, 11(2), 188–209. Henrici, P. (1962). Discrete variable methods in ordinary differential equations. New York, NY: John Wiley and Sons. Hijazi, M., & Abdelrahim, R. (2017). The numerical computation of three-step hybrid block method for directly solving third order ordinary differential equations. Global Journal of Pure and Applied Mathematics, 13(1), 89–103. James, A., Adesanya, A., & Joshua, S. (2013). Continuous block method for the solution of second order initial value problems of ordinary differential equation. International Journal of Pure and Applied Mathematics, 83(3), 405–416. Jator, S. (2008a). Numerical integrators for fourth order initial and boundary value problems. International Journal of Pure and Applied Mathematics, 47(4), 563–576. Jator, S. (2008b). On the numerical integration of third order boundary value problems by a linear multi-step method. International Journal of Pure and Applied Mathematics, 46(3), 375–388. Jator, S., Akinfenwa, A., Okunuga, S. A., & Sofoluwe, A. B. (2013). High-order continuous third derivative formulas with block extensions for y00 = f (x;y;y0). International Journal of Computer Mathematics, 90(9), 1899–1914. Jator, S., & Li, J. (2012). An algorithm for second order initial and boundary value problems with an automatic error estimate based on a third derivative method. Numerical Algorithms, 59(3), 333–346. Kayode, S. (2011). A class of one-point zero-stable continuous hybrid methods for direct solution of second-order differential equations. African Journal of Mathematics and Computer Science Research, 4(3), 93–99. Kayode, S., & Adeyeye, O. (2011). A 3-step hybrid method for direct solution of second order initial value problems. Australian Journal of Basic and Applied Sciences, 5(12), 2121–2126. Kayode, S., Duromola, M., & Bolaji, B. (2014). Direct solution of initial value problems of fourth order ordinary differential equations using modified implicit hybrid block method. Journal of Scientific Research & Reports, 3(21), 2792-2800. Kayode, S., & Obarhua, F. (2015). 3-step y- function hybrid methods for direct numerical integration of second order IVPs in ODEs. Theoretical Mathematics & Applications, 5(1), 39–51. Kelesoglu, O. (2014). The solution of fourth order boundary value problem arising out of the beam-column theory using Adomian decomposition method. Mathematical Problems in Engineering, 2014, 1–6. Keller, H. B. (1976). Numerical solution of two point boundary value problems. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics. Kolawole, F., Adesanya, A., Momoh, A., & Emmanuel, N. (2014). Continuous hybrid block Stomer Cowell methods for solutions of second order ordinary differential equations. Journal of Mathematical and Computational Science, 4(1), 118-127. Kolawole, F., Fasansi, M., Remilekun, O. M., & Adesanya, A. (2013). Onestep, three hybrid block predictor-corrector method for the solution of y000 = f (x;y;y0;y00). Journal of Applied & Computational Mathematics, 2(4), 137-142. Kuboye, J., & Omar, Z. (2015a). Numerical solution of third order ordinary differential equations using a seven-step block method. International Journal of Mathematical Analysis, 9(15), 743–745. Kuboye, J., & Omar, Z. (2015b). Numerical solution of third order ordinary differential equations using a seven-step block method. International Journal of Mathematical Analysis, 9(15), 743–745. Lambert, J. (1973). Computational methods in ordinary differential equations. Chichester, West Sussex: John Wiley and Sons. Lambert, J. (1991). Numerical methods for ordinary differential systems:the initial value problem. New York, NY: John Wiley and Sons. Lambert, J., & Watson, I. (1976). Symmetric multistip methods for periodic initial value problems. IMA Journal of Applied Mathematics, 18(2), 189–202. Liu, L. B., Liu, H. W., & Chen, Y. (2011). Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions. Applied Mathematics and Computation, 217(16), 6872–6882. Milne, W. (1953). Numerical solution of differential equations. New York, NY: Wiley. Mohammed, U., & Adeniyi, R. (2014). A three-step implicit hybrid linear multi-step method for the solution of third order ordinary differential equations. Gen. Math. Notes, 25(1), 62–74. Morrison, D. D., Riley, J. D., & Zancanaro, J. F. (1962). Multiple shooting method for two-point boundary value problems. Communications of the ACM, 5(12), 613–614. Ngwane, F., & Jator, S. (2012). Block hybrid-second derivative method for stiff systems. International Journal of Pure and Applied Mathematics, 80(4), 543–559. Olabode, B. (2007). Some linear multi-step methods for special and general third order initial value problems of ordinary differential equations (Unpublished doctoral dissertation). Federal University of Technology, Akure, Nigeria. Olabode, B. (2014). Block multi-step method for the direct solution of third order of ordinary differential equations. Futa Journal of Research in Sciences, 9(2), 194–200. Olabode, B., & Omole, E. (2015). Implicit hybrid block numerov-type method for the direct solution of fourth-order ordinary differential equations. American Journal of Computational and Applied Mathematics, 5(5), 129–139. Olabode, B., & Yusuph, Y. (2009). A new block method for special third order ordinary differential equations. Journal of Mathematics and statistics, 5(3), 167. Omar, Z. (1999). Parallel block methods for solving higher order ordinary differential equations directly (Unpublished doctoral dissertation). Universiti Putra Malaysia, Malaysia. Omar, Z. (2004). Developing parallel 3-point implicit block method for solving second order ordinary differential equations directly. International Journal of Management Studies, 11(1), 91–103. Omar, Z., & Abdelrahim, R. (2015). Developing a single-step hybrid block method with generalized three off-step points for the direct solution of second order ordinary differential equations. International Journal of Mathematical Analysis, 9(46), 2257–2272. Omar, Z., & Abdelrahim, R. (2016). Direct solution of fourth order ordinary differential equations using a one-step hybrid block method of order five. International Journal of Pure and Applied Mathematics, 109(4), 763–777. Omar, Z., & Abdelrahim, R. (2017). A four-step implicit block method with three generalized off-step points for solving fourth order initial value problems directly. Journal of King Saud University-Science, 29(2017), 401–412. Omar, Z., & Adeyeye, O. (2016). Solving two-point second order boundary value problems using two-step block method with starting and non-starting values. International Journal of Applied Engineering Research, 11(4), 2407–2410. Omar, Z., & Kuboye, J. (2015). Derivation of block methods for solving second order ordinary differential equations directly using direct integration and collocation approaches. Indian Journal of Science and Technology, 8(12), 1–4. Omar, Z., & Suleiman, M. (1999). Solving second order ODEs directly using parallel 2-point explicit block method. Prosiding Kolokium Kebangsaan Pengintegrasian Teknologi Dalam Sains Matematik, Universiti Sains Malaysia, 390–395. Phang, P., Majid, Z., Suleiman, M., & Ismail, F. (2013). Solving boundary value problems with neumann conditions using direct method. World Applied Sciences Journal, 129–133. doi: 10.5829/idosi.wasj.2013.21.mae.99937 Ramos, H., Kalogiratou, Z., Monovasilis, T., & Simos, T. E. (2015). An optimized two-step hybrid block method for solving general second order initial-value problems. Numerical Algorithms, 1–14. doi: 10.1007/s11075-015-0081-8 Sagir, A. (2013a). 2–block 3-point modified numerov block methods for solving ordinary differential equations. World Academy of Science, Engineering and Technology, 7(1), 700–704. Sagir, A. (2013b). An accurate computation of block hybrid method for solving stiff ordinary differential equations. World Academy of Science, Engineering and Technology, 7(4), 321–324. Sahi, R., Jator, S., & Khan, N. (2013). Continuous fourth derivative method for third order boundary value problems. International Journal of Pure and Applied Mathematics, 85(5), 907–923. Taiwo, O. A., & Ogunlaran, O. M. (2011). A non-polynomial spline method for solving linear fourth-order boundary-value problems. International Journal of Physical Sciences, 6(13), 3246–3254. Vigo-Aguiar, J., & Ramos, H. (2006). Variable stepsize implementation of multi-step methods for y00 = f (x;y;y0). Journal of Computational and Applied Mathematics, 192(1), 114–131. Wend, D. V. (1969). Existence and uniqueness of solutions of ordinary differential equations. Proceedings of the American Mathematical Society, 23(1), 27–33. Yahaya, Y., Sagir, A., & Tech, M. (2013). An order five implicit 3-stepblock method for solving ordinary differential equations. The Pacific Journal of Science and Technology, 14, 176–181. Yap, L. K., & Ismail, F. (2015). Block hybrid collocation method with application to fourth order differential equations. Mathematical Problems in Engineering, 2015, 1–6. doi: 10.1155/2015/561489 Yap, L. K., Ismail, F., & Senu, N. (2014). An accurate block hybrid collocation method for third order ordinary differential equations. Journal of Applied Mathematics, 2014, 1-9. doi: 10.1155/2014/549597 Yun, J. H. (2008). A note on three-step iterative method for nonlinear equations. Applied Mathematics and Computation, 202(1), 401–405. Zhang,W. (2012). Improved implementation of multiple shooting for BVPs. (Master’s thesis, Computer Science Department, University of Toronto). Retrieved from http://www.cs.toronto.edu/pub/reports/na/ Weidong_Zhang_Thesis.pdf.

Similar Items