Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach

Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach

Countless problems in real life situations involve rates of change of one or more independent variables. These rates of change can be expressed in terms of derivatives which lead to differential equations. Conventionally, initial value problems of higher order ordinary differential equations are sol...

Full description

Saved in:
Bibliographic Details
Main Author: Olusola, Kuboye John
Format: Thesis
Language:eng
eng
Published: 2015
Subjects:
QA273-280 Probabilities
Mathematical statistics
Online Access:https://etd.uum.edu.my/5789/1/depositpermission_s95332.pdf
https://etd.uum.edu.my/5789/2/s95332_01.pdf
Tags: Add Tag
No Tags, Be the first to tag this record!
id my-uum-etd.5789
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
Olusola, Kuboye John
Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach
description Countless problems in real life situations involve rates of change of one or more independent variables. These rates of change can be expressed in terms of derivatives which lead to differential equations. Conventionally, initial value problems of higher order ordinary differential equations are solved by first reducing the equations to their equivalent systems of first order ordinary differential equations. Then, suitable existing numerical methods for first order ordinary differential equations will be employed to solve the resulting equations. However, this approach will enlarge the equations and thus increases computational burden which may jeopardise the accuracy of the solution. In overcoming the setbacks, direct methods were proposed. Disappointedly, most of the existing direct methods approximate the numerical solution at one point at a time. Block methods were then introduced with the aim of approximating numerical solutions at many points concurrently. Several new block methods using interpolation and collocation approach for solving initial value problems of higher order ordinary differential equations directly were developed in this study to increase the accuracy of the solution. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order d. The power series was interpolated at d points before the last two points while its highest derivative was collocated at all grid points in deriving the new block methods. In addition, the properties of the new methods such as order, error constant, zerostability, consistency, convergence and region of absolute stability were also investigated. The developed methods were then applied to solve several initial value problems of higher order ordinary differential equations. The numerical results indicated that the new methods produced better accuracy than the existing methods when solving the same problems. Therefore, this study has successfully produced new methods for solving initial value problems of higher order ordinary differential equations.
format Thesis
qualification_name Ph.D.
qualification_level Doctorate
author Olusola, Kuboye John
author_facet Olusola, Kuboye John
author_sort Olusola, Kuboye John
title Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach
title_short Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach
title_full Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach
title_fullStr Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach
title_full_unstemmed Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach
title_sort block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach
granting_institution Universiti Utara Malaysia
granting_department Awang Had Salleh Graduate School of Arts & Sciences
publishDate 2015
url https://etd.uum.edu.my/5789/1/depositpermission_s95332.pdf
https://etd.uum.edu.my/5789/2/s95332_01.pdf
_version_ 1747827982414643200
spelling my-uum-etd.57892016-07-20T10:17:26Z Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach 2015 Olusola, Kuboye John Omar, Zurni Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts and Sciences QA273-280 Probabilities. Mathematical statistics Countless problems in real life situations involve rates of change of one or more independent variables. These rates of change can be expressed in terms of derivatives which lead to differential equations. Conventionally, initial value problems of higher order ordinary differential equations are solved by first reducing the equations to their equivalent systems of first order ordinary differential equations. Then, suitable existing numerical methods for first order ordinary differential equations will be employed to solve the resulting equations. However, this approach will enlarge the equations and thus increases computational burden which may jeopardise the accuracy of the solution. In overcoming the setbacks, direct methods were proposed. Disappointedly, most of the existing direct methods approximate the numerical solution at one point at a time. Block methods were then introduced with the aim of approximating numerical solutions at many points concurrently. Several new block methods using interpolation and collocation approach for solving initial value problems of higher order ordinary differential equations directly were developed in this study to increase the accuracy of the solution. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order d. The power series was interpolated at d points before the last two points while its highest derivative was collocated at all grid points in deriving the new block methods. In addition, the properties of the new methods such as order, error constant, zerostability, consistency, convergence and region of absolute stability were also investigated. The developed methods were then applied to solve several initial value problems of higher order ordinary differential equations. The numerical results indicated that the new methods produced better accuracy than the existing methods when solving the same problems. Therefore, this study has successfully produced new methods for solving initial value problems of higher order ordinary differential equations. 2015 Thesis https://etd.uum.edu.my/5789/ https://etd.uum.edu.my/5789/1/depositpermission_s95332.pdf text eng staffonly https://etd.uum.edu.my/5789/2/s95332_01.pdf text eng public Ph.D. doctoral Universiti Utara Malaysia Adeniyi, R. B., & Alabi, M. (2011). A collocation method for direct numerical integration of initial value problems in higher order ordinary differential equations. Analele Stiintifice Ale Universitatii “AL. I. Cuza” Din Iasi (SN), Matematica, Tomul LV, f, 2, 311-321. Adesanya, A. O., Anake, T. A., & Oghonyon, G. J. (2009). Continuous implicit method for the solution of general second order ordinary differential equations. Journal of the Nigerian Association of Mathematical Physics, 15, 71-78. Adesanya, A. O., Odekunle, M. R. and Alkali, M. A. (2012). Three steps block predictorblock corrector method for solution of general second order ordinary differential equations. 2(4), 2297-2301. Adesanya, A. O., Udo, M. O., & Alkali, A. M. (2012). A New Block-Predictor Corrector Algorithm for the Solution of y’’’= f (x, y, y’, y’’). American Journal of Computational Mathematics, 2(4), 341 – 344. Adesanya, A. O., Odekunle, M. R., Alkali, M. A., & Abubakar, A. B. (2013). Starting five steps Stormer Cowell method by Adams-Bashforth method for the solution of first order ordinary differential equations. African J. of Mathematics and Computer Science, 6(5).89-93. Anake, T. A., Awoyemi, D. O., & Adesanya, A. A. (2012). 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. A. (1989): An introduction to Numerical Analysis. New York: John Wiley & Sons, ISBN 978-0-471-50023-0. Awari, Y. S., & Abada, A. A. (2014). A Class of Seven Point Zero Stable Continuous Block Method for Solution of Second Order Ordinary Differential Equation. International Journal and Mathematics and Statistics Invention (IJMSI), 2(1), 47 – 54. Awari, Y.S. Chima, E. E., Kamoh, N. M., Oladele, F. L. (2014). A family of implicit uniformly accurate order block integrators for the solution of second order ordinary differential equations. International Journal of Mathematics and Statistics Invention. 2(1). 33 – 46. Awoyemi, D. O. (1998). A class of Continuous Stormer–Cowell Type Methods for Special Second Order Ordinary Differential Equations. Journal of Nigerian Mathematical Society, 5(1), 100-108. Awoyemi, D. O. (1999). A class of Continuous methods for general second order initial value problems in ordinary differential equations. International journal of computer mathematics, 72(1), 29-37. Awoyemi, D. O. (2001). A new sixth-order algorithm for general second order ordinary differential equations. International Journal of Computer Mathematics, 77(1), 117-124. Awoyemi, D. O. (2003). A P-stable linear multistep method for solving third order ordinary differential equations. Inter. J. Computer Math. 80(8), 85 – 991. Awoyemi, D. O. (2005). An algorithm collocation method for direct solution of special and general fourth order initial value problems of ordinary differential equation. J. of N. A. M. P. 6, 271 – 238 Awoyemi, D. O., Adebile, E. A., Adesanya, A. O., & 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. O., Adesanya, A. O. and Ogunjebi, S. N. (2009): Construction of self-starting Numerov method for the solution of Initial value problems of General second order ordinary differential equation. J. Num. Math. University of Ado Ekiti, 4(2), 267 – 278. Awoyemi, D. O. and Kayode (2002). An optimal order continuous multistep algorithm for initial value problems of special second order differential equations. J. Nig. Assoc. Math. Phys, 6: 285-292. Awoyemi, D. O., & Kayode, S. J. (2004). Maximal order multi-derivative collocation method for direct solution of fourth order initial value problems of ordinary differential equations. Awoyemi, D. O., & Kayode, S. J. (2005). An implicit collocation method for direct solution of second order ordinary differential equations. J. of the Nigerian Mathematical Society, 24. Awoyemi, D. O., & Kayode, S. J. (2005). A maximal order collocation method for direct solution of initial value problems of general second order ordinary differential equation. In Proceedings of the conference organized by the National mathematical centre, Abuja, Nigeria. Awoyemi, D. O., Kayode, S. J., & Adoghe, L. O. (2014). A four-point fully implicit method for the numerical integration of third-order ordinary differential equations. International Journal of Physical Sciences, 9(1), 7 – 12. Badmus, A. M., and Yahaya, Y. A. (2009). An accurate uniform order 6 block method for direct solution of general second order ordinary differential equations. Pacific Journal of Science and Technology, 10(2), 248-254. Bolarinwa B, Ademiluyi R. A, Awoyemi D. O and Ogundele J. O (2012): A Single Step Implicit Hybrid Numerical Method for the Numerical Integration of Initial Value Problems of Third Order Ordinary Differential Equations. Canadian Journal on Science and Engineering Mathematics Vol. 3 No 4. Brown, R. L. (1977). Some characteristics of implicit multistep multi-derivative integration formulas. SIAM Journal on Numerical Analysis, 14(6), 982-993. Brugnano, L., & Trigiante, D. (1998). Solving Differential Equations by Multistep Initial and Boundary Value Methods. CRC Press. Chakravarti, P. C., & Worland, P. B. (1971). A class of self-starting methods for the numerical solution of y ″= f (x, y). BIT Numerical Mathematics, 11(4), 368-383. Chawla, M. M., & Sharma, S. R. (1981). Families of fifth order Nystrom methods for y ″= f (x, y) and intervals of periodicity. Computing, 26(3), 247-256. Chu, M. T., & Hamilton, H. (1987). Parallel solution of ODE's by multiblock methods. SIAM journal on scientific and statistical computing, 8(3), 342-353. Dahlquist, G. (1959): Convergence and stability in the numerical integration of ordinary differential equations. Mathematica Scandinavia, 4, 33 – 53. Dormand, J. R., EL-Mikkawy, M. E. A. and Prince, P. J. (2003): Families of Runge Kutta Nystrom formula, IMA, Journal of Numerical analysis, 7, 235 – 250. Ehigie, J. O., Okunuga, S. A., & Sofoluwe, A. B. (2011). 3-point block methods for direct integration of general second-order ordinary differential equations. Advances in Numerical Analysis.10:1– 14 Fair weather, G., & Meade, D. (1989). A survey of spline collocation methods for the numerical solution of differential equations. Mathematics for large scale computing, Lecture notes in pure and applied mathematics, 120, 297-341. Fatunla, S. O. (1988): Numerical Methods for Initial Value Problems in Ordinary Differential Equations, Academic press inc. Harcourt Brace Jovanovich Publishers, New York. Fatunla, S. O. (1991). Block methods for second order ODEs. International journal of computer mathematics, 41(1-2), 55-63. Fatunla, S. O. (1992): Parallel Methods for Second Order Ordinary Differential Equation, Proceedings of the National Conference on Computational Mathematics, University of Benin, Nigeria, 87 – 99. Fatunla, S. O. (1994): A Class of Block Methods for Second Order IVPs. International Journal of Computer Mathematics. 55:119-133. Fox, L. and Parker, I. B. (1968): Chebyshev Polynomial in Numerical Analysis Oxford University press, New York. Frazer, R. A., Duncan, W. J., & Collar, A. R. (1988). Elementary matrices and some applications to dynamics and differential equations. Cambridge University Press. Gear, C. (1971). Simultaneous numerical solution of differential-algebraic equations. Circuit Theory, IEEE Transactions on, 18(1), 89-95. Gerald, C. T. and Wheatley, P. O (1994). Applied numerical analysis. Addison-Wesley Publishing Company, Inc. Goult, R. A. Hoskins, R. F. Milier and Pratt, M. J. (1973): Applicable Mathematics for Engineers and Scientists. Macmillan press Ltd. London. Hall, G., & Suleiman, M. B. (1981). Stability of Adams-type formulae for second order ordinary differential equations. IMA Journal of Numerical Analysis, 1(4), 427-438. Hairer, E., & Wanner, G. (1975). A theory for Nyström methods. Numerische Mathematik, 25(4), 383-400. Hairer, E; Norsett, S. P., Wanner, G. (1993): Solving ordinary differential equations; nonstiff problems. Berlin Springer Verlag. ISBN 978-3-540-56670. Henrici, P. (1962). Discrete variable methods in ordinary differential equations. Ibrahim, Z. B., Othman, K. I., & Suleiman, M. (2007). Implicit r-point block backward differentiation formula for solving first-order stiff ODEs. Applied Mathematics and Computation, 186(1), 558-565. Ibrahim, Z. B., Suleiman, M., & Othman, K. I. (2008). Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential e quations. European Journal of Scientific Research, 21(3), 508-520. Ibrahim, Z. B., Suleiman, M., Nasir, N. A. A. M., & Othman, K. I. (2011). Convergence of the 2-point block backward differentiation formulas. Applied Mathematical Sciences, 5(70), 3473-3480. Ismail, F., Ken, Y. L., & Othman, M. (2009). Explicit and implicit 3-point block methods for solving special second order ordinary differential equations directly. Int. Journal of Math. Analysis, 3, 239-254. Jain, R. K., Kambo, N. S., & Goel, R. (1984). A sixth-order P-stable symmetric multistep method for periodic initial-value problems of second-order differential equations. IMA journal of numerical analysis, 4(1), 117-125. James, A. A., Adesanya, A. O., Sunday, J., Yakubu, D. G. (2013). Half- step continuous block for solution of modeled problems of ordinary differential equations. American Journal of Computational Mathematics 3, 261-269. Jator, S. N. (2001). Improvements in Adams-Moulton Methods for the first order initial value problems. Journal of the Tennessee Academy of Science, 76(2), 57- 60. Jator, S. N. (2007). A sixth order linear multistep method for the direct solution of y"= f (x, y, y'). International Journal of Pure and Applied Mathematics, 40(4), 457-472. Jator, S. N and Li, J. (2009). A self-stationary linear multistep method for a direct solution of the general second order initial value problems. Inter. Journal of Computer Math. 86(5), 817-836. Jeltsh, R (1976). Note on a – stability of multi-step multi derivative methods. BIT, 16, 7 – 78. Jaun, A. (2002). Numerical methods for partial differential equations. Kayode, S. J. (2011). A class of one-point zero-stable continuous hybrid methods for direct solution of second-order differential equations. Afr. J. Math. Compt. Sci. Res, 4(3), 93-99. Kayode, S. J. (2004). A class of maximal order linear multistep collocation methods for direct solution of ordinary differential equations. Unpublished doctoral dissertation, Federal University of Technology, Akure, Nigeria. Kayode S. J. (2005). An improved numerov method for direct solution of general second order initial value problems of ordinary differential equations. Proceedings of the Seminar Organized by the National Mathematical Centre, Abuja, Nigeria. Kayode, S. J. (2008a). An efficient zero-stable numerical method for fourth-order differential equations. International Journal of Mathematics and Mathematical Sciences.1 – 10. Kayode, S. J. (2008b). An order six zero-stable method for direct solution of fourth order ordinary differential equations. American Journal of Applied Sciences, 5(11), 1461 – 1466. Kayode, S. J. and Adeyeye, O (2011): A 3-Step Hybrid Methods for Direct Solution of Second Order Initial Value Problems. Aust. Basic Applied. Sci. 5(12): 2121-2126. Kayode, S. J., & Awoyemi, D. O. (2008). A 5-step maximal order method for direct solution of second order ordinary differential equations. Journal of the Nigerian Association of Mathematical Physics, 9(1), 279-284. Kayode, S. J. and Adeyeye, O (2013): A 2-Step Two-Point Hybrid Methods for Direct Solution of Second Order Initial Value Problems. Afric. J. Math. Comp. Sci. 6(10):191-196. Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations. Introductory Mathematics for Scientists and Engineers. Wiley. Lambert, J. D., & Watson, I. A. (1976). Symmetric multistep methods for periodic initial value problems. IMA Journal of Applied Mathematics, 18(2), 189-202. Lambert, J. B. (1988): Numerical methods for initial value problems in ordinary differential equation. Academic Press Inc, New York. Lambert, J. D., & Lambert, J. D. (1991). Numerical methods for ordinary differential systems: the initial value problem (pp. 104-107). Chichester: Wiley. Lanczos, C. (1965). Applied Analysis, Prentice Hall, New Jersey. Majid, Z. A. (2004). Parallel block methods for solving ordinary differential equations. PhD thesis, Universiti Putra Malaysia. (unpublished). Majid, Z. A., Bin Suleiman, M., & Omar, Z. (2006). 3-point implicit block method for solving ordinary differential equations. Bulletin of the Malaysian Mathematical Sciences Society. Second Series, 29(1), 23-31. Majid, Z.A., Azimi, N.A. & Suleiman, M. (2009). Solving special second order ordinary differential equations. European Journal of Scientific Research, 31(1), 29-36. Milne, W. E., & Milne, W. E. (1953). Numerical solution of differential equations (Vol. 19, No. 3). New York: Wiley. Mohammed, U. (2010): A six step block method for solution of fourth order ordinary differential equations. The Pacific Journal of Science and Technology. 11(1): 259 – 265. Mohammed, U., Tech, M., Jiya, M., Mohammed, A. A., & Tech, M. (2010). A Class of Six Step Block Method for Solution of General Second Order Ordinary Differential Equations. Pacific Journal of Science and Technology, 11(2), 273 – 277. Mohammed, U. (2011). A class of implicit five – step block method for general second order ordinary differential equations. Journal of Nigeria Mathematical Society (JNMS), 30: 25 – 39. Mohammed, U., & Adeniyi, R. B. (2014). Derivation of Five Step Block Hybrid Backward Differential Formulas (HBDF) through the Continuous Multi-Step Collocation for Solving Second Order Differential Equation. Pacific Journal of Science and Technology, 15(2), 89 – 95. Odekunle, M. R., Egwurube, M. O., Adesanya, A. O., & Udo, M. O. (2014). Five Steps Block Predictor-Block Corrector Method for the Solution of y''= f (x, y, y'). Applied Mathematics, 5(08), 1252 – 1256. Olabode, B. T. (2007). Some Linear Multistep Methods for Special and General third Order Initial Value Problems of Ordinary Differential Equation, PhD Thesis, Federal University of Technology, Akure. (Unpublished). Olabode, B. T. (2009a). An accurate scheme by block method for the third order ordinary differential equation. Pacific journal of science and technology, 10(1):136 – 142. Olabode, B. T. (2009b). A six-step scheme for the solution of fourth order ordinary differential equations. The Pacific Journal of Science and Technology, 10(1), 143-148. Olabode, B. T., & Yusuph, Y. (2009). A new block method for special third order ordinary differential equations. Journal of Mathematics and statistics, 5(3), 167 – 170. Olabode, B. T. (2013). Block multistep method for the direct solution of third order of ordinary differential equations. Futa Journal of Research In Sciences, 9(2), 194-200. Omar, Z. (1999). Parallel block methods for solving higher order ordinary differential equations directly. Thesis Submitted in Fulfilment of the Requirement for the Degree of Doctor of Philosophy in the Faculty of Science and Environmental studies Universiti Putra Malaysia (unpublished) Omar, Z. B. and Suleiman, M. B. (1999). Solving second order odes directly using parallel 2 point explicit block method. Prosiding Kolokium Kebangsaan Pengintegrasian Teknologi Dalam Sains Matematik, Universiti SainsMalaysia, 390-395. Omar, Z and Suleiman, M. (2003). Parallel r-point implicit block method for solving higher order ordinary differential equation directly, Journal of ICT. 3(1), 53 – 66. Omar, Z. (2004). Developing parallel 3-point implicit block method for solving second order ordinary differential equations directly. IJMS 11 (1), 91-103. Omar, Z., & Suleiman, M. (2004). Parallel r-point implicit block method for solving higher order ordinary differential equations directly. Journal of ICT, 3(1), 53-66. Omar, Z. B. and Suleiman, M. B. (2005). Solving higher order odes directly using parallel 2- point explicit block method. Matematika, 21(1), 15-23. Omolehin, J. O., Ibiejugba, M. A., Alabi, M. O., & Evans, D. J. (2003). A new class of adams-bashforth schemes for ODEs. International journal of computer mathematics, 80(5), 629-638. Onumanyi, P. Sirisena, U. W. and Dauda, Y. (2001). Toward uniformly accurate continuous finite difference approximation of ordinary differential equation. Bagalel J. of Pure and Applied Science 1, 5 – 8. Ross, S. L. (1989). Introduction to Ordinary Differential Equations. Fourth Edition. Singapore: John Wiley & Sons, Inc. Rosser, J. B. (1967). A Runge-Kutta for all seasons. Siam Review, 9(3), 417-452. Sagir, A. M. (2014). Numerical treatment of block method for the solution of ordinary differential equations. International Journal of Mathematical, Computational, Natural and Physical Engineering, 8(2), 259 – 263. Sarafyan, D. (1990). New algorithms for the continuous approximate solutions of ordinary differential equations and the upgrading of the order of the processes. Computers & Mathematics with Applications, 20(1), 77-100. Sharp, P. W., & Fine, J. M. (1992). Some Nyström pairs for the general second-order initialvalue problem. Journal of computational and applied mathematics, 42(3), 279-291. Spiegel R Murray (1971). Theory and Problems of Advance Mathematics for Engineers and scientist, McGraw Hill, Inc. New York. Taiwo, O. A., & Onumanyi, P. (1991). A collocation approximation of singularly perturbed second order ordinary differential equation. International Journal of Computer Mathematics, 39(3-4), 205-211. Twizell, G. H. and Khaliq, A. Q. M. (1984). Multiderivative method for periodic IVPs, SIAM Journal of Numerical Analysis. 21, 111 – 121. Udoh, M. O., Ademiluyi, R. A., Adesanya, A.O. and Ekpenyang, B. O. (2007). An implicit zero stable linear multistep method for direct solution of general third initial value problem of ordinary differential equation. Journal of Science and Technology, 6(2), 153 – 158. 3Voss, D., & Abbas, S. (1997). Block predictor-corrector schemes for the parallel solution of ODEs. Computers & Mathematics with Applications, 33(6), 65-72. Wend, D. V. (1969). Existence and uniqueness of solutions of ordinary differential equations. Proceedings of the American Mathematical Society, 27-33. Yahaya, Y. A. (2007). A note on the construction of Numerov method through a quadratic continuous polynomial for the general second order ordinary differential equation. Journal of Nigerian Association of mathematical physics, II, 253-260. Yahaya, Y. A., & Sagir, A. M. (2013). Accurate and Efficient Implementation of Block Hybrid Method for Solving Special Second Order Ordinary Differential Equations. International Journal of Mathematics & Computation™, 21(4), 15-22. Yap, L. K., Ismail, F., & Senu, N. (2014). An accurate block hybrid collocation method for third order ordinary differential equations. Journal of Applied Mathematics, 1- 9 .

Similar Items