International Journal of Applied Mathematics and Theoretical Physics

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Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points

Received: May 30, 2021    Accepted: Oct. 15, 2021    Published: Feb. 05, 2022
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Abstract

This paper presents a direct approach for numerical solution of special second order delay differential equations (DDEs) directly without reduction to systems of low orders. The methods were generated using collocation approach via a combination of power series and exponential function. The approximate basis functions are interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve second order Delay Differential Equations (DDEs) directly without the rigor of developing separate predictors. The required methods were obtained for step lengths of five with generalized number of hybrid points (3k). The basic properties of the methods were examined, the methods were found to have high order of accuracy of 21, low error constant, gives large interval of absolute stability, zero stable, consistence and convergent. The developed methods were applied to solve some special second order Delay Differential Equations. The methods also solve an engineering problem namely Matheiu’s equation in order to test for the efficiency and accuracy of the new methods. The results obtained were compared with existing methods in the literature. The results obtained showed better performance than some existing methods. The stability domain of the method is showed in figure 1 whereas the efficiency curve of the application problem for linear and nonlinear is presented in figure 2.

DOI 10.11648/j.ijamtp.20220801.11
Published in International Journal of Applied Mathematics and Theoretical Physics ( Volume 8, Issue 1, March 2022 )
Page(s) 1-23
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Block Method, Special Second Order Delay Differential Equations, Five-step, Off-grid Points, Power Series, Exponential Function

References
[1] Awoyemi, D. O., and Kayode, S. J. (2005): A maximal order collocation Method for initial value problems of General Second order ordinary differential equation, Proceedings of the Conference organized by the National Mathematical Centre, Abuja, Nigeria.
[2] Evans D. J., and Raslan, K. R., (2005): The Adomian decomposition method for solving delay differential equations. International journal of computer mathematics, vol. 82, no 1, pp 49–54.
[3] Fatunla, S. O (1991): Block method for second order ordinary differential equation. International Journal of computer mathematics, 41 (1 & 2), 55-63.
[4] Kayode S. J. and Obarhua F. O. (2013), Continuous y-function Hybrid Methods for Direct Solutions of Differential equations, International Journal of Differential Equations and Application 12 (1), 37–48.
[5] Kuang Y. (1993): Delay differential equations with applications in population dynamics, vol. 191 of Mathematics in Science and Engineering, Academics Press, Boston, Mass, USA.
[6] Familua A. B., Areo E. A., Olabode B. T., Owolabi M. K. (2019): A class of numerical integrators of order 13 for solving special second order delay differential equations, International Journal of Physics and Mathematics, 1 (2), pp. 01-17.
[7] Zanariah A. M and Hoo Y. S. (2013): Direct method for solving second order delay difference equation, proceedings of the 20th national symposium on mathematical sciences, AIP Conference Preceding. 676-680.
[8] Mechee M., Ismail F., Senu N., and Siri Z., (2013): Directly solving special second order delay differential equations using Runge-kutta-Nystrom method. Journal of Mathematical Problems in Engineering, Article ID 830317, page 1-9.
[9] Morisson T. M. and Rand R. H, (2007): Resonance in the delay nonlinear Mathieu equation, “Nonlinear Dynamics: An international Journal of Nonlinear Dynamics and chaos in Engineering Systems, vol. 50, no 1-2, pp. 341-352.
[10] San, H. C., Majid Z. A., and Othman M. (2011): Solving delay differential equations using coupled block method. In preceding of the 4thk international conference on Modelling, Simulation and Applied Optimization (ICMSAO 11).
[11] Shampine, L. F &Thompson S. T. (2001): Numerical Solution of Delay Differential Equations, Application of Numerical Mathematics, 37, pp 441-458.
[12] Skip Thompson (2007): Delay differential equations- (Scholarpedia, 2 (3): 2367): doi: 10.4249/scholarpediahttp://www.scholarpedia.org/org/article/Delay-differential_equations.
[13] Suleiman M. B., and Ishak (2010): Numerical solution and stability of multistep method for solving delay differential equations: Japan Journal of Industrial and applied mathematics, vol. 27, pp. 395–410.
[14] Smith H. (2011): An Introduction to delay differential equations with applications to the life sciences, vol. 57, Texts in Applied Mathematics, Springer, New York.
[15] Mohammed S. Mechee, F. Ismail, Z. Siri and N. Senu (2014). A third order direct integrators of Rungekutta type for special third order ordinary and delay differential equations. Asian Journal of Applied Sciences 7 (3): 102–116, ISSN 1996-334/DOI: 10.3923/ajaps.
[16] Baker C. T. H, Paul, C. A. H., and Wille (1994): Issues in the Numerical solution of Evolutionary Delay Differential Equation, Numerical Analysis Report No. 248, University of Manchester, United Kingdom.
[17] Ibijola E. A, Skywame Y, Kumleng G. (2011): Formulation of the continuous multistep collocation. Americal Journal of Scientific and Industrial Research, 2: 161-173.
[18] Ogunfiditimi F. O. (2015), Numerical solution of Delay Differential Equations using Adomain Decomposition method (ADM) The International Journal Of Engineering And Science (IJES). 4 (5): 18-23.
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  • APA Style

    Familua A. B. (2022). Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points. International Journal of Applied Mathematics and Theoretical Physics, 8(1), 1-23. https://doi.org/10.11648/j.ijamtp.20220801.11

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    ACS Style

    Familua A. B. Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points. Int. J. Appl. Math. Theor. Phys. 2022, 8(1), 1-23. doi: 10.11648/j.ijamtp.20220801.11

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    AMA Style

    Familua A. B. Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points. Int J Appl Math Theor Phys. 2022;8(1):1-23. doi: 10.11648/j.ijamtp.20220801.11

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  • @article{10.11648/j.ijamtp.20220801.11,
      author = {Familua A. B.},
      title = {Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points},
      journal = {International Journal of Applied Mathematics and Theoretical Physics},
      volume = {8},
      number = {1},
      pages = {1-23},
      doi = {10.11648/j.ijamtp.20220801.11},
      url = {https://doi.org/10.11648/j.ijamtp.20220801.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ijamtp.20220801.11},
      abstract = {This paper presents a direct approach for numerical solution of special second order delay differential equations (DDEs) directly without reduction to systems of low orders. The methods were generated using collocation approach via a combination of power series and exponential function. The approximate basis functions are interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve second order Delay Differential Equations (DDEs) directly without the rigor of developing separate predictors. The required methods were obtained for step lengths of five with generalized number of hybrid points (3k). The basic properties of the methods were examined, the methods were found to have high order of accuracy of 21, low error constant, gives large interval of absolute stability, zero stable, consistence and convergent. The developed methods were applied to solve some special second order Delay Differential Equations. The methods also solve an engineering problem namely Matheiu’s equation in order to test for the efficiency and accuracy of the new methods. The results obtained were compared with existing methods in the literature. The results obtained showed better performance than some existing methods. The stability domain of the method is showed in figure 1 whereas the efficiency curve of the application problem for linear and nonlinear is presented in figure 2.},
     year = {2022}
    }
    

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    AU  - Familua A. B.
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    T2  - International Journal of Applied Mathematics and Theoretical Physics
    JF  - International Journal of Applied Mathematics and Theoretical Physics
    JO  - International Journal of Applied Mathematics and Theoretical Physics
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    AB  - This paper presents a direct approach for numerical solution of special second order delay differential equations (DDEs) directly without reduction to systems of low orders. The methods were generated using collocation approach via a combination of power series and exponential function. The approximate basis functions are interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve second order Delay Differential Equations (DDEs) directly without the rigor of developing separate predictors. The required methods were obtained for step lengths of five with generalized number of hybrid points (3k). The basic properties of the methods were examined, the methods were found to have high order of accuracy of 21, low error constant, gives large interval of absolute stability, zero stable, consistence and convergent. The developed methods were applied to solve some special second order Delay Differential Equations. The methods also solve an engineering problem namely Matheiu’s equation in order to test for the efficiency and accuracy of the new methods. The results obtained were compared with existing methods in the literature. The results obtained showed better performance than some existing methods. The stability domain of the method is showed in figure 1 whereas the efficiency curve of the application problem for linear and nonlinear is presented in figure 2.
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Author Information
  • Department of Mathematics and Statistics, First Technical University, Ibadan, Nigeria

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