Research Article

Year 2021,
Volume: 18 Issue: 2, 87 - 100, 01.11.2021
### Abstract

### References

- [1] D. O. Awoyemi, “On some continuous linear multistep methods for initial value problems,” Unpublished doctoral dissertation, University of Ilorin, Ilorin, Nigeria, 1992.
- [2] A. O. Adesanya, M.R. Odekunle, and A. O. Adeyeye, “Continuous block hybrid-predictor-corrector method for the solution of y''= f (x, y, y'),” International Journal of Mathematics and Soft Computing, vol.2, no. 2, pp. 35-42, 2012.
- [3] S. O. Fatunla, “Block Methods for Second Order IVPs,” International Journal of Computer Mathematics, vol. 42, no. 9, pp. 55-63, 1991.
- [4] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley, New York, 1991.
- [5] B. T. Olabode, “An accurate scheme by block method for third order ordinary differential equations,” Pacific Journal of Science and Technology, vol. 10, no. 1, pp. 136-142 , 2009.
- [6] A. Jajarmi, B. Ghanbari, and D. Baleanu, “A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, no. 9, pp. 54-67, 2019.
- [7] Z. A. Majid, Azmi, M. Suleiman, and Z. B. Ibrahim,‘’Solving directly general third order ordinary differential equations using two-point four step block method,’’ Sains Malaysiana, vol. 41, no. 5, pp. 623—632, 2012.
- [8] T. Aliya, A. A. Shaikh, and S. Qureshi, “Development of a nonlinear hybrid numerical method,” Advances in Differential Equations and Control Processes, vol. 19, no.3, pp. 275-285, 2018.
- [9] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, “Fractional calculus: models and numerical methods,” World Scientific Journal, vol. 3, pp. 98-105, 2012.
- [10] S. Qureshi, and F. S. Emmanuel, “Convergence of a numerical technique via interpolating function to approximate physical dynamical systems,” Journal of Advanced Physics, vol. 7, no .3, pp. 446-450, 2018.
- [11] S. Qureshi, and H. Ramos, “L-stable explicit nonlinear method with constant and variable step-size formulation for solving initial value problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 19, no. (7- 8), pp. 741-751, 2018.
- [12] S. Qureshi, and A. Yusuf, “A new third order convergent numerical solver for continuous dynamical systems,” Journal of King Saud University-Science, vol. 32, no. 2, pp. 1409-1416, 2020.
- [13] H. Ramos, R. Abduganiyu, R. Olowe and S. Jatto, “A family of functionally-fitted third derivative block Falkner methods for solving second-order initial-value problems with oscillating solutions,” Mathematics, vol.9, no.2, pp. 713, 2021.
- [14] H. Ramos, S. N. Jator and M.I. Modebei, “Efficient k-step linear block methods to solve second order initial Value problems directly,’’ Mathematics, vol.8, no.10, pp.1752, 2020.
- [15] M. A. Rufai and H. Ramos, “One-step hybrid block method containing third derivatives and improving strategies for solving Bratu's and Troesch's problems,” Numerical Mathematics: Theory, Methods & Applications, vol. 13, no. 4, 2020.
- [16] A. O. Adesanya, T. A. Anake and G. J. Oghonyon, “Continuous implicit method for the solution of general second order ordinary differential equations,” Journal of the Nigeria Association of Mathematical Physics, vol. 15, pp. 71-78, 2009.
- [17] S. O. Fatunla, “Block Methods for Second Order Initial Value Problems,” International Journal of Computer Mathematics, vol. 41, pp.55-63, 1994.
- [18]J. D. Lambert, Computational methods in Ordinary Differential System, John Wiley, New York, 1973.
- [19] E. O. Adeyefa , “Orthogonal based Hybrid Block Method for solving general second order initial value problems,” Italian Journal of Pure and Applied Mathematics, vol. 37, pp. 659-672, 2017.
- [20] L. Brugnano, and D. Trigiante, “Solving Differential Problems by Multistep Initial and Boundary Value Methods,” Amsterdam: Gordon and Breach Science Publishers, 1998.
- [21] R. A. Bun, and Y. D. Vasil’Yev, “A Numerical Method for Solving Differential Equations of Any Order,” Computational Mathematics and Mathematical Physics, vol. 32, no.3, pp. 317-330, 1992.
- [22] P. Henrichi, “Discrete variable methods in ODE,” John Wiley and Sons, New York, 1962.
- [23] P. Onumanyi, D. O. Awoyemi, S. N. Jator, and U. W. Sirisena, “New linear multistep methods with continuous coefficients for first order IVPs,” Journal of the Nigerian Mathematical Society, vol. 13, pp. 37-51, 1994.
- [24] A. O. Adesanya, D. M. Udoh, and A. M. Ajileye, “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, vol. 86, no.2, pp. 365-375, 2013.
- [25] M. A. Rufai and H. Ramos, “One-step hybrid block method containing third derivatives and improving strategies for solving Bratu's and Troesch's problems,” Numerical Mathematics: Theory, Methods & Applications, vol.13, no.4, 2020

Year 2021,
Volume: 18 Issue: 2, 87 - 100, 01.11.2021
### Abstract

### Thanks

### References

The formulation of hybrids block method as integrator of third-order Initial Value Problems in Ordinary Differential Equations is our focus in this paper. Chebyshev polynomials were used as trial function to develop a hybrid One-step Method (HBOSM3) adopting collocation and interpolation technique. The basic properties of HBOSM3 were integrated and findings revealed that the method was accurate and convergent. One of desirable features of these methods is the production of exact solutions at the grid points.

The authors are grateful to the handling Editor and, also appreciate the anonymous reviewers for their views and suggestions that greatly improve the manuscript.

- [1] D. O. Awoyemi, “On some continuous linear multistep methods for initial value problems,” Unpublished doctoral dissertation, University of Ilorin, Ilorin, Nigeria, 1992.
- [2] A. O. Adesanya, M.R. Odekunle, and A. O. Adeyeye, “Continuous block hybrid-predictor-corrector method for the solution of y''= f (x, y, y'),” International Journal of Mathematics and Soft Computing, vol.2, no. 2, pp. 35-42, 2012.
- [3] S. O. Fatunla, “Block Methods for Second Order IVPs,” International Journal of Computer Mathematics, vol. 42, no. 9, pp. 55-63, 1991.
- [4] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley, New York, 1991.
- [5] B. T. Olabode, “An accurate scheme by block method for third order ordinary differential equations,” Pacific Journal of Science and Technology, vol. 10, no. 1, pp. 136-142 , 2009.
- [6] A. Jajarmi, B. Ghanbari, and D. Baleanu, “A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, no. 9, pp. 54-67, 2019.
- [7] Z. A. Majid, Azmi, M. Suleiman, and Z. B. Ibrahim,‘’Solving directly general third order ordinary differential equations using two-point four step block method,’’ Sains Malaysiana, vol. 41, no. 5, pp. 623—632, 2012.
- [8] T. Aliya, A. A. Shaikh, and S. Qureshi, “Development of a nonlinear hybrid numerical method,” Advances in Differential Equations and Control Processes, vol. 19, no.3, pp. 275-285, 2018.
- [9] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, “Fractional calculus: models and numerical methods,” World Scientific Journal, vol. 3, pp. 98-105, 2012.
- [10] S. Qureshi, and F. S. Emmanuel, “Convergence of a numerical technique via interpolating function to approximate physical dynamical systems,” Journal of Advanced Physics, vol. 7, no .3, pp. 446-450, 2018.
- [11] S. Qureshi, and H. Ramos, “L-stable explicit nonlinear method with constant and variable step-size formulation for solving initial value problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 19, no. (7- 8), pp. 741-751, 2018.
- [12] S. Qureshi, and A. Yusuf, “A new third order convergent numerical solver for continuous dynamical systems,” Journal of King Saud University-Science, vol. 32, no. 2, pp. 1409-1416, 2020.
- [13] H. Ramos, R. Abduganiyu, R. Olowe and S. Jatto, “A family of functionally-fitted third derivative block Falkner methods for solving second-order initial-value problems with oscillating solutions,” Mathematics, vol.9, no.2, pp. 713, 2021.
- [14] H. Ramos, S. N. Jator and M.I. Modebei, “Efficient k-step linear block methods to solve second order initial Value problems directly,’’ Mathematics, vol.8, no.10, pp.1752, 2020.
- [15] M. A. Rufai and H. Ramos, “One-step hybrid block method containing third derivatives and improving strategies for solving Bratu's and Troesch's problems,” Numerical Mathematics: Theory, Methods & Applications, vol. 13, no. 4, 2020.
- [16] A. O. Adesanya, T. A. Anake and G. J. Oghonyon, “Continuous implicit method for the solution of general second order ordinary differential equations,” Journal of the Nigeria Association of Mathematical Physics, vol. 15, pp. 71-78, 2009.
- [17] S. O. Fatunla, “Block Methods for Second Order Initial Value Problems,” International Journal of Computer Mathematics, vol. 41, pp.55-63, 1994.
- [18]J. D. Lambert, Computational methods in Ordinary Differential System, John Wiley, New York, 1973.
- [19] E. O. Adeyefa , “Orthogonal based Hybrid Block Method for solving general second order initial value problems,” Italian Journal of Pure and Applied Mathematics, vol. 37, pp. 659-672, 2017.
- [20] L. Brugnano, and D. Trigiante, “Solving Differential Problems by Multistep Initial and Boundary Value Methods,” Amsterdam: Gordon and Breach Science Publishers, 1998.
- [21] R. A. Bun, and Y. D. Vasil’Yev, “A Numerical Method for Solving Differential Equations of Any Order,” Computational Mathematics and Mathematical Physics, vol. 32, no.3, pp. 317-330, 1992.
- [22] P. Henrichi, “Discrete variable methods in ODE,” John Wiley and Sons, New York, 1962.
- [23] P. Onumanyi, D. O. Awoyemi, S. N. Jator, and U. W. Sirisena, “New linear multistep methods with continuous coefficients for first order IVPs,” Journal of the Nigerian Mathematical Society, vol. 13, pp. 37-51, 1994.
- [24] A. O. Adesanya, D. M. Udoh, and A. M. Ajileye, “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, vol. 86, no.2, pp. 365-375, 2013.
- [25] M. A. Rufai and H. Ramos, “One-step hybrid block method containing third derivatives and improving strategies for solving Bratu's and Troesch's problems,” Numerical Mathematics: Theory, Methods & Applications, vol.13, no.4, 2020

There are 25 citations in total.

Primary Language | English |
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Subjects | Engineering |

Journal Section | Articles |

Authors | |

Publication Date | November 1, 2021 |

Published in Issue | Year 2021 Volume: 18 Issue: 2 |