Numerical Solution of Second Order Fuzzy Ordinary Differential Equations using Two-Step Block Method with Third and Fourth Derivatives

https://doi.org/10.46481/jnsps.2023.1087

Authors

  • Kashif Hussain School of Quantitative Sciences, Universiti Utara Malaysia, Kedah, Malaysia
  • Oluwaseun Adeyeye School of Quantitative Sciences, Universiti Utara Malaysia, Kedah, Malaysia
  • Nazihah Ahmad School of Quantitative Sciences, Universiti Utara Malaysia, Kedah, Malaysia

Keywords:

Fuzzy initial value problem, Fuzzy boundary value problem, Second order, Two-step, Block method, Linear, Nonlinear

Abstract

Fuzzy differential equation models are suitable where uncertainty exists for real-world phenomena. Numerical techniques are used to provide an approximate solution to these models in the absence of an exact solution. However, existing studies that have developed numerical techniques for solving second-order fuzzy ordinary differential equations (FODEs) possess an absolute error accuracy that could be improved. Therefore, this article developed a more accurate higher derivative self-starting block scheme for the numerical solution of second-order FODEs with fuzzy initial and boundary conditions imposed. Linear block approach using Taylor series expansion is adopted for the derivation of the proposed method and the basic properties are established using the definitions of stability and consistency for block methods. According to the numerical results, when compared to the exact solution in terms of absolute error, the new method proposed in this article outperformed existing numerical methods. It is thus concluded that the proposed method is effective for solving second-order FODEs directly.

Dimensions

S. S. Mohammed, “On fuzzy soft set-valued maps with application”, J. Nig. Soc. Phy. Sci. 2 (2020) 26. https://doi.org/10.46481/jnsps.2020.48 DOI: https://doi.org/10.46481/jnsps.2020.48

L. A. Zadeh & S. Chang, “On fuzzy mapping and control”, IEEE Trans. Syst. Man Cybern. 2 (1972) 30. https://doi.org/10.1142/9789814261302 0012 DOI: https://doi.org/10.1109/TSMC.1972.5408553

M. L. Puri & D. A. Ralescu, “Differentials of fuzzy functions”, J. Math. Anal. Appl. 91 (1983) 552. https://doi.org/10.1016/0022-247X(83)90169-5 DOI: https://doi.org/10.1016/0022-247X(83)90169-5

S. Seikkala, “On the fuzzy initial value problem”, Fuzzy Sets Syst. 24 (1987) 319. https://doi.org/10.1016/0165-0114(87)90030-3 DOI: https://doi.org/10.1016/0165-0114(87)90030-3

B. Bede & S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to

fuzzy differential equations”, Fuzzy Sets Syst. 151 (2005) 581. https://doi.org/10.1016/j.fss.2004.08.001 DOI: https://doi.org/10.1016/j.fss.2004.08.001

A. F. Jameel, M. Ghoreishi, & A. I. M. Ismail, “Approximate solution of high order fuzzy initial value problems”, J. Uncertain Syst. 8 (2014) 149.

A. Jameel, A. Shather, N. Anakira, A. Alomari, & A. Saaban, “Comparison for the approximate solution of the second-order fuzzy nonlinear differential equation with fuzzy initial conditions”, Math. Stat. 8 (2020) 527. https://doi.org/10.13189/ms.2020.080505 DOI: https://doi.org/10.13189/ms.2020.080505

O¨. Ak?n, T. Khaniyev, O¨ . Oruc¸, & I. Tu¨rks¸en, “An algorithm for the solution of second order fuzzy initial value problems”, Expert Syst. Appl. 40 (2013) 953. https://doi.org/10.1016/j.eswa.2012.05.052 DOI: https://doi.org/10.1016/j.eswa.2012.05.052

E. ElJaoui, S. Melliani, & L. S. Chadli, “Solving second-order fuzzy differential equations by the fuzzy Laplace transform method”, Adv. Differ. Equ. 2015 (2015) 1. https://doi.org/10.1186/s13662-015-0414-x DOI: https://doi.org/10.1186/s13662-015-0414-x

N. Salamat, M. Mustahsan, & M. M. Saad Missen, “Switching point solution of second-order fuzzy differential equations using differential transformation method”, Math. 7 (2019) 231. https://doi.org/10.3390/math7030231 DOI: https://doi.org/10.3390/math7030231

N. J. B. Pinto, E. Esmi, V. F. Wasques, & L. C. Barros, “Least square method with quasi linearly interactive fuzzy data: fitting an HIV dataset”, Proceedings of the International Fuzzy Systems Association World Congress, Springer (2019) 177. https://doi.org/10.1007/978-3-030-21920-8 17 DOI: https://doi.org/10.1007/978-3-030-21920-8_17

A. G. Fatullayev, E. Can, & C. K¨oro?glu, “Numerical solution of a boundary value problem for a second order fuzzy differential equation”, TWMS J. Pure Appl. Math. 4 (2013) 169.

T. Jayakumar, K. Kanagarajan, & S. Indrakumar, “Numerical solution of Nth-order fuzzy differential equation by Runge-Kutta method of order five”, Int. J. Math. Anal. 6 (2012) 2885.

A. Jameel, N. Anakira, A. Alomari, I. Hashim, & M. Shakhatreh, “Numerical solution of n-th order fuzzy initial value problems by six stages Range Kutta method of order five”, Int. J. Electr. Comput. Eng. 10 (2019) 6497. http://dx.doi.org/10.22436/jnsa.009.02.26 DOI: https://doi.org/10.22436/jnsa.009.02.26

T. K. Fook, & Z. B. Ibrahim, “Block backward differentiation formulas for solving second order fuzzy differential equations”, MATEMATIKA: Malaysian J. Ind. Appl. Math. 33 (2017) 215. https://doi.org/10.11113/matematika.v33.n2.868 DOI: https://doi.org/10.11113/matematika.v33.n2.868

K. J. Audu, A. Ma’Ali, U. Mohammed, & A. Yusuf, “Extended block hybrid backward differentiation formula for second order fuzzy differential equations using legendre polynomial as basis function”, J. Sci. Tech. Math. Educ. 16 (2020) 100.

S. Al-Refai, M. I. Syam, & M. Al-Refai, “Optimization of one-step block method for solving second-order fuzzy initial value problems”, Complex. 2021 (2021) 1. https://doi.org/10.1155/2021/6650413 DOI: https://doi.org/10.1155/2021/6650413

B. Bede, “Fuzzy sets”, in Mathematics of Fuzzy Sets and Fuzzy Logic, Springer (2013) 1. https://doi.org/10.1007/978-3-642-35221-8 1 DOI: https://doi.org/10.1007/978-3-642-35221-8_1

S. S. Devi & K. Ganesan, “An approximate solution by fuzzy Taylor’s method”, Int. J. Pure Appl. Math. 113 (2017) 236.

J. D. Lambert, Computational methods in ordinary differential equations, New York: John Wiley and Sons Inc, (1973). https://doi.org/10.1002/zamm.19740540726 DOI: https://doi.org/10.1002/zamm.19740540726

J. C. Butcher, Numerical methods for ordinary differential equations, New York: John Wiley and Sons Ltd (2008). DOI: https://doi.org/10.1002/9780470753767

https://doi.org/10.1002/9781119121534 DOI: https://doi.org/10.1002/9781119121534

E. S¨uli, Numerical solution of ordinary differential equations, Mathematical Institute, University of Oxford (2010).

J. O. Ehigie, S. A. Okunuga & A. B. Sofoluwe, “3-point block methods for direct integration of general second-order ordinary differential equations”, Advances in Numerical Analysis (2011). DOI: https://doi.org/10.1155/2011/513148

M. O. Ogunniran, “A class of block multi-derivative numerical techniques for singular advection equations”, J. Nig. Soc. Phy. Sci. 1 (2019) 62. https://doi.org/10.46481/jnsps.2019.12 DOI: https://doi.org/10.46481/jnsps.2019.12

R. Abdulganiy, O. Akinfenwa, O. Yusu , O. Enobabor, & S. Okunuga, “Block third derivative trigonometrically-fitted methods for stiff and periodic problems”, J. Nig. Soc. Phy. Sci. 2 (2020) 12. https://doi.org/10.46481/jnsps.2020.33 DOI: https://doi.org/10.46481/jnsps.2020.33

Z. Omar & O. Adeyeye, “Numerical solution of first order initial value problems using a self-starting implicit two-step Obrechko -type block method”, J. Math. Stat. 12 (2016) 127. http://doi.org/10.3844/jmssp.2016.127.134 DOI: https://doi.org/10.3844/jmssp.2016.127.134

Z. Zlatev, K. Georgiev, & I. Dimov, “Studying absolute stability properties of the Richardson Extrapolation combined with explicit Runge–Kutta methods”, Comput. Math. with Appl. 67 (2014) 2294. https://doi.org/10.1016/j.camwa.2014.02.025 DOI: https://doi.org/10.1016/j.camwa.2014.02.025

A. O. Adesanya, D. M. Udoh, & A. M. Ajileye, “A new hybrid block method for the solution of general third order initial value problems of ordinary differential equations”, Int. J. Pure Appl. Math. 86 (2013) 365. http://dx.doi.org/10.12732/ijpam.v86i2.11 DOI: https://doi.org/10.12732/ijpam.v86i2.11

J. O. Kuboye & Z. Omar, “Numerical solution of third order ordinary differential equations using a seven-step block method”, Int. J. Math. Anal. 9 (2014) 743. http://dx.doi.org/10.12988/ijma.2015.5125 DOI: https://doi.org/10.12988/ijma.2015.5125

A. F. Jameel, A. Saaban, & H. H. Zureigat, “Numerical solution of second-order fuzzy nonlinear two-point boundary value problems using combination of finite difference and Newton’s methods”, Neural. Comput. Appl. 30 (2018) 3167. https://doi.org/10.1007/s00521-017-2893-z DOI: https://doi.org/10.1007/s00521-017-2893-z

Published

2023-04-04

How to Cite

Hussain, K., Adeyeye, O., & Ahmad, N. (2023). Numerical Solution of Second Order Fuzzy Ordinary Differential Equations using Two-Step Block Method with Third and Fourth Derivatives. Journal of the Nigerian Society of Physical Sciences, 5(2), 1087. https://doi.org/10.46481/jnsps.2023.1087

Issue

Section

Original Research