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

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

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

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

S. Seikkala, “On the fuzzy initial value problem”, Fuzzy Sets Syst. 24 (1987) 319. 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

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

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

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

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

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

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

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

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

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

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

J. C. Butcher, Numerical methods for ordinary differential equations, New York: John Wiley and Sons Ltd (2008).

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).

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

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

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

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

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

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

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

Published

2023-04-04

How to Cite

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

Issue

Volume 5, Issue 2, May 2023

Section

Original Research
Copyright & Licensing

Copyright (c) 2023 Kashif Hussain, Oluwaseun Adeyeye, Nazihah Ahmad

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

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

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