A fuzzy approach using modified adomian decomposition method for solving fuzzy Duffing differential equation
*Nuha Mohammed DabwanCorresponding authornuhamdm@gmail.comAffiliation 1Department of MathematicsFaculty of Education-AdenUniversity of AdenAden, YemenAffiliation 2Faculty of Education-AL-MekhlafTaiz UniversityTaiz, Yemen0009-0000-8686-0535View full profile → , Yahya Qaid Hasanqaid.yahya@tu.edu.yeDepartment of MathematicsUniversity of Saba RegionMarib, Yemen0000-0002-8071-3785View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Nov 2024
- Published Online:
- 11 Mar 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2274
- Pages:
- 2159–2170
Abstract
This study presents a novel application of the fuzzy Adomian decomposition method (FADM) to address the fuzzy Duffing differential equation, a significant non-linear problem frequently encountered in mechanical systems. The traditional methods of solving such equations often struggle to accommodate the inherent uncertainties present within the system parameters. By leveraging fuzzy logic, the FADM enables a more accurate representation of these uncertainties, resulting in robust approximate solutions. The method is systematically outlined, highlighting its effectiveness in producing convergent series solutions. Numerical examples are provided to illustrate the method’s application and efficacy in approximating solutions under varying fuzzy conditions. Results indicate that the fuzzy approach significantly improves solution accuracy and stability compared to traditional deterministic methods, making it a valuable tool in nonlinear dynamics.
Keywords
Subject Classifications
References
[1] M. Mazandarani and L. Xiu, “A Review on Fuzzy Differential Equations,” IEEE Access, vol. 9, pp. 62195-62211 (2021).
[2] M. A. K. Azad, M. S. Rahman, and M. H. U. Molla, “A New Technique for Obtaining Approximate Solution of Higher Order Nonlinear Differential Equation,” Journal of Interdisciplinary Mathematics, vol. 22, no. 5, pp. 797-809 (2019), doi: 10.1080/09720502.2019.1675291.
[3] M. Alshammari, M. Al-Smadi, O. AbuArqub, I. Hashim, and M. A. Alias, “Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations,” Symmetry, vol. 12, no. 4, pp. 572 (2020), doi: 10.3390/sym1204057.
[4] M. A. Karim and A. Y. Gunawan, “Parameter Estimations of Fuzzy Forced Duffing Equation: Numerical Performances by the Extended Runge-Kutta Method,” Abstract and Applied Analysis, vol. 2020, Article ID 6179591 (2020), doi: 10.1155/2020/6179591.
[5] N. Ahmad, M. Mamat, J. Kavikumar, and N. S. A. Hamzah, “Solving Fuzzy Duffing’s Equation by the Laplace Transform Decomposition,” Applied Mathematical Sciences, vol. 6, no. 59, pp. 2935-2944 (2012).
[6] A. F. Jameel, “The Variational Iteration Method for Solving Fuzzy Duffing’s Equation,” Journal of Interpolation and Approximation in Scientific Computing (2014), doi: 10.5899/2014/jiasc-00053.
[7] A. A. M. Fawze and L. H. Al-Taee, “Approximate Solutions of a Certain Initial Value Problem,” Journal of Interdisciplinary Mathematics, vol. 26, pp. 1471-1476 (2023).
[8] M. H. Suhhiem and R. I. Khwayyit, “Semi Analytical Solution for Fuzzy Autonomous Differential Equations,” International Journal of Analysis and Applications, vol. 20, no. 61 (2022), doi: 10.28924/2291-8639-20-2022-61.
[9] L. Wang and S. Guo, “Adomian Method for Second-Order Fuzzy Differential Equation,” World Academy of Science, Engineering and Technology, vol. 52, pp. 979-982 (2011).
[10] S. Biswas and T. K. Roy, “Adomian Decomposition Method for Fuzzy Differential Equations with Linear Differential Operator,” Journal of Information and Computing Science, vol. 4, no. 11, pp. 243-250 (2016).
[11] Z. Feng, “Duffing’s Equation and Its Applications to the Hirota Equation,” Physics Letters A, vol. 317, no. 1, pp. 115-119 (2003).
[12] G. Li, J. Cui, and H. Yang, “A New Detecting Method for Underwater Acoustic Weak Signal Based on Differential Double Coupling Oscillator,” IEEE Access, vol. 6, pp. 18842-18854 (2018).
[13] H. Nijmeijer and H. Berghuis, “On Lyapunov Control of the Duffing Equation,” IEEE Transactions on Circuits and Systems I, vol. 42, pp. 473-477 (1995).
[14] R. G. Junkera, R. Relana, and H. Madsena, “Physical-Stochastic Continuous-Time Identification of a Forced Duffing Oscillator,” ISA Transactions (2021), doi: 10.1016/j.isatra.2021.07.041.
[15] M. Ghanbari, “Solution of the First Order Linear Fuzzy Differential Equations by Some Reliable Methods,” Journal of Fuzzy Set and Analysis (2012), doi: 10.5899/2012/jfsva-00126.
[16] O. Kaleva, “A Note on Fuzzy Differential Equations,” Nonlinear Analysis, vol. 64, no. 1, pp. 895-900 (2006).
[17] M. L. Puri and D. A. Ralescu, “Differentials of Fuzzy Functions,” Journal of Mathematical Analysis and Applications, vol. 19, pp. 552-558 (1983).
[18] B. Bede and S. G. Gal, “Generalizations of the Differentiability of Functions with Applications to Fuzzy Differential Equations Fuzzy-Number-Valued,” Fuzzy Sets and Systems, vol. 151, pp. 581-599 (2005).
[19] G. Adomian and R. Rach, “Modified Adomian Polynomials,” Mathematical and Computer Modelling, vol. 24, no. 11, pp. 39-46 (1996).
[20] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, London: Kluwer Academic Publishers (1994).
[21] G. Adomian, “A Review of the Decomposition in Applied Mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, pp. 501-544 (1988).
[22] N. M. Dabwan and Y. Q. Hasan, “Solving Second Order Ordinary Differential Equation Using a New Modified Adomian Method,” Advances in Mathematics: Scientific Journal, vol. 3, no. 9, pp. 937-943 (2020).
[23] B. Berna and S. Mehmet, “Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method,” Journal of Applied Mathematics, vol. 2013, Article ID 691614 (2013), doi: 10.1155/2013/691614.
[24] J. C. Jiao, Y. Yamamoto, C. Dang, and Y. Ha, “An After Treatment Technique for Improving the Accuracy of Adomian’s Decomposition Method,” Computers Mathematics with Applications, vol. 43, pp. 783-798, 2002.
