Characterization Theorem for the Numerical Solution of Fuzzy Differential Inclusions (FDIs)

Document Type : Research Article

Authors

1 Phd Student, Department of Mathematics and Computer Science, SR.C, Islamic Azad University, Tehran, Iran

2 Department of Mathematics and Computer Science, SR.C, Islamic Azad University, Tehran, Iran

3 University of Applied Science and Technology, Central of Tehran Shahdad Milk Industries, Iran

4 Department of Mathematics Qazvin Branch, Islamic Azad University, Qazvin, Iran

Abstract

In this paper, we investigate the numerical solution of fuzzy differential inclusions (FDIs) using characterization results for fuzzy differential equations (FDEs) based on Hukuhara differentiability. By employing the characterization theorem introduced by Bede and the construction of solutions via differential inclusions developed by Kaleva, we establish a rigorous connection among fuzzy differential equations, fuzzy differential inclusions, and systems of ordinary differential equations (ODEs). In particular, under suitable regularity and monotonicity assumptions, it is shown that the solution of a fuzzy differential inclusion coincides with the solution of the corresponding fuzzy differential equation and can be equivalently represented by a system of ODEs defined on the $\alpha$-level sets. This equivalence enables the reduction of fuzzy-valued problems to classical real-valued systems, thereby allowing the direct application of standard numerical methods for ordinary differential equations. Based on this framework, we propose a numerical approach for solving FDIs by first transforming the fuzzy problem into a parametric family of ODEs and then applying the Euler method to approximate the solutions. The validity of the proposed approach is illustrated through a numerical example, in which the approximate fuzzy solution is compared with the exact solution. The results demonstrate that classical numerical schemes can be effectively employed for fuzzy differential inclusions without reformulating them within a fully fuzzy numerical framework.

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References

[1] B. Bede, Note on Numerical solutions of fuzzy differential equations by predictor corrector method, Information Sciences, 178, 1917–1922, (2008).
[2] F. Pederson, M. Sambandham, Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem, 1, (2008).
[3] O. Kaleva, A note on fuzzy differential equations, Nonlinear Analysis, 64, 895–900, (2006).
[4] S. Siahmansouri, E. Yousefi, M. Gholamian, Fuzzy stochastic differential system, Mathematical Analysis and Convex Optimization, (2025), e731888.
[5] S. Siahmansouri, E. Yousefi, M. Gholamian, S. Neyshabouri, Application of the Homotopy Perturbation Method for Approximating Solutions to Fuzzy Initial Value Problems under Generalized Differentiable, Analytical and Numerical Solutions for Nonlinear Equations, 9(2), 184–192, (2025). 
Analytical and Numerical Solutions for Nonlinear Equations
Volume 11, Issue 1
June 2026
Pages 84-92

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History

  • Receive Date: 24 January 2026
  • Revise Date: 18 February 2026
  • Accept Date: 21 February 2026
  • Publish Date: 07 June 2026

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