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A numerical method for solving first-order fully fuzzy differential equation under strongly generalized H-differentiability

Abstract

In this paper, a numerical solution (Euler method) for solving first-order fully fuzzy differential equations (FFDE) in the form \(y'(t)=a\otimes y(t),~y(0)=y_0,t\in [0,T]\) under strongly generalized H-differentiability is considered. First, we will show that under H-differentiability the FFDE can be divided into four differential equations. Then, we will prove that each of divided differential equations satisfies the Lipschitz condition, therefore, FFDE has a unique solution and Euler method can be used to find an approximate solution in each case. Convergence of this method is proved and an algorithm by which the exact solution can be approximated in each case will be provided.

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Authors

Corresponding author

Correspondence to P. Darabi.

Additional information

Communicated by V. Loia.

Appendices

Appendix A

MATLAB implementation for the Euler method to find approximate solutions of FFDE (3.1) in case (1):

figure a

Appendix B

MATLAB implementation for the Euler method to find approximate solutions of FFDE (3.1) in case (2):

figure b

Appendix C

MATLAB implementation for the Euler method to find approximate solutions of FFDE (3.1) in case (3):

figure c

Appendix D

MATLAB implementation for the Euler method to find approximate solutions of FFDE (3.1) in case (4):

figure d

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Darabi, P., Moloudzadeh, S. & Khandani, H. A numerical method for solving first-order fully fuzzy differential equation under strongly generalized H-differentiability. Soft Comput 20, 4085–4098 (2016). https://doi.org/10.1007/s00500-015-1743-0

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  • DOI: https://doi.org/10.1007/s00500-015-1743-0

Keywords

  • Approximate solution
  • Cross product
  • First-order fully fuzzy differential equation (FFDE)
  • Lipschitz condition
  • Strongly generalized H-differentiability