Autonomous block method for uncertainty analysis in first-order real-world models using fuzzy initial value problem

Kashif Hussain; Ala Amourah; Jamal Salah; Ali Fareed Jameel; Nidal Anakira; Kashif Hussain; Ala Amourah; Jamal Salah; Ali Fareed Jameel; Nidal Anakira

doi:10.3934/math.2025443

AIMS Mathematics

2025, Volume 10, Issue 4: 9614-9636. doi: 10.3934/math.2025443

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Research article

Autonomous block method for uncertainty analysis in first-order real-world models using fuzzy initial value problem

1.
School of Quantitative Sciences, Universiti Utara Malaysia, Sintok, Kedah, Malaysia
2.
Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman
3.
Jadara University Research Center, Jadara University, Irbid 21110, Jordan
4.
College of Applied and Health Sciences, A'Sharqiyah University, Post Box No. 42, Post Code No. 400 Ibra, Sultanate of Oman
5.
Applied Science Research Center. Applied Science Private University, Amman, Jordan

Received: 15 October 2024 Revised: 27 February 2025 Accepted: 26 March 2025 Published: 25 April 2025
MSC : 35A15, 45G15, 65H20, 49M27

This article employs fuzzy derivatives and fuzzy differential equations (FDEs) to handle uncertainty in real-world applications. When exact answers are unavailable, numerical approaches are utilized to derive approximations for FDE. The autonomous two-step block method (TBM) with two higher fuzzy derivatives is used to discover optimum solutions to first-order FDEs with greater absolute accuracy. The technique competency is evaluated by analyzing first-order real-world models with fuzzy initial value problems (FIVPs). Using fuzzy calculus principles, we establish a novel universal fuzzification formulation of the TBM approach with the Taylor series. TBM is a convergent, zero-stable, and absolute stability region approach for solving linear and nonlinear fuzzy models, with a focus on regulating the convergence of approximate solutions. The developed method offers approximations for difficulties encountered in real life and is a transformational and workable method for solving first-order FIVPs.
- fuzzy derivative,
- first-order,
- two-step,
- block method,
- zero-stability,
- real-life models
Citation: Kashif Hussain, Ala Amourah, Jamal Salah, Ali Fareed Jameel, Nidal Anakira. Autonomous block method for uncertainty analysis in first-order real-world models using fuzzy initial value problem[J]. AIMS Mathematics, 2025, 10(4): 9614-9636. doi: 10.3934/math.2025443

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Abstract

This article employs fuzzy derivatives and fuzzy differential equations (FDEs) to handle uncertainty in real-world applications. When exact answers are unavailable, numerical approaches are utilized to derive approximations for FDE. The autonomous two-step block method (TBM) with two higher fuzzy derivatives is used to discover optimum solutions to first-order FDEs with greater absolute accuracy. The technique competency is evaluated by analyzing first-order real-world models with fuzzy initial value problems (FIVPs). Using fuzzy calculus principles, we establish a novel universal fuzzification formulation of the TBM approach with the Taylor series. TBM is a convergent, zero-stable, and absolute stability region approach for solving linear and nonlinear fuzzy models, with a focus on regulating the convergence of approximate solutions. The developed method offers approximations for difficulties encountered in real life and is a transformational and workable method for solving first-order FIVPs.

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