New Solutions of Fuzzy-Fractional Fisher Models via Optimal He–Laplace Algorithm

Abstract

Fuzzy differential equations have gained significant attention in recent years due to their ability to model complex systems in the presence of uncertainty or imprecise information. These equations find applications in various fields, such as biomathematics, horological processes, production inventory models, epidemic models, fluid models, and economic investments. The Fisher model is one such example, which studies the dynamics of a population with uncertain growth rates. This study proposes a dual parametric extension of the He–Laplace algorithm for solving time-fractional fuzzy Fisher models. The proposed methodology uses triangular fuzzy numbers to introduce uncertainty in highly nonlinear fractional differential equations under the generalized Hukuhara differentiability concept. The obtained solutions are validated against existing results in crisp form and are found to be more accurate. The results are analyzed by finding solutions with different values of spatial coordinate $\xi$ , time $\eta$ , and $\varsigma$ -cut for both upper and lower bounds, and they are presented graphically. The analysis reveals that the uncertain probability density function gradually decreases at left and right boundaries when the fractional order is increased. The study provides valuable insights into the behavior of population growth under uncertain conditions and demonstrates the effectiveness of the proposed methodology in solving fuzzy-fractional models.