Fuzzy Differential Equations for Real-World Systems with Uncertainty: Modeling Financial Markets, Biological Dynamics, and Engineering Processes
Abstract
Fuzzy differential equations (FDEs) provide a valuable framework for modeling systems that involve uncertainty and imprecision, which are commonly found in real-world scenarios. Traditional differential equations assume precise parameters, which are often unrealistic in applications where data is vague or uncertain. FDEs extend classical differential equations by incorporating fuzzy logic, allowing for the representation of uncertainty in system parameters and functions. This paper explores the application of fuzzy differential equations in modeling real-world systems across various domains, particularly in financial markets, biological systems, and engineering processes, where uncertainty plays a crucial role. The study develops and applies numerical methods for solving fuzzy differential equations, focusing on both first-order and higher-order nonlinear systems. The modified Runge-Kutta method and fuzzy finite difference method are implemented and tested for solving fuzzy differential equations and fuzzy partial differential equations (FPDEs), respectively. The paper demonstrates the application of these methods to real-world problems such as option pricing in uncertain financial markets, population dynamics with fuzzy birth and death rates in biology, and heat conduction in engineering systems with fuzzy thermal conductivity. The results highlight the effectiveness of fuzzy differential equations in capturing the inherent uncertainty in these systems. In financial modeling, fuzzy equations provided a range of possible asset prices rather than single deterministic values, offering a more realistic depiction of market volatility. In biological and engineering applications, fuzzy models allowed for the consideration of imprecise system parameters, yielding solutions that account for a broader range of possible outcomes. This research underscores the potential of fuzzy differential equations as a powerful tool for modeling and solving real-world systems with uncertainty, contributing to the field of fuzzy mathematics and its applications in engineering, finance, and biology. Through a series of numerical experiments and case studies, the paper evaluates the performance of the proposed methods, comparing them to existing approaches. The results show that the developed methods offer significant improvements in both accuracy and computational efficiency, making them valuable tools for solving real-world problems in engineering and finance.
How to Cite
Manisha, Pardeep Malhan, Vishal Saxena, "Fuzzy Differential Equations for Real-World Systems with Uncertainty: Modeling Financial Markets, Biological Dynamics, and Engineering Processes", Vol. 3, Issue 10, 21-01-2026, pp. 39-54.
