Advancements in Numerical Analysis of Fuzzy Differential Equations: Solving Nonlinear Systems in Engineering and Finance
Abstract
Fuzzy differential equations (FDEs) have become a critical tool for modeling systems in engineering and finance that involve uncertainty and imprecision. These equations extend traditional differential equations by incorporating fuzzy sets to account for vague or incomplete information, which is often encountered in real-world applications. However, solving FDEs, especially higher-order and nonlinear systems, remains a significant challenge due to the inherent complexity of dealing with fuzzy parameters and the need for efficient numerical methods. This research paper aims to develop and refine numerical techniques for solving fuzzy differential equations, with a particular focus on extending these methods to fuzzy partial differential equations (FPDEs). The objective is to address the limitations of existing methods and propose efficient and accurate numerical approaches for both FDEs and FPDEs. The first part of the study presents an in-depth exploration of numerical methods for higher-order and nonlinear fuzzy differential equations, including advancements in modified Runge-Kutta methods and stability analysis. By improving the accuracy and efficiency of these methods, the research seeks to provide reliable solutions for systems in engineering and finance, where uncertainty plays a crucial role. The second part of the paper extends these numerical techniques to fuzzy partial differential equations, which model multi-dimensional systems with fuzzy parameters. The study addresses the complexities involved in solving FPDEs by modifying existing fuzzy differential equation solution techniques to handle partial derivatives and multi-dimensionality. Applications in engineering, such as heat conduction and fluid dynamics, and in finance, such as option pricing and risk management, are explored to demonstrate the practical applicability of the proposed methods.
How to Cite
Manisha, Pardeep Malhan, Vishal Saxena, "Advancements in Numerical Analysis of Fuzzy Differential Equations: Solving Nonlinear Systems in Engineering and Finance", Vol. 3, Issue 10, 19-01-2026, pp. 22-38.
