Numerical methods for m-polar fuzzy initial value problems

Some problems in science and technology are modeled using ambiguous, imprecise, or lacking contextual data. In the modeling of some real-world problems, differential equations often involve multi-agent, multi-index, multi-objective, multi-attribute, multi-polar information or uncertainty, rather than single bits. These types of differentials are not well represented by fuzzy differential equations or bipolar fuzzy differential equations. Therefore, m-pole fuzzy set theory can be applied to differential equations to deal with problems with multi-polar information. In this paper, we study differential equations in m extremely fuzzy environments. We introduce the concept gH-derivative of m-polar fuzzy valued function. By considering different types of differentiability, we propose some properties of the gH-differentiability of m-polar fuzzy-valued functions. We consider the m-polar fuzzy Taylor expansion. Using Taylor expansion, Euler's method and modified Euler's method for solving the m-polar fuzzy initial value problem are proposed. We discuss the convergence analysis of these methods. Some numerical examples are described to see the convergence and stability of the proposed method. We compare these methods by computing the global truncation error. From the numerical results, it can be seen that the modified Euler method converges to the exact solution faster than the Euler method.