An efficient numerical method for solving m-polar fuzzy initial value problems

Abstract

Several problems in the field of science and technology are modeled with information about the situation that is ambiguous, imprecise, or incomplete. That is, the information about values of parameters, functional relationships, or initial conditions is not given in precise. In these circumstance, existing analytic or numerical methods can be applied only to the selected behavior of the system. For example, by fixing the values of unknown parameters to some credible values. On the basis of partial knowledge, it is impossible to describe the behavior of complete system. Thus, fuzzy differential equations arise in many dynamical models. In modeling of several real-world problems, differential equations frequently involve multi-agent, multi-index, multi-objective, multi-attribute, multi-polar information or uncertainty rather than a single bit. These type of differential equations cannot be well represented by means of fuzzy differential equations or bipolar fuzzy differential equations. Therefore, the theory of m-polar fuzzy sets can be applied to differential equations to handle the problems which have multi-polar information. The aim of this paper is to study differential equation in m-polar fuzzy environment. A fourth-order Runge–Kutta method to solve m-polar FIVPs is presented. The consistency, stability and convergence of suggested method are discussed to ensure its efficiency and validity. Since it requires no higher order function derivatives, the suggested method is straightforward to implement. Euler and Euler modified methods have global truncations errors of O(h) and $O(h^{2})$ respectively whereas the suggested Runge–Kutta’s global truncation errors of $O(h^{4})$. Numerical examples are provided to compare the proposed method with Euler and modified Euler methods in terms of global truncation errors (GTE). The numerical findings suggest that the purposed method has an adequate level of accuracy.