A Runge–Kutta numerical method to approximate the solution of bipolar fuzzy initial value problems

Abstract

In many fields of science, engineering and social sciences, fuzzy differential equations occur as it is the simplest way to model unpredictable dynamical systems. In many contexts and scientific disciplines, certain aspects of uncertainty in data are likely to occur. In applied fields, various types of vagueness were recognized. Some are caused by incomplete or contradictory information, as well as different interpretations of the same phenomenon. In various applications like belief systems, expert systems, and information fusion, one real value from the interval [0, 1] in favor of a certain property can not be utilized conveniently, as it is much useful to consider its counter-property. Many domains have bipolarity as a central characteristic. This paper deals with bipolar fuzzy initial value problems (BFIVPs). Numerical methods have much importance because we have many problems which cannot be solved analytically or it is much complicated to solve them analytically. We introduce a Runge–Kutta method to solve BFIVPs. To check efficiency and validity of proposed method, we prove the consistency, convergence and stability of the method. The proposed method is simple to implement as it does not require higher order derivatives of function. The local truncation errors of Euler and Modified Euler methods are $O(h^{2})$ and $O(h^{3})$, respectively, while local truncation errors in proposed Runge–Kutta method is $O(h^{5})$. We apply an introduced method to solve a few numerical examples. We give the comparison of our proposed method with the Euler and Modified Euler methods by finding global truncation errors. Numerical results show the acceptable accuracy of proposed method.