Abstract
Uncertain differential equations have been widely applied to many fields especially to uncertain finance. Unfortunately, we cannot always get the analytic solution of uncertain differential equations. Early researchers have put up a numerical method based on the Euler method. This paper designs a new numerical method for solving uncertain differential equations via the widely-used Runge-Kutta method. Some examples are given to illustrate the effectiveness of the Runge-Kutta method when calculating the uncertainty distribution, expected value, extreme value, and time integral of solution of uncertain differential equations.
Highlights
In real research programs, there are many new problems which lack empirical data
The belief degree has a larger variance than the real probability because human beings usually overweight unlikely events (Kahneman and Tversky [1]) and human beings usually estimate a much wider range of values than the object takes (Liu [2])
Uncertain differential equations have been widely applied in many fields such as uncertain finance (Liu [7], Yao [8]), uncertain optimal control (Zhu [9]), and uncertain differential game (Yang and Gao [10])
Summary
There are many new problems which lack empirical data. In these situations, we cannot obtain the probability distribution of the variables, and instead, we usually invite several experts to give their “belief degree” that each event will occur. The existence and uniqueness theorem of solution of the uncertain differential equation was proved by Chen and Liu [11] under linear growth condition and Lipschitz continuous condition. An uncertain differential equation dXt = f (t, Xt) dt + g (t, Xt) dCt is said to have an α-path Xtα if it solves the corresponding ordinary differential equation dXtα = f t, Xtα dt + g t, Xtα −1(α)dt where −1(α) is the inverse uncertainty distribution of standard normal uncertain variable, i.e.,. We obtain the inverse uncertainty distribution of Xs. Numerical Experiments Based on the Runge-Kutta method, we will give some numerical experiments to calculate uncertainty distribution, expected value, extreme value and time integral of solution of uncertain differential equation. Runge-Kutta Method for Uncertainty Distribution of Solution For linear uncertain differential equation, Chen and Liu [11] proved an analytic solution. The error of Runge-Kutta method is 3.6784, the errors
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