Research Paper Runge Kutta Method Pdf

In real research programs, 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 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 actually takes (Liu [2]). In order to deal with these problems, Liu [3] put up the uncertainty theory in 2007 and refined [4] it in 2010. Nowadays, the uncertainty theory has become a new branch of mathematics for modeling nondeterministic phenomena.

Liu [5] proposed the concept of canonical Liu process in 2009. The canonical Liu process is a process with stationary and independent increments, and its every increment is a normal uncertain variable. It begins with time 0 and almost all sample paths are Lipschitz-continuous. Based on canonical Liu process, Liu [5] developed uncertain calculus to deal with differentiation and integration of an uncertain process. The concept of uncertain differential equations was proposed by Liu [6] in 2008. 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]).

The existence and uniqueness of an uncertain differential equation was studied by Chen and Liu [11] in 2010. An uncertain differential equation has a unique solution if its coefficients satisfy Lipschitz condition and linear growth condition. The definition of stability was given by Liu [5] in 2009. After that, Yao et al. [12] gave a sufficient condition for stability. An uncertain differential equation is stable if its coefficients satisfy the linear growth condition and the strong Lipschitz condition. Furthermore, Yao et al. [13] gave the concept of stability in mean for an uncertain differential equation and proved the sufficient and necessary condition for the linear uncertain differential equation being stable in mean. Based on those works, other types of stability were extended, like that, stability in moment (Sheng and Wang [14]), almost sure stability (Liu et al. [15]), and exponential stability (Sheng and Gao [16]).

Chen and Liu [11] figured out the analytic solution of the linear uncertain differential equation. Liu [17] and Yao [18] considered a spectrum of analytic methods to solve some special classes of nonlinear uncertain differential equations. Unfortunately, we cannot obtain the analytic solution of every uncertain differential equation. Then, it is sufficient to obtain the numerical results in most situations. Yao and Chen [19] found a way to transfer uncertain differential equations into a spectrum of ordinary differential equations. They put up a Yao-Chen formula to calculate the inverse distribution of solution at a given time. Based on the Yao-Chen formula, a numerical method was designed for giving the solution to uncertain differential equations via the Euler method. Yao [20] also studied the extreme value, first hitting time and time integral of solution of uncertain differential equations.

The Runge-kutta method is wide-used in solving ordinary differential equations, and it is more accurate than the Euler method. In this paper, we will present a way to solve uncertain differential equations with the Runge-Kutta method. The rest of the paper is organized as follows. The “Preliminaries” section presents some basic concepts and properties in uncertainty theory, including uncertain calculus, uncertain differential equations, and α-path. The “Runge-Kutta Method” section shows a new numerical method using the Runge-Kutta method. The “Numerical Experiments” section gives some numerical experiments to illustrate the new method and to calculate the uncertainty distribution, expected value, extreme value, and time integral of solution of the uncertain differential equation.

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