Parameter estimation for uncertain fractional differential equations

Abstract

Since the concept of uncertain fractional differential equations was proposed, its wide range of applications have urged us to consider parameter estimation for uncertain fractional differential equations. In this paper, based on the definition of Liu process, we construct a function of unknown parameters which follows a standard normal uncertainty distribution. Then the method of moments is used to build a system of equations whose solutions are the estimated values of unknown parameters. After that, an algorithm of parameter estimation for a special uncertain fractional differential equation is proposed. Finally, the algorithm is applied to two numerical examples and the acceptability of the estimated parameters is proved by using uncertain hypothesis test.

Fractional order derivatives

In this section, two types of fractional order derivatives of a function are reviewed. Let p be a real positive number with $0 \le n-1< p \le n$, where n is a positive integer. Suppose that $f: [a, b] \rightarrow R$ is a continuous function. The fractional primitive of order p of f(t) is defined by

$$\begin{aligned} I_{a+}^pf(t)=\frac{1}{\varGamma (p)}\int _{a}^{t}(t-s)^{p-1}f(s)ds. \end{aligned}$$

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The p-th Riemann-Liouville fractional order derivative of f(t) is defined by

$$\begin{aligned} D_{a+}^pf(t)=\frac{1}{\varGamma (n-p)}\frac{d^n}{dt^n}\int _{a}^{t}(t-s)^{n-p-1}f(s)ds. \end{aligned}$$

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The p-th Caputo fractional order derivative of f(t) is defined by

$$\begin{aligned} ^cD_{a+}^pf(t)=\frac{1}{\varGamma (n-p)}\int _{a}^{t}(t-s)^{n-p-1}f^{(n)}(s)ds \end{aligned}$$

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where $f^{(n)}(s)$ represents the n-th order derivative of f(s), provided that f is a differentiable function of at least order n.

The relationship between these two types of fractional order derivatives satisfies

$$\begin{aligned} ^cD^p_{a+}f(t)&=D^p_{a+}f(t)- \sum _{k=0}^{n-1} \frac{(t-a)^{k-p}}{\varGamma (k-p+1)}f^{(k)}(a). \end{aligned}$$

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For the sake of convenience, denote the $D_{0+}^p$ and $^cD_{0+}^p$ by the abbreviations $D^p$ and $^cD^p$, respectively.