# Improved Short Memory Principle Method for Solving Fractional Damped Vibration Equations
## 2 Memory Effect of Fractional Differential  
Unlike the integer order differential, time fractional differential has a historical memory which becomes stronger or weaker with the change of the fractional order. For the fractional differential based on the Grunwald-Letnikov definition, this change is directly reflected in the binomial coefficients. The Grunwald-Letnikov definition is  
\[ \begin{split}\underset{t_{0}}{}^{GL}D_{t}^{\alpha}f(t)=\underset{h \to 0}{\lim}\frac{1}{h^{\alpha}}\sum_{j=0}^{[(t-t_{0})/h]}(-1)^{j}\bigg{(} \begin{array}{c}\alpha\\ j\end{array}\bigg{)}f(t-jh)\end{split}  \tag{1}\]  
where \(h\) is the time step size. If \(h\) is sufficiently small, then the limit operation can be ignored and changed into the following form:  
\[ \begin{split}\underset{t_{0}}{}^{GL}D_{t}^{\alpha}f(t)=\frac{1}{h ^{\alpha}}\sum_{j=0}^{[(t-t_{0})/h]}w_{j}f(t-jh)\end{split}  \tag{2}\]where \(w_{j}=(-1)^{j}\left( \begin{array}{c}\alpha\\ j\end{array} \right)\) is the coefficient of binomial \((1-z)^{\alpha}\):  
\[w_{j}=\frac{\prod_{i=1}^{j}(n-\alpha-1)}{j!},\ j=1,\ 2,\ \cdots \tag{3}\]  
To avoid the calculation of the Gamma function, the recursive form is adopted:  
\[w_{j}=\left(1-\frac{\alpha+1}{j}\right)w_{j-1},\ j=1,\ 2,\ \cdots \tag{4}\]  
The initial term is \(w_{0}=\frac{(-1)^{0}\Gamma(\alpha+1)}{\Gamma(0+1)\Gamma(\alpha-0+1)}=1\). \(w_{j}\) is the weighted coefficient of the function value of the time interval of \(j\) steps from the current time, i.e., the binomial coefficient. The coefficient on the left side of the equal sign of the recurrence formula which is less than 1 indicates that this weighting coefficient decreases with the increase of the time interval. The weighting coefficient is also related to the order of derivation. At the same step size, the coefficient decreases with the increase of \(\alpha\). In other words, within \(0<\alpha<1\), the memory of the fractional differential decreases with the increase of the derivative order; on the contrary, when \(\alpha\) tends to 0, the memory is enhanced. Tables 1 and 2 provide some binomial coefficients when \(\alpha=0.8\) and \(\alpha=0.2\), respectively. The two tables indicate that the absolute value of \(w_{j}\) tends to decrease as \(j\) increases, and the rate of decrease decelerates. Table 1 shows that it takes only approximately 60 items to decay from \(w_{0}=1\) to \(-1.00\times 10^{-4}\), and the items whose absolute value decreases by approximately 10 times are \(w_{200}\), \(w_{800}\), \(w_{2930}\), \(w_{10530}\) and \(w_{37870}\) and attenuate to \(-1.00\times 10^{-10}\), requiring approximately 136,140 items. Although the speed of attenuation decelerates, the limited items intercepted by the SMP are sufficient to ensure calculation accuracy. In Table 2, 70 items are needed to attenuate from \(w_{0}=1\) to \(-1.00\times 10^{-3}\), and then the items whose absolute value decreases by approximately 10 times are \(w_{450}\), \(w_{3380}\), \(w_{23040}\), \(w_{156970}\), \(w_{1069400}\), \(w_{7225900}\) and \(w_{45849200}\). The attenuation to \(-1.00\times 10^{-10}\) requires 45,849,200 items. More than 38 million items from \(-1.00\times 10^{-9}\) to \(-1.00\times 10^{-10}\), i.e., the truncated items with only \(10^{-10}\) impact on the current moment will also cause approximately \(10^{-2}\) effects. Therefore, the SMP must be improved. Given the initial conditions, the fractional damped single-degree-of-freedom vibration equation is as follows:  
\[\left\{ \begin{array}{c}m\ddot{x}+cD^{\alpha}x+kx=0\\ x_{0}=0,\ \dot{x}_{0}=1\end{array} \right. \tag{5}\]  
where \(m\), \(c\), \(k\) and \(\alpha\) are the mass, the damping coefficient, the stiffness coefficient and the fractional order, respectively. The Taylor auxiliary function is introduced to change the non-zero initial condition into zero initial condition. The Taylor auxiliary function form is as follows:  
\[T(t)=\sum_{k=0}^{q}\frac{x_{t0}^{(k)}}{k!}(t-t_{0}) \tag{6}\]  
where \(q\) is the order of the equation. Obviously, \(T(t)=t\) in this example.