## 2 Definition and Properties of Distributed Order Calculus  
Due to the forms, the distributed order calculus includes derivatives and integrals, whose definitions and basic properties are as follows. (1) Definitions and properties of distributed order derivatives  
Let \(\mu\) be a non-negative continuous function between [0, 1], the distributed order derivative \((D^{(\mu)}f)(t)\) of \(\mu\in[0,T]\) could be defined as follows:  
\[(D^{(\mu)}f)(t)=\int_{0}^{1}(D^{(\alpha)}f)(t)\mu(\alpha)d\alpha \tag{1}\]  
where \(D^{(\alpha)}\) is the Caputo-Dzhrbashyan regularized distributed order derivative of the order of \(\alpha\). \[ \begin{array}{l}(D^{(\alpha)}f)(t)\\ =\frac{1}{\Gamma(1-\alpha)}\Big{[}\frac{d}{dt}\int_{0}^{t}(t-\tau)^{-\alpha}f( \tau)d\tau-t^{-\alpha}f(0)\Big{]}\ (0<t<T)\end{array}  \tag{2}\]  
Let  
\[k(s)=\int_{0}^{1}\frac{s^{-\alpha}}{\Gamma(1-\alpha)}\mu(\alpha)d\alpha\ \ \ (s>0) \tag{3}\]  
\(k(s)\) is a non-negative decreasing function. In the existence of \(\frac{d}{dt}\int_{0}^{t}k(t-\tau)f(\tau)d\tau\), Formulas (1) and (2) could be transformed into Equation (4). \[(D^{(\mu)}f)(t)=\frac{d}{dt}\int_{0}^{t}k(t-\tau)f(\tau)d\tau-k(t)f(0) \tag{4}\]  
If \(f(\tau)\) is continuous, Equation (4) can be changed into Equation (5). \[(D^{(\mu)}f)(t)=\int_{0}^{t}k(t-\tau)f^{\prime}(\tau)d\tau \tag{5}\]  
The distributed order derivative contains two lemmas, as follows:  
**Lemma 1**. : _Let \(\mu\in C^{3}[0,1],\mu(1)\
eq 0,\,\mu(0)\
eq 0\,\,or\,\,\,\mu(\alpha)\sim a \alpha^{v},\ a,v>0,\ \alpha\to 0,\ s\to 0,\)_  
\[k(s)\,\sim\,s^{-1}(\log s)^{-2}\mu(1),k^{\prime}(s)\,\sim-s^{-2}(\log s)^{-2} \mu(1),\]  
_Equation (6) could be obtained through Laplace transforms of \(K(p)=\int_{0}^{\infty}k(s)e^{-ps}ds,\mathrm{Re}\,p>0\) and \(\int_{0}^{\infty}s^{-\alpha}e^{-ps}ds=\frac{\Gamma(1-\alpha)}{p^{1-\alpha}}.\)_  
\[ \begin{array}{l}K(p)=\int_{0}^{\infty}k(s)e^{-ps}ds=\int_{0}^{\infty}e^{-ps} \int_{0}^{1}\frac{s^{-\alpha}}{\Gamma(1-\alpha)}\mu(\alpha)d\alpha\ \ \ (\mathrm{Re}p>0)\\ =\int_{0}^{1}\frac{\Gamma(1-\alpha)}{p^{1-\alpha}}\frac{\mu(s)}{\Gamma(1- \alpha)}d\alpha=\int_{0}^{1}p^{\alpha-1}\mu(\alpha)d\alpha\ \ \ (\mathrm{Re}p>0)\end{array}  \tag{6}\]  
**Lemma 2**. :
1. _Let_ \(\mu\in C^{2}[0,1]\)_, if_ \(p\in C\backslash R_{-},|p|\to\infty\) _and_ \(K(p)=\frac{\mu(1)}{\log p}+O((\log|p|)^{-2})\)_, if_ \(\mu\in C^{3}[0,1]\)_, Equation (_7_) could be obtained._\[K(p)=\frac{\mu(1)}{\log p}-\frac{\mu^{\prime}(1)}{\left(\log p\right)^{2}}+O(( \left(\log|p|\right)^{-3})\] (7) (ii) _Let \(\mu\in C[0,1],\mu(0)\
eq 0\), if \(p\in C\backslash R_{-},|p|\to 0\), Equation (8) could be obtained._  
\[K(p)=p^{-1}\bigg{(}\log\frac{1}{p}\bigg{)}^{-1}\mu(0)\] (8) (iii) _Let \(\mu\in C[0,1],\mu(\alpha)\sim a\alpha^{\lambda},a>0,\lambda>0\), if \(p\in C\backslash R_{-},|p|\to 0\), Equation (9) could be obtained._  
\[K(p)=a\Gamma(1+\lambda)p^{-1}\bigg{(}\log\frac{1}{p}\bigg{)}^{-1-\lambda}\] (9) (2)  
Definitions and properties of distributed order integrals  
Suppose that the inverse Laplace transform of arbitrary function \(x(t)\) could be obtained by the Laplace transform formula of \(p\rightarrow\frac{1}{pK(p)}\), as follows:  
\[x(t)=\frac{d}{dt}\frac{1}{2\pi i}\int_{\delta-i\infty}^{\delta-i\infty}\frac{ ep^{t}}{p}\frac{1}{pK(p)}dp,\delta>0 \tag{10}\]  
If \((D^{(\mu)}f)(t)=u,f(0)=0\), \(\widetilde{f}(p)=\frac{1}{pK(p)}\widetilde{u}(p)\) could be obtained by the Laplace transform formula of \(f\rightarrow\widetilde{f}\). Then the form of the distributed order integral could be calculated as follows:  
\[(I^{(\mu)}u)(t)=\int_{0}^{t}x(t-\tau)u(\tau)d\tau \tag{11}\]  
The following lemma can express the properties of distributed order integrals. Suppose that \(\mu\in C^{3}[0,1],\mu(1)\
eq 0\), \(\mu(0)\
eq 0\) or \(\mu(\alpha)\sim a\alpha^{v},a,v>0,\alpha\to 0\)  
(i) \(x(t)\in C^{\infty}(0,\infty)\) and it is an entirely monotonic function;  
(ii) For any t, \(x(t)\leq C\log\frac{1}{t},|x^{\prime}(t)|\leq Ct^{-1}\log\frac{1}{t}\). **3.