## 2 Notations and Preliminaries  
In this section, the fundamentals resulting of conformable fractional calculus theory are presented briefly, and the most important definitions and theories of the fractional power series approach are also given in conformable sense. **Definition 1**. : _From [26], let \(f\) be \(n\)-differentiable at \(t>s\). The conformable fractional derivative starting from \(s\) of a function \(f:[s,\infty)\to\mathbb{R}\) of order \(\alpha>0\) is defined by_  
\[T^{a}f(t)=\lim_{\epsilon\to 0}\frac{f^{([\alpha]-1)}\big{(}t+\epsilon(t-s)^{ [\alpha]-\alpha}\big{)}-f^{([\alpha]-1)}(t)}{\epsilon},\ \alpha\in(n-1,n]\,\ t>s, \tag{3}\]  
_and \(T^{a}f(s)=\lim_{t\to s^{+}}T^{a}f(t)\) provided \(f(t)\) is \(\alpha\)-differentiable in some \((0,s)\), \(s>0\), and \(\lim_{t\to s^{+}}T^{a}f(t)\) exists, where \(\lceil\alpha\rceil\) is the smallest integer greater than or equal to \(\alpha\). The function \(f\) is called \(\alpha\)-differentiable at a point \(t\) whenever \(f\) has a conformable fractional derivative of order \(\alpha\)at the point \(t\)._  
**Theorem 1**. : _From [26], let \(\alpha\in(0,\ 1]\) and assume \(f\), \(g\) be \(\alpha\)-differentiable at a point \(t>s\). Then_  
1. \(\frac{d^{a}}{d\alpha^{a}}(kf+hg)=kf^{(\alpha)}+h\ g^{(\alpha)},\,\forall k,\ h \in\mathbb{R}.\)__
2. \(\frac{d^{a}}{d\alpha^{a}}(\lambda)=0\)_,_ \(\lambda\) _is a constant._
3. \(\frac{d^{a}}{d\alpha^{a}}(\lambda f)=\lambda\frac{d^{a}}{d\alpha^{a}}(f)\)_._
4. \(\frac{d^{a}}{d\alpha^{a}}((t-\alpha)^{p})=p(t-a)^{p-a},\,\forall p\in\mathbb{R}.\)__
5. _If_ \(f\) _is differentiable, then_ \(\frac{d^{a}f}{d\alpha^{a}}(t)=(t-a)^{1-a}\frac{df}{dt}(t)\)  
**Remark 1**. : _For \(\alpha\in\ (n-1,n]\), if \(f\) is \(n\)-differentiable at \(t>s\), then \(\frac{d^{\alpha}f}{dt^{\alpha}}(t)=(t-s)^{n-\alpha}\frac{d^{\alpha}f}{dt^{\alpha}}(t)\)._  
**Definition 2**. : _From [27], the conformable fractional integral starting from \(s\) of order \(\alpha\in(n-1,n]\) of \(f(t)\) is defined as_  
\[I_{s}^{\alpha}f(t)=\frac{1}{(n-1)!}\int_{s}^{t}\frac{(t-\tau)^{\alpha-1}f(\tau )}{(\tau-s)^{n-\alpha}}d\tau\,t>\tau\geq s\geq 0. \tag{4}\]  
**Definition 3**. : _From [27], Let \(\partial^{k}u\partial^{k}\) and \(\partial^{k}u\partial x^{k}\), \(k=1,2,\ldots,n-1,\) be defined on \(I\times[s,\infty)\), then the conformable time-fractional differential operator of order \(\alpha\) of a function \(u(x,t):I\times[s,\infty)\to\mathbb{R}\) is defined by_  
\[T_{t}^{\alpha}u(x,t)=\frac{\partial^{\alpha}u(x,t)}{\partial t^{\alpha}}=\lim _{t\to 0}^{u(e^{-1})\left(x+t(e-s)^{n-\alpha}\right)-u_{t}^{(e-1)}(x,t)} \frac{}{\epsilon},\ \alpha\in(n-1,n], \tag{5}\] \[t\geq 0.\]  
**Definition 4**. : _From [27], the conformable fractional integral starting from \(s\) of order \(\alpha\in(n-1,n]\) of a function \(u(x,t):I\times[s,\infty)\to\mathbb{R}\) is defined by_  
\[I_{s}^{\alpha}u(x,t)=\left\{ \begin{array}{ll}\frac{1}{(n-1)!}\int_{s}^{t} \frac{(t-\tau)^{\alpha-1}u(x,x)}{(\tau-s)^{\alpha}}d\tau,&x\in I,\ t>\tau\geq s \geq 0,\\ u(x,t),&\alpha=0.\end{array} \right. \tag{6}\]  
**Definition 5**. : _From [29], for \(0\leq n-1<\alpha\leq n\), a power series (PS) of the form_  
\[ \begin{array}{ll}\sum\limits_{k=0}^{\infty}f_{k}(x)\big{(}t-t_{0}\big{)}^{ k\alpha}=f_{0}(x)+f_{1}(x)\big{(}t-t_{0}\big{)}^{\alpha}+f_{2}(x)\big{(}t-t_{0} \big{)}^{2\alpha}+\ldots\,\ x\in I,\ t_{0}\leq t\\ \hskip 113.811024 pt<t_{0}+R^{1/\alpha},\ R>0,\end{array}  \tag{7}\]  
_is called a multiple fractional PS at \(t=t_{0}\), wheret is a variable and the function \(f_{k}(x)\) is called the coefficients of the PS._  
As the classical power series [36; 37; 38; 39; 40; 41], it is clear that all terms of the multiple fractional PS (Equation (7)) vanish as soon as \(t=t_{0}\) except the first term, which means the multiple fractional PS is convergent when \(t=t_{0}\). Furthermore, for \(t\geq t_{0}\), this multiple fractional series is definitely convergent for \(\big{|}t-t_{0}\big{|}<R^{1/\alpha}\), ( \(R>0\)), where \(R^{1/\alpha}\) is the radius of convergence of the series. **Theorem 2**.