A definition of the UFFDE with Riemann–Liouville type will be exhibited in this section. The analytic solutions to a class of the linear UFFDEs with Riemann–Liouville type will be provided along with ones for linear uncertain first order forward difference equations.

### Definition 5

A fractional difference equation is called an uncertain fractional difference equation if it is driven by an uncertain sequence. Further, an uncertain fractional forward difference equation for Riemann–Liouville type is the uncertain fractional difference equation with Riemann–Liouville-like forward difference.

We will just explore the solution of the following UFFDE with the initial condition in this paper:

$$\begin{aligned} &\Delta _{\mu -1}^{\mu }Y(t)=G\bigl(t+\mu ,Y(t+\mu )\bigr)+H \bigl(t+\mu ,Y(t+\mu )\bigr) \xi _{t+\mu }, \end{aligned}$$

(5)

$$\begin{aligned} &\Delta _{\mu -1}^{\mu -1}Y(t)|_{t=0}=Y_{0}, \end{aligned}$$

(6)

where \(\Delta _{\mu -1}^{\mu }\) denotes fractional Riemann–Liouville-like forward difference, \(G,H:[0,\infty )\times R \to R \) are two functions. \(t\in {\mathbb{N} _{0}}\cap [0,T]\), \(\mu \in (0,1]\), \(Y_{0}\) is a crisp number, and \(\xi _{\mu },\xi _{\mu +1},\ldots , \xi _{\mu +t}\) are \(t+1\) i.i.d. uncertain variables that have linear uncertainty distribution \(\mathcal{L}(a,b)\). A solution of UFFDE (5) with the initial condition (6) is an uncertain sequence \(Y(t)\) that satisfies Eq. (5) uniformly in *t*.

Note that i.i.d. uncertain variables means that they are independent and have the same uncertainty distribution. More details can be seen in [17].

### Remark 1

According to Definition 3 and Lemma 4, UFFDE (5) is equivalent to the uncertain fractional sum equation

$$\begin{aligned} Y(t)= {}&\frac{1}{\varGamma (\mu )}\sum_{l=0}^{t-\mu } \bigl(t-\sigma (l)\bigr)^{( \mu -1)}\bigl(G\bigl(l+\mu ,Y(l+\mu )\bigr)+H \bigl(l+\mu ,Y(l+\mu )\bigr)\xi _{l+\mu }\bigr) \\ &{}+\frac{Y_{0}}{\varGamma (\mu )}t^{(\mu -1)} \end{aligned}$$

for \(t\in {\mathbb{N}_{\mu }}\cap [0,T]\).

The following special linear UFFDE will be considered in the sequel:

$$\begin{aligned} &\Delta _{\mu -1}^{\mu }Y(t)=\lambda Y(t+\mu )+\lambda \xi _{t+\mu }, \end{aligned}$$

(7)

$$\begin{aligned} &\Delta _{\mu -1}^{\mu -1}Y(t)|_{t=0}=Y_{0} \end{aligned}$$

(8)

for \(t\in {\mathbb{N}_{0}}\cap [0,T]\), where \(\Delta _{\mu -1}^{\mu }\) denotes the fractional Riemann–Liouville-like forward difference, \(\mu \in (0,1]\), \(0<\lambda <1\), \(\xi _{\mu },\xi _{\mu +1}, \ldots ,\xi _{\mu +t}\) are \(t+1\) i.i.d. uncertain variables that have linear uncertainty distribution \(\mathcal{L}(a,b)\), \(Y_{0}\) is a crisp number.

### Theorem 1

*UFFDE* (7) *with the initial value condition* (8) *has a solution*

$$ Y(t)=Y_{0}F_{\mu ,\lambda }(t)+\zeta _{t}, \quad t\in { \mathbb{N}_{\mu }\cap [0,T]} $$

(9)

*for*
\(0<\lambda <1\), *where*
\(\zeta _{t}\)
*is an uncertain sequence with the linear uncertainty distribution*
\(\mathcal{L}(a\cdot e_{\mu ,\lambda }(t),b \cdot e_{\mu ,\lambda }(t))\), *and*

$$ F_{\mu ,\lambda }(t)=\sum_{k=0}^{\infty }\lambda ^{k}\frac{(t+k \mu -1)^{((k+1)\mu -1)}}{\varGamma ((k+1)\mu )}, $$

(10)

*and*

$$ e_{\mu ,\lambda }(t)=\sum_{k=1}^{\infty }\lambda ^{k}\frac{(t+(k-1) \mu )^{(k\mu )}}{\varGamma (k\mu +1)}. $$

(11)

### Proof

The Picard approximation can be adopted to derive the solution. First, apply the \(\Delta _{0}^{-\mu }=\Delta ^{-\mu }\) operator to Eq. (7) to get

$$ \Delta ^{-\mu }\Delta _{\mu -1}^{\mu }Y(t)= \lambda \Delta ^{-\mu } Y(t+ \mu )+\lambda \Delta ^{-\mu }\xi _{t+\mu }, \quad t\in {\mathbb{N}_{\mu }}\cap [0,T]. $$

(12)

Apply Lemma 2, Lemma 3, and Lemma 4 to the left-hand side of Eq. (12) to obtain

$$\begin{aligned} \Delta ^{-\mu }\Delta _{\mu -1}^{\mu }Y(t) &=\Delta ^{-\mu }\Delta \Delta _{\mu -1}^{-(1-\mu )}Y(t) \\ &=\Delta \Delta ^{-\mu }\Delta _{\mu -1}^{-(1-\mu )}Y(t)- \frac{t^{( \mu -1)}}{\varGamma (\mu )}Y_{0} \\ &=\Delta \Delta ^{-1}Y(t)-\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0} \\ &=Y(t)-\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}, \quad t\in {\mathbb{N}_{\mu }} \cap [0,T]. \end{aligned}$$

According to Eq. (12), we have

$$ Y(t)-\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}=\lambda \Delta ^{-\mu } Y(t+ \mu )+\lambda \Delta ^{-\mu }\xi _{t+\mu }. $$

That is, the solution of UFFDE (7) is the solution of the sum equation as follows:

$$ Y(t)=\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}+\lambda \Delta ^{-\mu } Y(t+ \mu )+\lambda \Delta ^{-\mu }\xi _{t+\mu }, \quad t\in {\mathbb{N}_{\mu }}\cap [0,T]. $$

(13)

Define \(Y_{0}(t)=\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}\) for \(t\in {\mathbb{N}_{\mu -1}}\cap [0,T] \) and

$$ Y_{n}(t)=\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}+\lambda \Delta ^{- \mu } Y_{n-1}(t+\mu )+\lambda \Delta ^{-\mu }\xi _{t+\mu },\quad t\in {\mathbb{N} _{\mu }}\cap [0,T],n\in \mathbb{N}_{0}. $$

(14)

Since \(\xi _{\mu },\xi _{\mu +1},\ldots ,\xi _{\mu +t}\) are \(t+1\) i.i.d. uncertain variables, write \(\xi _{\mu +t}=\xi \) in distribution. By Lemma 1 and the fact (Theorem 1.21 [17]) that the linear combination of finite independent linear uncertain variables is a linear uncertain variable with positive linear combination coefficient, we get

$$\begin{aligned} Y_{1}(t) ={}&\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}+\lambda \Delta ^{- \mu } Y_{0}(t+\mu -1)+\lambda \Delta ^{-\mu }\xi \\ ={}&\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}+\lambda \frac{(t+\mu -1)^{(2 \mu -1)}}{\varGamma (2\mu )}Y_{0} +\lambda \frac{t^{(\mu )}}{\varGamma ( \mu +1)}\xi , \\ Y_{2}(t) ={}&\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}+\lambda \Delta ^{- \mu } Y_{1}(t+\mu )+\lambda \Delta ^{-\mu }\xi _{t+\mu } \\ ={}&\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}+\lambda \frac{(t+\mu -1)^{(2 \mu -1)}}{\varGamma (2\mu )}Y_{0} +\lambda ^{2}\frac{(t+2\mu -1)^{(3 \mu -1)}}{\varGamma (3\mu )}Y_{0} \\ &{}+\lambda \frac{t^{(\mu )}}{\varGamma (\mu +1)}\xi +\lambda ^{2}\frac{(t+ \mu )^{(2\mu )}}{\varGamma (2\mu +1)}\xi , \\ &\vdots \\ Y_{n}(t) ={}&\sum_{k=1}^{n}\lambda ^{k}\frac{(t+(k-1)\mu )^{(k \mu )}}{\varGamma (k\mu +1)}\xi +\sum_{k=0}^{n} \lambda ^{k}\frac{(t+k \mu -1)^{((k+1)\mu -1)}}{\varGamma ((k+1)\mu )}Y_{0}, \quad t\in { \mathbb{N}_{\mu }}\cap [0,T], \\ &\vdots \end{aligned}$$

Since the series

$$ \sum_{k=0}^{\infty }\lambda ^{k} \frac{(t+(k-1)\mu )^{(k\mu )}}{ \varGamma (k\mu +1)} $$

and

$$ \sum_{k=0}^{\infty }\lambda ^{k} \frac{(t+k\mu -1)^{((k+1)\mu -1)}}{ \varGamma ((k+1)\mu )} $$

are absolutely convergent for \(\vert \lambda \vert <1\) by the d’Alembert ratio comparison test, the limitation \(\lim_{n\to \infty }Y_{n}\) exists. Write \(Y_{n}(t)\to \bar{Y}(t)\) as \(n\to \infty \). We have

$$\begin{aligned} \bar{Y}(t)=\sum_{k=1}^{\infty }\lambda ^{k}\frac{(t+(k-1)\mu )^{(k \mu )}}{\varGamma (k\mu +1)}\xi +\sum_{k=0}^{\infty } \lambda ^{k}\frac{(t+k \mu -1)^{((k+1)\mu -1)}}{\varGamma ((k+1)\mu )}Y_{0}, \quad t\in { \mathbb{N}_{\mu }}\cap [0,T]. \end{aligned}$$

Taking limit on both sides of Eq. (14) yields

$$ \bar{Y}(t)=\frac{t^{(\mu -1)}}{\varGamma (\mu )}Y_{0}+\lambda \Delta ^{- \mu } \bar{Y}(t+\mu )+\lambda \Delta ^{-\mu }\xi _{t+\mu },\quad t\in {\mathbb{N} _{\mu }}\cap [0,T],n\in \mathbb{N}_{0}. $$

That is, \(\bar{Y}(t)\) satisfies Eq. (13). Hence \(\bar{Y}(t)\) is a solution of Eq. (7) with the initial value condition (8). The proof is completed. □

The following corollary will give the solutions of linear uncertain first order forward difference equations in light of Theorem 1.

### Corollary 1

*Uncertain first order forward difference equation*

$$\begin{aligned} &\Delta Y(t)=\alpha Y(t+1)+\alpha \xi _{t+1} , \quad t\in { \mathbb{N}_{0}}\cap [0,T] , \end{aligned}$$

(15)

$$\begin{aligned} &Y(t)|_{t=0}=Y_{0} \end{aligned}$$

(16)

*has a solution*

$$ Y(t)=Y_{0}\frac{1}{(1-\alpha )^{t}}+\eta _{t}, \quad t\in { \mathbb{N}_{1}}\cap [0,T] $$

*for*
\(0<\alpha <1\), *where*
\(\eta _{t}\)
*is an uncertain sequence with linear uncertainty distribution*
\(\mathcal{L}(\frac{a}{(1-\alpha )^{t}}-a, \frac{b}{(1- \alpha )^{t}}-b)\).

### Proof

Since \(\sum_{k=0}^{\infty }\alpha ^{k} \frac{(t+k-1)^{(k)}}{ \varGamma (k+1)}= \frac{1}{(1-\alpha )^{t}}\), uncertain first order forward difference equation (15) with initial value condition (16) has a solution \(Y(t)=\frac{1}{(1-\alpha )^{t}}Y_{0}+\eta (t)\) by Theorem 1, where \(\eta (t)\) is an uncertain sequence with linear uncertainty distribution \(\mathcal{L}(\frac{a}{(1-\alpha )^{t}}-a, \frac{b}{(1- \alpha )^{t}}-b)\). The conclusion is proved. □