### Integer-order solution

### Theorem 1

*Let* \(\mathcal{L}\) *be the Laplace operator*, *applying this operator to Eq*. (4) *along with initial and boundary conditions*, *the exact solution of velocity is given in Eq*. (19).

### Proof

Applying the Laplace transform to Eq. (4), we obtain:

$$\begin{aligned}& \mathcal{L} \biggl[ \frac{\partial \bar{W}}{\partial \zeta } + \lambda \frac{\partial ^{2}\bar{W}}{\partial \zeta ^{2}} \biggr] \\& \quad = \mathcal{L} \biggl[ \frac{\partial ^{2}\bar{W}}{\partial \zeta ^{2}} + Gr \biggl( \bar{\Upsilon } + \lambda \frac{\partial \bar{\Upsilon }}{\partial \zeta } \biggr) + Gm \biggl( \bar{\Lambda } + \frac{\partial \bar{\Lambda }}{\partial \zeta } \biggr) - M \biggl( \bar{W} + \lambda \frac{\partial \bar{W}}{\partial \zeta } \biggr) \biggr]. \end{aligned}$$

(13)

After some simplifications, we obtain:

$$\begin{aligned}& ( 1 + \lambda q )q\bar{W} = \frac{\partial ^{2}\bar{W}}{\partial \zeta ^{2}} + Gr ( 1 + \lambda q )\bar{ \Upsilon } + Gm ( 1 + \lambda q )\bar{\Lambda } - M ( 1 + \lambda q ) \bar{W}, \end{aligned}$$

(14)

$$\begin{aligned}& \biggl[ \frac{\partial ^{2}}{\partial \zeta ^{2}} - \lambda q^{2} - ( 1 + \lambda M )q - M \biggr]\bar{W} = Gr \bigl[ ( 1 + \lambda q )\bar{\Upsilon } \bigr] + Gm \bigl[ ( 1 + \lambda q )\bar{\Lambda } \bigr]. \end{aligned}$$

(15)

Equation (15) is a 2nd-order nonhomogeneous linear equation.

The standard solution of Eq. (15) is

$$ \bar{W} ( \zeta ,q ) = \bar{W}_{1} ( \zeta ,q ) + \bar{W}_{2} ( \zeta ,q ). $$

(16)

where \(\bar{W}_{1} ( \zeta ,q )\) and \(\bar{W}_{2} ( \zeta ,q )\) are the complementary and particular solutions, respectively.

$$\begin{aligned}& \bar{W}_{1} ( \zeta ,q ) = c_{1}e^{\zeta \sqrt{\lambda q^{2} + ( 1 + \lambda M )q + M}} + c_{2}e^{ - \zeta \sqrt{\lambda q^{2} + ( 1 + \lambda M )q + M}}, \end{aligned}$$

(17)

$$\begin{aligned}& \begin{aligned}[b] \bar{W}_{2} ( \zeta ,q ) &= Gr \biggl[ \frac{1 + \lambda q}{\lambda q ( \sqrt{\Pr q} - 1 ) ( q^{2} - a_{1}q + a_{2} )} \biggr]e^{ - \zeta \sqrt{\Pr q}} \\ &\quad {}+ Gm \biggl[ \frac{1 + \lambda q}{\lambda q^{2} ( q^{2} - b_{1}q + a_{2} )} \biggr]e^{ - \zeta \sqrt{scq}}. \end{aligned} \end{aligned}$$

(18)

Here, \(a_{1} = \frac{\Pr - ( 1 + \lambda M )}{\lambda } \), \(a_{2} = \frac{M}{\lambda } \) and \(b_{1} = \frac{Sc - ( 1 + \lambda M )}{\lambda } \).

Putting Eqs. (17) and (18) into Eq. (16), we obtain:

$$ \begin{aligned} \bar{W} ( \zeta ,q ) &= c_{1}e^{\zeta \sqrt{\lambda q^{2} + ( 1 + \lambda M )q + M}} + c_{2}e^{ - \zeta \sqrt{\lambda q^{2} + ( 1 + \lambda M )q + M}} \\ &\quad {}+ Gr \biggl[ \frac{1 + \lambda q}{\lambda q ( \sqrt{\Pr q} - 1 ) ( q^{2} - a_{1}q + a_{2} )} \biggr]e^{ - \zeta \sqrt{\Pr q}} \\ &\quad {}+ Gm \biggl[ \frac{1 + \lambda q}{\lambda q^{2} ( q^{2} - b_{1}q + a_{2} )} \biggr]e^{ - \zeta \sqrt{scq}}. \end{aligned} $$

After applying initial and boundary conditions to the above equation, we obtain the Laplace solution for velocity:

$$ \begin{aligned}[b] \bar{W} ( \zeta ,q ) &= Gr \biggl[ \frac{1 + \lambda q}{\lambda ( \sqrt{\Pr q} - 1 )} \biggl( \frac{4}{A_{3}q} + \frac{1}{A_{4} ( q - A_{2} )} + \frac{1}{A_{5} ( q - A_{1} )} \biggr) \\ &\quad {}\times \bigl( e^{ - \zeta \sqrt{\Pr q}} - e^{ - \zeta \sqrt{\lambda \{ ( q - a_{3} )^{2} - a_{4}^{2} \} }} \bigr) \biggr] \\ &\quad {}+ Gm \biggl[ \frac{1 + \lambda q}{\lambda } \biggl( \frac{2}{q} \biggl( \frac{1}{B_{3}} - \frac{1}{B_{4}} \biggr) + \frac{4}{B_{5}q^{2}} + \frac{2}{B_{2} ( q - B_{2} )} + \frac{2}{B_{4} ( q - B_{1} )} \biggr) \\ &\quad {}\times\bigl( e^{ - \zeta \sqrt{scq}} - e^{ - \zeta \sqrt{\lambda \{ ( q - a_{3} )^{2} - a_{4}^{2} \} }} \bigr) \biggr], \end{aligned} $$

(19)

where \(a_{3} = \frac{1 + \lambda M}{2\lambda } \), \(a_{4} = \sqrt{\frac{ ( 1 + \lambda M )^{2} - 4\lambda M}{4\lambda ^{2}}}\), \(A_{1} = \frac{a_{1}}{2} - z\), \(A_{2} = \frac{a_{1}}{2} + z\), \(A_{3} = a_{1}^{2} - 4z^{2}\), \(A_{4} = 2z^{2} - a{}_{1}z\), \(A_{5} = 2z^{2} + a{}_{1}z\), \(z = \sqrt{\frac{a_{1}^{2} - 4a_{2}}{4}}\), \(B_{1} = \frac{b_{1}}{2} - z_{1}\), \(B_{2} = \frac{b_{1}}{2} + z_{1}\), \(B_{3} = z_{1}(b_{1} - 2z_{1})^{2}\), \(B_{4} = z_{1}(b_{1} + 2z_{1})^{2}\), \(B_{5} = b_{1}^{2} - 4z_{1}^{2}\) and \(z_{1} = \sqrt{\frac{b_{1}^{2} - 4a_{2}}{4}}\). □

### Caputo fractional derivative

### Theorem 2

*Let* \({}^{C}D_{t}^{\gamma } i(\zeta ,t)\) *be the Caputo fractional derivative and* *L* *be the Laplace operator*, *applying these operators to Eq*. (4) *along with initial and boundary conditions*, *the exact solution of velocity is given in Eq*. (26).

### Proof

Applying the Caputo time derivative Eq. (7) to the nondimensional velocity Eq. (4) and taking the Laplace transform, we obtain:

$$\begin{aligned}& \begin{aligned}[b] &\mathcal{L} \biggl[ \bigl( 1 + \lambda {}^{C}D_{t}^{\gamma } \bigr)\frac{\partial \bar{W}}{\partial \zeta } \biggr] \\ &\quad = \mathcal{L} \biggl[ \frac{\partial ^{2}\bar{W}}{\partial \zeta ^{2}} + Gr \bigl( 1 + \lambda {}^{C}D_{t}^{\gamma } \bigr) \bar{\Upsilon } + Gm \bigl( 1 + {}^{C}D_{t}^{\gamma } \bigr)\bar{\Lambda } - M \bigl( 1 + \lambda {}^{C}D_{t}^{\gamma } \bigr)\bar{W} \biggr], \end{aligned} \end{aligned}$$

(20)

$$\begin{aligned}& \bigl( 1 + \lambda q^{\gamma } \bigr)q\bar{W}_{C} = \frac{\partial ^{2}\bar{W}_{C}}{\partial \zeta ^{2}} + Gr \bigl( 1 + \lambda q^{\gamma } \bigr)\bar{ \Upsilon }_{C} + Gm \bigl( 1 + \lambda q^{\gamma } \bigr)\bar{ \Lambda }_{C} - M \bigl( 1 + \lambda q^{\gamma } \bigr) \bar{W}_{C}, \end{aligned}$$

(21)

$$\begin{aligned}& \begin{aligned}[b] &\biggl[ \frac{\partial ^{2}}{\partial \zeta ^{2}} - \bigl( 1 + \lambda q^{\gamma } \bigr)q - M \bigl( 1 + \lambda q^{\gamma } \bigr) \biggr]\bar{W}_{C} \\ &\quad = Gr \bigl[ - \bigl( 1 + \lambda q^{\gamma } \bigr)\bar{\Upsilon }_{C} \bigr] + Gm \bigl[ - \bigl( 1 + \lambda q^{\gamma } \bigr)\bar{\Lambda }_{C} \bigr]. \end{aligned} \end{aligned}$$

(22)

The general solution of the nonhomogeneous linear equation, Eq. (22), is:

$$ \bar{W}_{C} ( \zeta ,q ) = \bar{W}_{1} ( \zeta ,q ) + \bar{W}_{2} ( \zeta ,q ). $$

(23)

Here,

$$ \bar{W}_{1} ( \zeta ,q ) = c_{1}e^{\zeta \sqrt{ ( 1 + \lambda q^{\gamma } ) ( q + M )}} + c_{2}e^{ - \zeta \sqrt{ ( 1 + \lambda q^{\gamma } ) ( q + M )}}, $$

(24)

and

$$ \bar{W}_{2} ( \zeta ,q ) = Gr \biggl[ \frac{1 + \lambda q^{\gamma }}{q ( \sqrt{\Pr q^{\gamma }} - 1 ) ( pq^{\gamma } + m_{1} )}e^{ - \zeta \sqrt{\Pr q^{\gamma }}} \biggr] + Gm \biggl[ \frac{1 + \lambda q^{\gamma }}{q^{2} ( sq^{\gamma } + m_{1} )}e^{ - \zeta \sqrt{Scq^{\gamma }}} \biggr]. $$

(25)

Substituting Eq. (24) and Eq. (25) into Eq. (23) and applying initial and boundary conditions, the Laplace solution of Eq. (23) is

$$ \begin{aligned}[b] \bar{W}_{C}(\zeta ,q) &= Gr \biggl[ \frac{1 + \lambda q^{\gamma }}{q ( \sqrt{\Pr q^{\gamma }} - 1 ) ( pq^{\gamma } + m_{1} )} \bigl( e^{ - \zeta \sqrt{\Pr q^{\gamma }}} - e^{ - \zeta \sqrt{m_{1} ( 1 + \lambda q^{\gamma } )}} \bigr) \biggr] \\ &\quad {}+ Gm \biggl[ \frac{1 + \lambda q^{\gamma }}{q^{2} ( sq^{\gamma } + m_{1} )} \bigl( e^{ - \zeta \sqrt{Scq^{\gamma }}} - e^{ - \zeta \sqrt{m_{1} ( 1 + \lambda q^{\gamma } )}} \bigr) \biggr], \end{aligned} $$

(26)

where \(m_{1} = q + M\), \(p = \lambda ( q + M ) - \Pr \) and \(s = \lambda ( q + M ) - Sc\). □

### Caputo–Fabrizio fractional derivative

### Theorem 3

*Let* \({}^{CF}D_{t}^{\gamma } i(\zeta ,t)\) *be the Caputo–Fabrizio fractional derivative and* *L* *be the Laplace operator*, *applying these operators to Eq*. (4) *along with initial and boundary conditions*, *the exact solution of velocity is given in Eq*. (33).

### Proof

Applying the Caputo–Fabrizio time derivative Eq. (9) and then its Laplace transform Eq. (10) to Eq. (4), we obtain:

$$\begin{aligned}& \mathcal{L} \biggl[ \bigl( 1 + \lambda {}^{CF}D_{t}^{\gamma } \bigr)\frac{\partial \bar{W}}{\partial \zeta } \biggr] \\& \quad = \mathcal{L} \biggl[ \frac{\partial ^{2}\bar{W}}{\partial \zeta ^{2}} + Gr \bigl( 1 + \lambda {}^{CF}D_{t}^{\gamma } \bigr) \bar{\Upsilon } + Gm \bigl( 1 + {}^{CF}D_{t}^{\gamma } \bigr)\bar{\Lambda } - M \bigl( 1 + \lambda {}^{CF}D_{t}^{\gamma } \bigr)\bar{W} \biggr], \end{aligned}$$

(27)

$$\begin{aligned}& \biggl( 1 + \frac{\lambda q}{ ( 1 - \gamma )q + \gamma } \biggr)q\bar{W}_{CF} \\& \quad = \frac{\partial ^{2}\bar{W}_{CF}}{\partial \zeta ^{2}} + Gr \biggl( 1 + \frac{\lambda q}{ ( 1 - \gamma )q + \gamma } \biggr)\bar{\Upsilon }_{CF} \\& \qquad {}+ Gm \biggl( 1 + \frac{\lambda q}{ ( 1 - \gamma )q + \gamma } \biggr)\bar{\Lambda }_{CF} - M \biggl( 1 + \frac{\lambda q}{ ( 1 - \gamma )q + \gamma } \biggr) \bar{W}_{CF}. \end{aligned}$$

(28)

After some calculations we obtain a second-order nonhomogeneous linear equation:

$$\begin{aligned}& \biggl[ \frac{\partial ^{2}}{\partial \zeta ^{2}} - \biggl( 1 + \frac{\lambda q}{ ( 1 - \gamma )q + \gamma } \biggr)q - M \biggl( 1 + \frac{\lambda q}{ ( 1 - \gamma )q + \gamma } \biggr) \biggr]\bar{W}_{CF} \\& \quad = - \biggl( 1 + \frac{\lambda q}{ ( 1 - \gamma )q + \gamma } \biggr) ( Gr\bar{\Upsilon }_{CF} + Gm \bar{\Lambda }_{CF} ). \end{aligned}$$

(29)

The solution of Eq. (29) is

$$ \bar{W}_{CF} ( \zeta ,q ) = \bar{W}_{1} ( \zeta ,q ) + \bar{W}_{2} ( \zeta ,q ), $$

(30)

where

$$ \bar{W}_{1} ( \zeta ,q ) = c_{1}e^{\zeta \sqrt{ ( \frac{dq + \gamma }{lq + \gamma } )m_{1}}} + c_{2}e^{ - \zeta \sqrt{ ( \frac{dq + \gamma }{lq + \gamma } )m_{1}}} , $$

(31)

and

$$\begin{aligned} \bar{W}_{1} ( \zeta ,q ) =& Gr \biggl[ \frac{dq + \gamma }{q ( \sqrt{\frac{\Pr q}{lq + \gamma }} - 1 ) ( p_{2}q + m_{1}\gamma )}e^{ - \zeta \sqrt{\frac{\Pr q}{lq + \gamma }}} \biggr] \\ &{}+ Gm \biggl[ \frac{dq + \gamma }{q^{2} ( s_{2}q + m_{1}\gamma )}e^{ - \zeta \sqrt{\frac{Scq}{lq + \gamma }}} \biggr]. \end{aligned}$$

(32)

Inserting the values of Eq. (31) and Eq. (32) into Eq. (30) and applying initial and boundary conditions, we have:

$$\begin{aligned} \bar{W}_{\mathbf{C}F} ( \zeta ,q ) =& Gr \biggl[ \frac{dq + \gamma }{q ( p_{2}q + m_{1}\gamma ) ( \sqrt{\frac{\Pr q}{lq + \gamma }} )} \bigl( e^{ - \zeta \sqrt{\frac{\Pr q}{lq + \gamma }}} - e^{ - \zeta \sqrt{m_{1} ( \frac{dq + \gamma }{lq + \gamma } )}} \bigr) \biggr] \\ &{}+ Gm \biggl[ \frac{dq + \gamma }{q^{2} ( s_{2}q + m_{1}\gamma )} \bigl( e^{ - \zeta \sqrt{\frac{Scq}{lq + \gamma }}} - e^{ - \zeta \sqrt{m_{1} ( \frac{dq + \gamma }{lq + \gamma } )}} \bigr) \biggr], \end{aligned}$$

(33)

where \(m_{1} = q + M\), \(l = 1 - \gamma \), \(d = 1 - \gamma + \lambda \), \(p_{2} = m_{1}d - \Pr \) and \(s_{2} = m_{1}d - Sc\). □

### Atangana–Baleanu fractional derivative

### Theorem 4

*Let* \({}^{ABC}D_{t}^{\gamma } i(\zeta ,t)\) *be the Atangana–Baleanu fractional derivative and* *L* *be the Laplace operator*, *applying these operators to Eq*. (4) *along with initial and boundary conditions*, *the exact solution of velocity is given in Eq*. (40).

### Proof

Applying the Atangana–Baleanu time derivative Eq. (11) and then its Laplace transform Eq. (12) to Eq. (4), we obtain:

$$\begin{aligned}& \mathcal{L} \biggl[ \bigl( 1 + \lambda {}^{ABC}D_{t}^{\gamma } \bigr)\frac{\partial \bar{W}}{\partial \zeta } \biggr] \\& \quad = \mathcal{L} \biggl[ \frac{\partial ^{2}\bar{W}}{\partial \zeta ^{2}} + Gr \bigl( 1 + \lambda {}^{ABC}D_{t}^{\gamma } \bigr) \bar{\Upsilon } + Gm \bigl( 1 + {}^{ABC}D_{t}^{\gamma } \bigr)\bar{\Lambda } - M \bigl( 1 + \lambda {}^{ABC}D_{t}^{\gamma } \bigr)\bar{W} \biggr], \end{aligned}$$

(34)

$$\begin{aligned}& \biggl( 1 + \frac{\lambda q^{\gamma }}{ ( 1 - \gamma )q^{\gamma } + \gamma } \biggr)q\bar{W}_{ABC} \\& \quad = \frac{\partial ^{2}\bar{W}_{ABC}}{\partial \zeta ^{2}} + Gr \biggl( 1 + \frac{\lambda q^{\gamma }}{ ( 1 - \gamma )q^{\gamma } + \gamma } \biggr)\bar{ \Upsilon }_{ABC} \\& \qquad {} + Gm \biggl( 1 + \frac{\lambda q^{\gamma }}{ ( 1 - \gamma )q^{\gamma } + \gamma } \biggr)\bar{ \Lambda }_{ABC} - M \biggl( 1 + \frac{\lambda q^{\gamma }}{ ( 1 - \gamma )q^{\gamma } + \gamma } \biggr)\bar{W}_{ABC}, \end{aligned}$$

(35)

$$\begin{aligned}& \biggl[ \frac{\partial ^{2}}{\partial \zeta ^{2}} - \biggl( 1 + \frac{\lambda q^{\gamma }}{lq^{\gamma } + \gamma } \biggr)q - M \biggl( 1 + \frac{\lambda q^{\gamma }}{lq^{\gamma } + \gamma } \biggr) \biggr]\bar{W}_{ABC} \\& \quad = - \biggl( 1 + \frac{\lambda q^{\gamma }}{lq^{\gamma } + \gamma } \biggr) ( Gr\bar{\Upsilon }_{ABC} + Gm\bar{\Lambda }_{ABC} ). \end{aligned}$$

(36)

Equation (36) is a 2nd-order nonhomogeneous linear equation. So, its exact solution is:

$$ \bar{W}_{ABC} ( \zeta ,q ) = \bar{W}_{1} ( \zeta ,q ) + \bar{W}_{2} ( \zeta ,q ). $$

(37)

Here,

$$ \bar{W}_{1} ( \zeta ,q ) = c_{1}e^{\zeta \sqrt{ ( \frac{dq^{\gamma } + \gamma }{lq^{\gamma } + \gamma } )m_{1}}} + c_{2}e^{ - \zeta \sqrt{ ( \frac{dq^{\gamma } + \gamma }{lq^{\gamma } + \gamma } )m_{1}}} , $$

(38)

and

$$\begin{aligned} \bar{W}_{2} ( \zeta ,q ) =& Gr \biggl[ \frac{dq^{\gamma } + \gamma }{q ( \sqrt{\frac{\Pr q^{\gamma }}{lq^{\gamma } + \gamma }} - 1 ) ( p_{3}q^{\gamma } + m_{1}\gamma )}e^{ - \zeta \sqrt{\frac{\Pr q^{\gamma }}{lq^{\gamma } + \gamma }}} \biggr] \\ &{}+ Gm \biggl[ \frac{dq^{\gamma } + \gamma }{q^{2} ( s_{3}q^{\gamma } + m_{1}\gamma )}e^{ - \zeta \sqrt{\frac{Scq^{\gamma }}{lq^{\gamma } + \gamma }}} \biggr]. \end{aligned}$$

(39)

Putting values of the solution of \(\bar{W}_{1} ( \zeta ,q )\) and \(\bar{W}_{2} ( \zeta ,q )\) into Eq. (37) and after some simplifications, we obtain:

$$\begin{aligned} \bar{W}_{ABC} ( \zeta ,q ) =& Gr \biggl[ \frac{dq^{\gamma } + \gamma }{q ( \sqrt{\frac{\Pr q^{\gamma }}{lq^{\gamma } + \gamma }} ) ( p_{3}q^{\gamma } + m_{1}\gamma )} \bigl( e^{ - \zeta \sqrt{\frac{\Pr q^{\gamma }}{lq^{\gamma } + \gamma }}} - e^{ - \zeta \sqrt{m_{1} ( \frac{dq^{\gamma } + \gamma }{lq^{\gamma } + \gamma } )}} \bigr) \biggr] \\ &{}+ Gm \biggl[ \frac{dq^{\gamma } + \gamma }{q^{2} ( s_{3}q^{\gamma } + m_{1}\gamma )} \bigl( e^{ - \zeta \sqrt{\frac{Scq^{\gamma }}{lq^{\gamma } + \gamma }}} - e^{ - \zeta \sqrt{m_{1} ( \frac{dq^{\gamma } + \gamma }{lq^{\gamma } + \gamma } )}} \bigr) \biggr], \end{aligned}$$

(40)

where \(m_{1} = q + M\), \(l = 1 - \gamma \), \(d = 1 - \gamma + \lambda \), \(p_{3} = m_{1}d - \Pr \) and \(s_{3} = m_{1}d - Sc\). □

Stehfest’s formula [22] is one of the simplest algorithms we use to determine the inverse Laplace transform:

$$ W ( r,t ) = \frac{e^{4.7}}{t} \Biggl[ \frac{1}{2}\bar{W} \biggl( r,\frac{4.7}{t} \biggr) + \operatorname{Re} \Biggl\{ \sum _{k = 1}^{N_{1}} ( - 1 )^{k}\bar{W} \biggl( r,\frac{k\pi i + 4.7}{t} \biggr) \Biggr\} \Biggr], $$

where \(N_{1}\) is a natural number, \(\operatorname{Re} (\cdot)\) and *i* are the real part and the imaginary unit, respectively [22].