Let \(t_{m}=m\Delta t\), where \(\Delta t=\frac{T}{K}\) is the step size in the time direction for \(m=0,1,2,\ldots,K\). To discretize the Caputo fractional time derivative at \(t=t_{m+1}\), the usual central difference approach is used as follows [20]:
$$\begin{aligned}& \frac{\partial ^{\alpha } y(z,t_{m+1})}{\partial t^{\alpha } }=\frac{1}{ \varGamma (2-\alpha )} \int _{0}^{t_{k+1}}\frac{\partial ^{2} y(z,w)}{ \partial w^{2}}(t_{m+1}-w)^{-\alpha +1} \,dw, \\& \frac{\partial ^{\alpha } y(z,t_{m+1})}{\partial t^{\alpha } } \\& \quad =\frac{1}{\varGamma (2-\alpha )}\sum _{k=0}^{m} \int _{t_{k}}^{t_{k+1}}\frac{ \partial ^{2} y(z,w)}{\partial w^{2}}(t_{m+1}-w)^{-\alpha +1} \,dw \\& \quad = \frac{1}{\varGamma (2-\alpha )} \sum_{k=0}^{m} \frac{y(z,t_{k+1})-2y(z,t _{k})+y(z,t_{k-1})}{\Delta t^{2}} \int _{t_{k}}^{t_{k+1}}(t_{m+1}-w)^{-\alpha +1} \,dw+l_{ \Delta t}^{m+1} \\& \quad = \frac{1}{\varGamma (2-\alpha )} \sum_{k=0}^{m} \frac{y(z,t_{k+1})-2y(z,t _{k})+y(z,t_{k-1})}{\Delta t^{2}} \int _{t_{m-k}}^{t_{m-k+1}}(\upsilon )^{-\alpha +1}\,d \upsilon +l_{\Delta t}^{m+1} \\& \quad = \frac{1}{\varGamma (2-\alpha )}\sum_{k=0}^{m} \frac{y(z,t_{m-k+1})-2y(z,t _{m-k})+y(z,t_{m-k-1})}{\Delta t^{2}} \int _{t_{k}}^{t_{k+1}}(\upsilon )^{-\alpha }\,d\upsilon +l _{\Delta t}^{m+1} \\& \quad = \frac{1}{\varGamma (3-\alpha )}\sum_{k=0}^{m} \frac{y(z,t_{m-k+1})-2y(z,t _{m-k})+y(z,t_{m-k-1})}{\Delta t^{\alpha }}\bigl((k+1)^{2-\alpha }-k^{2- \alpha }\bigr) +l_{\Delta t}^{m+1} \\& \quad = \frac{1}{\varGamma (3-\alpha )}\sum_{k=0}^{m} d_{k}\frac{y(z,t_{m-k+1})-2y(z,t _{m-k})+y(z,t_{m-k-1})}{\Delta t^{\alpha }}+l_{\Delta t}^{m+1}, \end{aligned}$$
(18)
where \(d_{k}=(k+1)^{2-\alpha }-k^{2-\alpha }\) and \(\upsilon =(t_{m+1}-w)\).
Now, introduce a fractional differential operator \(\varOmega _{t}^{\alpha }\):
$$ \varOmega _{t}^{\alpha }y(z,t_{k+1})=\frac{1}{\varGamma (3-\alpha )} \sum_{k=0} ^{m} d_{k} \frac{y(z,t_{m-k+1})-2y(z,t_{m-k})+y(z,t_{m-k-1})}{\Delta t ^{\alpha }}. $$
Equation (18) can be rewritten as follows:
$$ \frac{\partial ^{\alpha } y(z,t_{m+1})}{\partial t^{\alpha }}=\varOmega _{t}^{\alpha }y(z,t_{k+1})+l_{\Delta t}^{m+1}. $$
(19)
Here, \(l_{\Delta t}^{m+1}\) denotes the truncation error between \(\frac{\partial ^{\alpha }}{\partial t^{\alpha }}y(z,t_{m+1})\) and \(\varOmega _{t}^{\alpha }y(z,t_{m+1})\). Equation (1) can be written as
$$ \varOmega _{t}^{\alpha }y(z,t_{m+1})+ \gamma \frac{\partial ^{4}}{\partial z^{4}}y(z,t_{m+1})=f(z,t_{m+1}), $$
(20)
where \(\varOmega _{t}^{\alpha }y(z,t_{m+1})\) denotes the Caputo fractional time derivative approximation at \(t=t_{m+1}\). Using (18), expression (20) takes the following form:
$$\begin{aligned} \begin{aligned}[b] y^{m+1}(z)+\beta \gamma y^{m+1}_{xxxx}&=-d_{m}y^{-1}(z)+(2d_{m}-d_{m-1})y ^{0}(z)+\sum_{k=1}^{m-1}(-d_{k-1}+2d_{k}-d_{k+1})y^{m-k}(z) \\ &\quad{} +(2d_{0}-d_{1})y^{m}(z)+\beta f^{m+1}(z), \quad m=1,2,3,\ldots,K-1, \end{aligned} \end{aligned}$$
(21)
where \(\beta =\varGamma (3-\alpha )\Delta t^{\alpha }\) and \(y^{m+1}(z)=y(z,t ^{m+1})\) and the initial conditions are imposed as follows:
$$\begin{aligned}& y(z,t_{0})=\phi (z), \quad 0\leq z \leq L, \end{aligned}$$
(22)
$$\begin{aligned}& \frac{\partial y(z,t_{0})}{\partial t}=\psi (z), \quad 0\leq z \leq L. \end{aligned}$$
(23)
Moreover, the constants \(d_{k}s\) appearing in (18) possess the following properties:
\(d_{0}=1\) and ∀k, \(d_{k}s>0\),
\((2-d_{1})- \sum_{k=1}^{m-1}(d_{k+1}-2d_{k+1}+d_{k-1})+(2d_{m}-d_{m-1})-d _{m}=1 \).
The truncation error in (19) is bounded, i.e.,
$$ \bigl\vert l^{m+1}_{\Delta t} \bigr\vert \leq \zeta \Delta t^{2}. $$
(24)
Here, ζ is a constant depending on y.
To implement this scheme, first we calculate \(y^{-1}\) as follows:
$$\begin{aligned}& y^{-1}(z)=y(z,t_{0})-\Delta t y_{t}(z,t_{0}), \\& y^{-1}(z)=\phi (z)-\Delta t \psi (z). \end{aligned}$$
For \(m=0\), (21) takes the following form:
$$ y^{1}(z)+\beta \gamma y^{1}_{zzzz}=-d_{0}y^{-1}(z)+2d_{0}y_{0}(z)+ \beta f^{1}(z). $$
(25)
Now, Eqs. (21) and (25) together with initial/boundary conditions become a complete set of semi-discrete problem for (1).
Also, \(l^{m+1}\), the error at \(t=t_{m+1}\), is given by [21]
$$ l^{m+1}=\beta \biggl(\frac{\partial ^{\alpha }}{\partial t^{\alpha }}y(z,t _{m+1})-G^{\alpha }_{t}y(z,t_{m+1}) \biggr). $$
(26)
From Eqs. (19) and (24), the above expression can be written as
$$ \bigl\vert l^{m+1} \bigr\vert = \bigl\vert l^{m+1}_{\Delta t} \bigr\vert \leq \zeta \Delta t^{2}. $$
(27)
Some relevant functional spaces, their inner product and standard norms are defined as follows:
$$\begin{aligned}& \varUpsilon ^{2}(\eta ) = \bigl\{ u\in L^{2}(\eta ), u_{z}, u_{zz} \in L^{2}( \eta )\bigr\} , \\& \varUpsilon _{0}^{2}(\eta ) = \bigl\{ u \in \varUpsilon ^{2}(\eta ), u| _{\partial \eta }=0, u_{z}| _{\partial \eta }=0 \bigr\} , \\& \varUpsilon ^{n}(\eta ) = \bigl\{ u \in L^{2}(\eta ), u^{(r)}_{z}, \forall r \leq N\bigr\} , \end{aligned}$$
where \(L^{2}(\eta )\) denotes the space of those measurable functions whose squares are Lebesgue integrable in η. The norm and inner product of \(L^{2}(\eta )\) are given by
$$ \langle v,u\rangle = \int _{\eta }vu \,dz, \quad \Vert u \Vert _{0}= \langle u,u\rangle ^{\frac{1}{2}}. $$
Also, for \(\varUpsilon ^{2}(\eta )\), we take
$$ \langle v,u\rangle _{2}=\langle v,u\rangle +\langle v_{z},u_{z}\rangle +\langle v_{zz},u_{zz} \rangle , \quad \Vert u \Vert _{2}=\sqrt{\langle u,u\rangle _{2}}. $$
The norm \(\Vert \cdot \Vert \) of the space \(\varUpsilon ^{N}(\eta )\) is defined as
$$ \Vert u \Vert _{N}=\sqrt{\sum _{r=0}^{N} \bigl\Vert u_{x}^{(r)} \bigr\Vert _{0}^{2}}. $$
(28)
\(\Vert \cdot \Vert _{2}\) is defined as
$$ \Vert u \Vert _{2}=\sqrt{ \Vert u \Vert _{0}^{2}+\beta \gamma \bigl\Vert u_{x}^{(2)} \bigr\Vert _{0} ^{2}}. $$
(29)
Now, to carry out the stability and convergence analysis, we need to find \(y^{m+1}\in \varUpsilon _{0}^{2}(\eta )\) such that \(\forall u\in \varUpsilon _{0}^{2}(\eta )\). From (21) and (25), we have
$$\begin{aligned} \begin{aligned}[b] & \bigl\langle y^{m+1},u\bigr\rangle + \beta \gamma \bigl\langle y_{zzzz}^{m+1},u\bigr\rangle \\ &\quad =-d_{m}\bigl\langle y^{-1},u\bigr\rangle +(2d _{m}-d_{m-1})\bigl\langle y^{0},u\bigr\rangle \\ &\quad \quad{} +\sum_{k=1}^{m-1}(-d_{k-1}+2d_{k}-d_{k+1}) \bigl\langle y^{m-k},u\bigr\rangle +(2d_{0}-d_{1}) \bigl\langle y^{m},u\bigr\rangle +\beta \bigl\langle f^{m+1},u \bigr\rangle \end{aligned} \end{aligned}$$
(30)
and
$$ \bigl\langle y^{1},u\bigr\rangle +\beta \gamma \bigl\langle y_{zzzz}^{1},u\bigr\rangle =-d_{0}\bigl\langle y^{-1},u\bigr\rangle +2d_{0}\bigl\langle y^{0},u\bigr\rangle +\beta \bigl\langle f^{1},u\bigr\rangle . $$
(31)
Definition 1
Let \({g_{m}}\) and \(h_{m}\), \(m=1,2,\ldots,N\), be the sequences which satisfy the inequality
$$ g_{m}\leq \Biggl(\sum_{i=1}^{m-1}g_{i}h_{i}+ \kappa \Biggr), \quad m=1,2,\ldots, N, $$
where \(g_{m}\geq 0\), \(\kappa \geq 0\), then the following discrete Gronwall inequality holds:
$$ g_{m}\leq \kappa . \exp \Biggl(\sum _{i=1}^{m-1}g_{i} \Biggr),\quad m=1,2,\ldots, N. $$
(32)