A finite difference scheme for the two-dimensional Gray-Scott equation with fractional Laplacian

Abstract

This paper studies numerical methods for the two-dimensional fractional Gray-Scott (GS) model with fractional Laplacian. A three-level linearized difference scheme for solving the fractional GS model is established. The boundedness, existence, uniqueness and convergence of the finite difference scheme are strictly proved by discrete energy analysis method. Furthermore, the convergence of the difference scheme in $l^{\infty }$-norm is of second order in space and time. Numerical experiments are implemented to verify the above theoretical results.

Appendix

We estimate the four equations $(\textrm{I})$, $(\textrm{II})$, $(\textrm{III})$, $(\textrm{IV})$ at follows.

$$\begin{aligned} (\textrm{I})= & {} \frac{4 \tau }{\mu } \sum _{k=1}^{n} \left( R^{(2), k}, (-\varDelta _{h})^{\frac{\alpha }{2}}D_{\hat{t}}e^{k}\right) ,\nonumber \\= & {} \frac{2}{\mu }\left( (R^{(2), n}, (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n+1})-(R^{(2), n}, (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n-1})+(R^{(2),1}, (-\varDelta _{h})^{\frac{\alpha }{2}}e^{2})\right) \nonumber \\{} & {} -\frac{4 \tau }{\mu } \sum _{k=2}^{n-1} \left( D_{\hat{t}}R^{(2), k}, (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k})\right) \nonumber \\\le & {} \frac{2}{\mu }\left( \Vert R^{(2), n}\Vert \cdot \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n+1}\Vert +\Vert R^{(2), n}\Vert \cdot \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n-1}\Vert +\Vert R^{(2),1}\Vert \cdot \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{2}\Vert \right) \quad \nonumber \\{} & {} +\frac{4 \tau }{\mu } \sum _{k=2}^{n-1} \Vert D_{\hat{t}}R^{(2), k}\Vert \cdot \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k}\Vert \nonumber \\\le & {} \frac{1}{4}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n+1}\Vert ^{2}+ \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n}\Vert ^{2}+ \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{1}\Vert ^{2}\right) \nonumber \\{} & {} + \tau \sum _{k=2}^{n-1} \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k}\Vert ^{2}+\frac{12+4T}{\mu ^{2}}C_{2}^{2}(\tau ^{2}+h^{2})^{2}, \end{aligned}$$

(A.1)

for $(\textrm{II})$, we have

$$\begin{aligned} (\textrm{II})= & {} -\frac{4 \tau }{\mu } \sum _{k=1}^{n} \left( G^{\bar{k}}, (-\varDelta _{h})^{\frac{\alpha }{2}}D_{\hat{t}}e^{k})\right) , \nonumber \\= & {} \frac{2}{\mu }\left( (G^{\bar{n}}, (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n+1})-(G^{\bar{n}}, (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n-1})+(G^{\bar{1}}, (-\varDelta _{h})^{\frac{\alpha }{2}}e^{2})\right) \nonumber \\{} & {} +\frac{4 \tau }{\mu } \sum _{k=2}^{n-1} \left( D_{\hat{t}}G^{\bar{k}}, (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k})\right) \nonumber \\\le & {} \frac{1}{4}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n+1}\Vert ^{2}+ \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{n}\Vert ^{2}+ \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{1}\Vert ^{2}\right) \nonumber \\{} & {} +\frac{2 \tau }{\mu } \sum _{k=2}^{n-1} \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k}\Vert ^{2}+\frac{2 \tau }{\mu } \sum _{k=2}^{n-1} \Vert D_{\hat{t}}G^{\bar{k}}\Vert ^{2}+\frac{12}{\mu ^{2}}C_{8}^{2}C_{11}^{2}(\tau ^{2}+h^{2})^{2}.\qquad \quad \end{aligned}$$

(A.2)

In the same way

$$\begin{aligned} \begin{aligned} (\textrm{III})&=\frac{4 \tau }{\mu } \sum _{k=1}^{n} \left( R^{(4), k} (-\varDelta _{h})^{\frac{\alpha }{2}}D_{\hat{t}}f^{k})\right) , \\&\le \frac{1}{4}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{n+1}\Vert ^{2}+ \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{n}\Vert ^{2}+ \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{1}\Vert ^{2}\right) \\&\quad +\tau \sum _{k=2}^{n-1} \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k}\Vert ^{2}+\frac{12+4T}{\mu ^{2}}C_{4}^{2}(\tau ^{2}+h^{2})^{2} \end{aligned} \end{aligned}$$

(A.3)

We make the following estimate for $(\textrm{IV})$

$$\begin{aligned} \begin{aligned} (\textrm{IV})&=-\frac{4 \tau }{\mu } \sum _{k=1}^{n} \left( G^{\bar{k}}, (-\varDelta _{h})^{\frac{\alpha }{2}}D_{\hat{t}}f^{k})\right) , \\&\le \frac{1}{4}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{n+1}\Vert ^{2}+ \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{n}\Vert ^{2}+ \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{1}\Vert ^{2}\right) \\&\quad +\frac{2 \tau }{\mu } \sum _{k=2}^{n-1} \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k}\Vert ^{2}+\frac{2 \tau }{\mu } \sum _{k=2}^{n-1} \Vert D_{\hat{t}}G^{\bar{k}}\Vert ^{2}+\frac{12}{\mu ^{2}}C_{8}^{2}C_{11}^{2}(\tau ^{2}+h^{2})^{2}, \end{aligned} \end{aligned}$$

(A.4)

combining with the expression of $G^{\bar{k}}$, we obtain

$$\begin{aligned} D_{\hat{t}} G^{\bar{k}}= D_{\hat{t}}\left( e^{\bar{k}}(V^{k})^{2}\right) +2 D_{\hat{t}}\left( U^{\bar{k}}f^{k}V^{k}\right) -D_{\hat{t}}\left( U^{\bar{k}}(f^{k})^{2}\right) -2D_{\hat{t}}\left( e^{\bar{k}} f^{k}V^{k}\right) +D_{\hat{t}}\left( e^{\bar{k}}( f^{k} )^{2}\right) , \end{aligned}$$

(A.5)

it can be obtained from the equation expression

$$\begin{aligned} D_{\hat{t}} e^{k}=-\mu _{u}(-\varDelta _{h})^{\frac{\alpha }{2}}e^{\bar{k}}-\sigma e^{\bar{k}} -G^{\bar{k}}+R^{(2),k}, \end{aligned}$$

(A.6)

such that

$$\begin{aligned} \Vert D_{\hat{t}} e^{k}\Vert ^{2}\le & {} 4\mu ^{2}\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{\bar{k}}\Vert ^{2}+4\sigma ^{2}\Vert e^{\bar{k}}\Vert ^{2} +4\Vert G^{\bar{k}}\Vert ^{2}+4|R^{(2),k}|^{2},\nonumber \\\le & {} 2\mu ^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k+1}\Vert ^{2}+\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k-1}\Vert ^{2}\right) \nonumber \\{} & {} +2\sigma ^{2}\left( \Vert e^{k+1}\Vert ^{2}+\Vert e^{k-1}\Vert ^{2}\right) +C_{21}^{2}(\tau ^{2}+h^{2})^{2}, \end{aligned}$$

(A.7)

where $C_{21}^{2}=4(C^{2}_{11}+C_{8}^{2}C_{11}^{2})$, similarly, we have

$$\begin{aligned} \Vert D_{\hat{t}} f^{k}\Vert ^{2}\le & {} 2\mu ^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k+1}\Vert ^{2}+\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k-1}\Vert ^{2}\right) \nonumber \\{} & {} +2(\sigma +\kappa )^{2}\left( \Vert f^{k+1}\Vert ^{2}+\Vert f^{k-1}\Vert ^{2}\right) +C_{22}^{2}(\tau ^{2}+h^{2})^{2}, \end{aligned}$$

(A.8)

where $C_{22}^{2}=4(C^{2}_{13}+C_{8}^{2}C_{11}^{2})$.

To simplify the calculation, the paper assumes $\Vert e^{k+1}\Vert ^{2}+\Vert e^{k-1}\Vert ^{2}\le 1,\Vert f^{k+1}\Vert ^{2}+\Vert f^{k-1}\Vert ^{2}\le 1$, which can be obtained

$$\begin{aligned} \begin{aligned} \Vert D_{\hat{t}} e^{\bar{k}}\Vert ^{2}&=\frac{1}{4} \Vert D_{\hat{t}} e^{k+1}+D_{\hat{t}} e^{k-1}\Vert ^{2}\le \frac{1}{2}\left( \Vert D_{\hat{t}} e^{k+1}\Vert ^{2}+\Vert D_{\hat{t}} e^{k-1}\Vert ^{2}\right) \\&\le 2\mu ^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k-2}\Vert ^{2}+2\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k}\Vert ^{2}+\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k+2}\Vert ^{2}\right) \\&\quad +2\sigma ^{2}(\Vert e^{k-2}\Vert ^{2}+\Vert e^{k}\Vert ^{2}+\Vert e^{k+2}\Vert ^{2})+2C_{21}^{2}(\tau ^{2}+h^{2})^{2}, \end{aligned} \end{aligned}$$

(A.9)

Denote $C^{2}_{0}=\max \{\Vert U^{k}\Vert ^{2},\Vert V^{k}\Vert ^{2},\Vert D_{\hat{t}}(V^{k})\Vert ^{2},\Vert D_{\hat{t}}(U^{k})\Vert ^{2}\}$, $C_{s}^{2}=24C^{4}_{0}\left( C^{2}_{29}+\right.$ $\left. C^{2}_{20}+C^{2}_{11}+C^{2}_{17}C^{2}_{20}+C^{2}_{13}+C^{2}_{17}C^{2}_{20}\right)$, so, we have

$$\begin{aligned} s_{1}= & {} \Vert D_{\hat{t}}\left( e^{\bar{k}}(V^{k})^{2}\right) \Vert ^{2}\nonumber \\= & {} \Vert D_{\hat{t}}(e^{\bar{k}})(V^{k+1})^{2}+D_{\hat{t}}(V^{k})e^{\overline{k-1}}V^{k+1}+D_{\hat{t}}(V^{k})e^{\overline{k-1}}V^{k-1}\Vert ^{2}\nonumber \\\le & {} 3C_{0}^{4}\Vert D_{\hat{t}}(e^{\bar{k}})\Vert ^{2}+6C_{0}^{4}\Vert (e^{\overline{k-1}})\Vert ^{2}\nonumber \\\le & {} C_{s}^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k-2}\Vert ^{2}\!+\!\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k}\Vert ^{2}\!+\!\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k+2}\Vert ^{2} +\Vert e^{k-2}\Vert ^{2}+\Vert e^{k}\Vert ^{2}+\Vert e^{k+2}\Vert ^{2}\right) \nonumber \\{} & {} +C_{s}^{2}(\tau ^{2}+h^{2})^{2}. \end{aligned}$$

(A.10)

$$\begin{aligned} s_{2}= & {} \Vert D_{\hat{t}}\left( U^{\bar{k}}f^{k}V^{k}\right) \Vert ^{2}\nonumber \\= & {} \Vert D_{\hat{t}}(U^{\bar{k}})f^{k+1}V^{k+1}+D_{\hat{t}}(f^{k})U^{\overline{k-1}}V^{k+1}+D_{\hat{t}}(V^{k})U^{\overline{k-1}}f^{k-1}\Vert ^{2}\nonumber \\\le & {} 3C_{0}^{4}\Vert D_{\hat{t}}(f^{k})\Vert ^{2}+3C_{0}^{4}\Vert f^{k+1}\Vert ^{2}++3C_{0}^{4}\Vert f^{k-1}\Vert ^{2}\nonumber \\\le & {} \!C_{s}^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k-1}\Vert ^{2}\!\!+\!\!\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k+1}\Vert ^{2}\! \!+\!\!\Vert f^{k-1}\Vert ^{2}\!\!+\!\!\Vert f^{k+1}\Vert ^{2}\!\!+\!\!(\tau ^{2}\!+\!h^{2})^{2}\right) . \end{aligned}$$

(A.11)

$$\begin{aligned} s_{3}= & {} \Vert D_{\hat{t}}\left( U^{\bar{k}}(f^{k})^{2}\right) \Vert ^{2}\nonumber \\= & {} \Vert D_{\hat{t}}(U^{\bar{k}})(f^{k+1})^{2}+D_{\hat{t}}(f^{k})U^{\overline{k-1}}f^{k+1}+D_{\hat{t}}(f^{k})U^{\overline{k-1}}f^{k-1}\Vert ^{2}\nonumber \\\le & {} 3C_{0}^{4}\Vert f^{k+1}\Vert ^{2}+3C_{0}^{4}\Vert D_{\hat{t}}(f^{k})(f^{k+1}+f^{k-1})\Vert ^{2}\nonumber \\\le & {} 3C_{0}^{4}\Vert f^{k+1}\Vert ^{2}+3C_{0}^{4}\Vert D_{\hat{t}}f^{k}\Vert ^{2}\cdot \Vert f^{k+1}+f^{k-1}\Vert ^{2}\nonumber \\\le & {} 3C_{0}^{4}\Vert f^{k+1}\Vert ^{2}+6C_{0}^{4}\Vert D_{\hat{t}}f^{k}\Vert ^{2}\cdot \left( \Vert f^{k+1}\Vert ^{2}+\Vert f^{k-1}\Vert ^{2}\right) \nonumber \\\le & {} C_{s}^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k-1}\Vert ^{2}\!\!+\!\!\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k+1}\Vert ^{2}\!\! +\!\!\Vert f^{k-1}\Vert ^{2}\!\!+\!\!\Vert f^{k+1}\Vert ^{2}\!+\!(\tau ^{2}\!\!+\!\!h^{2})^{2}\right) . \end{aligned}$$

(A.12)

$$\begin{aligned} s_{4}= & {} \Vert D_{\hat{t}}\left( e^{\bar{k}}f^{k}V^{k}\right) \Vert ^{2}\nonumber \\= & {} \Vert D_{\hat{t}}(e^{\bar{k}})f^{k+1}V^{k+1}+D_{\hat{t}}(f^{k})e^{\overline{k-1}}V^{k+1}+D_{\hat{t}}(V^{k})e^{\overline{k-1}}f^{k-1}\Vert ^{2}\nonumber \\\le & {} 6C_{0}^{4}\left( \Vert f^{k+1}\Vert ^{2}+\Vert D_{\hat{t}}e^{\bar{k}}\Vert ^{2}\right) +6C_{0}^{4}\left( \Vert D_{\hat{t}}f^{k}\Vert ^{2}+\Vert e^{\bar{k-1}}\Vert ^{2}\right) \nonumber \\{} & {} +6C_{0}^{4}\left( \Vert e^{k-2}\Vert ^{2}+\Vert e^{k}\Vert ^{2}+\Vert f^{k-1}\Vert ^{2}\right) \nonumber \\\le & {} C_{s}^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k-1}\Vert ^{2}+\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k+1}\Vert ^{2} +\Vert f^{k-1}\Vert ^{2}+\Vert f^{k+1}\Vert ^{2}\right) \nonumber \\{} & {} +C_{s}^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k-2}\Vert ^{2}+\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k}\Vert ^{2}+\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k+2}\Vert ^{2}\right. \nonumber \\{} & {} \left. +\Vert e^{k-2}\Vert ^{2}+\Vert e^{k}\Vert ^{2}+\Vert e^{k+2}\Vert ^{2}\right) \nonumber \\{} & {} +C_{s}^{2}(\tau ^{2}+h^{2})^{2}. \end{aligned}$$

(A.13)

$$\begin{aligned} s_{5}= & {} \Vert D_{\hat{t}}\left( e^{\bar{k}}(f^{k})^{2}\right) \Vert ^{2}\nonumber \\= & {} \Vert D_{\hat{t}}(e^{\bar{k}})(f^{k+1})^{2}+D_{\hat{t}}(f^{k})e^{\overline{k-1}}f^{k+1}+D_{\hat{t}}(f^{k})e^{\overline{k-1}}f^{k-1}\Vert ^{2}\nonumber \\\le & {} 6C_{0}^{4}\left( \Vert f^{k+1}\Vert ^{2}+\Vert D_{\hat{t}}e^{\bar{k}}\Vert ^{2}\right) +3C_{0}^{4}\Vert D_{\hat{t}}(f^{k})e^{\bar{k-1}}\Vert ^{2}\cdot \Vert f^{k+1}+f^{k-1}\Vert ^{2}\nonumber \\\le & {} 6C_{0}^{4}\left( \Vert f^{k+1}\Vert ^{2}\!+\!\Vert D_{\hat{t}}e^{\bar{k}}\Vert ^{2}\right) \!+\!3C_{0}^{4}\Vert D_{\hat{t}}f^{k}\Vert ^{2}\cdot \left( \Vert e^{k+1}\Vert ^{2}\!+\!\Vert e^{k-1}\Vert ^{2}\right) \cdot \left( \Vert f^{k+1}\Vert ^{2}\!+\!\Vert f^{k-1}\Vert ^{2}\right) \nonumber \\\le & {} C_{s}^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k-1}\Vert ^{2}+\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}f^{k+1}\Vert ^{2} +\Vert f^{k-1}\Vert ^{2}+\Vert f^{k+1}\Vert ^{2}\right) \nonumber \\{} & {} +C_{s}^{2}\left( \Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k-2}\Vert ^{2}\!+\!\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k}\Vert ^{2}\!+\!\Vert (-\varDelta _{h})^{\frac{\alpha }{2}}e^{k+2}\Vert ^{2} \!\!+\!\!\Vert e^{k-2}\Vert ^{2}\!+\!\Vert e^{k}\Vert ^{2}\!+\!\Vert e^{k+2}\Vert ^{2}\right) \nonumber \\{} & {} +C_{s}^{2}(\tau ^{2}+h^{2})^{2}. \end{aligned}$$

(A.14)

Combining the above five estimators, it can be obtained

$$\begin{aligned} \frac{2\tau }{\mu } \sum _{k=2}^{n}\left\| D_{\hat{t}} G^{\bar{k}}\right\| ^{2}\le & {} \frac{6}{\mu } C_{s}^{2} \tau \sum _{k=2}^{n-1}\left( \left\| e^{k-2}\right\| ^{2}+\left\| e^{k}\right\| ^{2}+\left\| e^{k+2}\right\| ^{2}\right) \nonumber \\{} & {} +\frac{6}{\mu } C_{s}^{2} \tau \sum _{k=2}^{n-1}\left( \left\| \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} e^{k-2}\right\| ^{2}+\left\| \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} e^{k}\right\| ^{2}+\left\| \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} e^{k+2}\right\| ^{2}\right) \nonumber \\{} & {} + \frac{8}{\mu } C_{s}^{2} \tau \sum _{k=2}^{n-1}\left( \left\| f^{k-1}\right\| ^{2}+\left\| f^{k+1}\right\| ^{2}\right) \nonumber \\{} & {} +\frac{8}{\mu } C_{s}^{2} \tau \sum _{k=2}^{n-1}\left( \left\| \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} f^{k-1}\right\| ^{2}+\left\| \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} f^{k+1}\right\| ^{2}\right) \nonumber \\{} & {} +\frac{10C_{s}^{2}T\tau }{\mu }(\tau ^{2}+h^{2})^{2}\nonumber \\\le & {} \frac{C_{16}^{2}}{2}\tau \sum _{k=1}^{n}\left( \left\| \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} e^{k}\right\| ^{2}+\left\| \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} f^{k}\right\| ^{2}\right) \nonumber \\{} & {} +C_{16}^{2}\tau \left\| \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} e^{n+1}\right\| ^{2}+\frac{C_{17}^{2}}{2}(\tau ^{2}+h^{2})^{2}. \end{aligned}$$

(A.15)

where $C_{16}^{2}=\frac{16}{\mu } C_{s}^{2}$, $C_{17}^{2}=\frac{10C_{s}^{2}T }{\mu }$, we obtain

$$\begin{aligned} \begin{aligned} (\textrm{II}) +(\textrm{IV})\le&\left( \frac{1}{2}+C_{16}^{2} \tau \right) E^{n+1}+\frac{2 \tau }{\mu } \sum _{k=2}^{n}\left( \Vert \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} e^{k}\Vert ^{2}+\Vert \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} f^{k}\Vert ^{2}\right) \\&+C_{16}^{2} \tau \sum _{k=1}^{n}\left( \Vert \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} e^{k}\Vert ^{2}+\Vert \left( -\varDelta _{h}\right) ^{\frac{\alpha }{2}} f^{k}\Vert ^{2}\right) +C_{17}^{2}\left( \tau ^{2}+h^{2}\right) ^{2} \end{aligned} \end{aligned}$$

(A.16)