Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg-Landau equation

Abstract

In this paper, we propose a three-level linearized implicit difference scheme for the two-dimensional spatial fractional nonlinear complex Ginzburg-Landau equation. We prove that the difference scheme is stable and convergent under mild conditions. The optimal convergence order $\mathcal {O}(\tau ^{2}+{h_{x}^{2}}+{h_{y}^{2}})$ is obtained in the pointwise sense by developing a new two-dimensional fractional Sobolev imbedding inequality based on the work in Kirkpatrick et al. (Commun. Math. Phys. 317, 563–591 2013), an energy argument and careful attention to the nonlinear term. Numerical examples are presented to verify the validity of the theoretical results for different choices of the fractional orders α and β.

Appendices

Appendix: A

Define $\hat {m}_{1}=M_{x}-1$, $\hat {m}_{2}=M_{y}-1$, and $\hat {m}=\hat {m}_{1}\hat {m}_{2}$. Let v and u be the $\hat {m}$-dimensional vectors defined by

$$v=\left[v_{1,1},\ldots,v_{\hat{m}_{1},1},v_{1,2}, v_{2,2}, \cdots, v_{\hat m_{1},2},\ldots, v_{1,\hat{m}_{2}},\ldots,v_{\hat{m}_{1},\hat{m}_{2}}\right]^{\intercal}$$

and

$$u=\left[u_{1,1},\ldots,u_{\hat{m}_{1},1},u_{1,2}, u_{2,2}, {\cdots} , u_{\hat m_{1},2},\ldots, u_{1,\hat{m}_{2}},\ldots,u_{\hat{m}_{1},\hat{m}_{2}}\right]^{\intercal}.$$

By Property 2.1 in [6], it is easy to know that

$$ \ G_{\mu,m}=\left[\begin{array}{cccccc} g_{0}^{(\mu)}& g_{-1}^{(\mu)}&g_{-2}^{(\mu)}&\cdots&g_{-m+2}^{(\mu)}&g_{-m+1}^{(\mu)}\\ g_{1}^{(\mu)}& g_{0}^{(\mu)}&g_{-1}^{(\mu)}&\ldots&\ldots&g_{-m+2}^{(\mu)}\\ g_{2}^{(\mu)}&g_{1}^{(\mu)}&g_{0}^{(\mu)}&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&g_{-2}^{(\mu)}\\ g_{m-2}^{(\mu)}&\ddots&\ddots&\ddots&g_{0}^{(\mu)}&g_{-1}^{(\mu)}\\ g_{m-1}^{(\mu)}&g_{m-2}^{(\mu)}&\cdots&g_{2}^{(\mu)}&g_{1}^{(\mu)}&g_{0}^{(\mu)} \end{array}\right] $$

is a symmetric positive definite Toeplitz matrix with $(\mu ,m)=(\alpha ,\hat {m}_{1})$ or $(\beta ,\hat {m}_{2})$. Then, the matrix form of the operator $-(\delta _{x}^{\alpha }+\delta _{y}^{\beta })v_{ij}$, (x_i, y_j) ∈Ω_h is Av, here

$$ A =I_{\hat{m}_{2}}\otimes \left( \frac{1}{h_{x}^{\alpha}}G_{\alpha,\hat{m}_{1}}\right)+\left( \frac{1}{h_{y}^{\beta}}G_{\beta,\hat{m}_{2}}\right)\otimes I_{\hat{m}_{1}}, $$

Table 4 The maximum norm errors versus spatial grid size reduction and convergence orders of the difference scheme (3.12)–(3.15) in space with fixed τ = 1/16, T = 1

Fig. 3

The numerical solutions and corresponding contour profiles. The parameters are taken with T = 0.1, 0.4, 1.6, 3.2 from left to right and the grid sizes τ = 0.01, h = 5/16, α = β = 1.5

Fig. 4

The numerical solutions at the fixed time T = 2 and corresponding contour profiles. The parameters are taken with (α,β) = (1.1, 1.1), (1.2, 1.8), (1.5, 1.5), (1.9, 1.9) from left to right and the grid sizes M = 64, N = 100

where ⊗ denotes the Kronecker product and I_m the identity matrix of the order m. Furthermore, we know that A is a symmetric positive definite matrix by the properties of the Kronecker product. Therefore, there exists a symmetric positive matrix denoted by Λ such that

$$ v^{\intercal}A v = v^{\intercal}{\varLambda}^{2}v =({\varLambda} v)^{\intercal} ({\varLambda} v). $$

Define the corresponding difference operator Λ^α,β as ${\varLambda }^{\alpha ,\beta }v_{i,j}=({\varLambda } v)_{(j-1)\hat {m}_{1}+i}$, then we have

$$ (-(\delta_{x}^{\alpha}+\delta_{y}^{\beta}) u,v) = u^{\intercal} A v =u^{\intercal} {\varLambda}^{2}v= ({\varLambda} u)^{\intercal} {\varLambda} v = ({\varLambda}^{\alpha,\beta}u, {\varLambda}^{\alpha,\beta} v). $$

Appendix: B

The following lemma plays an important role in deriving the local truncation errors $R_{ij}^{n}$ and △_tR_ij.

Lemma B.7

Let g(t) ∈ C³([t_n− 1, t_n+ 1]); then

$$ \begin{array}{@{}rcl@{}} && \frac{g(t_{n+1})-g(t_{n-1})}{2\tau} = g^{\prime}(t_{n}) +\frac{\tau^{2}}{4} {{\int}_{0}^{1}} [g^{(3)}(t_{n}+s\tau)+g^{(3)}(t_{n}-s\tau)](1-s)^{2}ds, \end{array} $$

(B.1)

$$ \begin{array}{@{}rcl@{}} && \frac{g(t_{n+1})+g(t_{n-1})}{2} = g(t_{n}) +\frac{\tau^{2}}{2} {{\int}_{0}^{1}} [g^{(2)}(t_{n}+s\tau)+g^{(2)}(t_{n}-s\tau)](1-s)ds. \end{array} $$

(B.2)

Proof

Based on the Taylor expansion with the integral remainder

$$ \begin{array}{@{}rcl@{}} g(t_{n}+\tau) = \sum\limits_{i=0}^{k} \frac{\tau^{i}}{i!}g^{(i)}(t_{n})+\frac{\tau^{k+1}}{k!}{{\int}_{0}^{1}} g^{(k+1)}(t_{n}+s\tau)(1-s)^{k} ds,\quad g\in C^{k+1}, \end{array} $$

the results follow. □

Lemma B.8

Suppose $\partial _{t} q(\cdot ,t_{n}) \in {\mathscr{C}}^{2+\alpha }(\mathbb {R})$. Then, it holds

$$ \begin{array}{@{}rcl@{}} \left( -h_{x}^{-\alpha}\delta_{x}^{\alpha}\triangle_{t} q(x,t_{n})\right) - \partial_{x}^{\alpha}\triangle_{t} q(x,t_{n}) = \mathcal{O}({h_{x}^{2}}) \end{array} $$

(B.3)

uniformly for $x\in \mathbb {R}$ and t_n ∈ [0,T].

Proof

Let p(x,t) = ∂_tq(x,t). A straightforward application of Lemma 2.2 gives

$$ \begin{array}{@{}rcl@{}} -h_{x}^{-\alpha} \delta_{x}^{\alpha} p(x,t) = \partial_{x}^{\alpha}p(x,t) + \mathcal{O}({h_{x}^{2}}), \end{array} $$

or

$$ \begin{array}{@{}rcl@{}} -h_{x}^{-\alpha} \delta_{x}^{\alpha} \partial_{t} q(x,t) = \partial_{x}^{\alpha}\partial_{t} q(x,t) + \mathcal{O}({h_{x}^{2}}). \end{array} $$

(B.4)

Let t = t_n− 1 + 2τ𝜃 in (B.4). Integrating the result from 0 to 1 for 𝜃, we have

$$ \begin{array}{@{}rcl@{}} -h_{x}^{-\alpha} \delta_{x}^{\alpha} {{\int}_{0}^{1}} \partial_{t} q(x,t_{n-1}+2\tau\theta)\mathrm{d}\theta = \partial_{x}^{\alpha} {{\int}_{0}^{1}} \partial_{t} q(x,t_{n-1}+2\theta \tau)\mathrm{d}\theta + {{\int}_{0}^{1}} \mathcal{O}({h_{x}^{2}})\mathrm{d}\theta. \end{array} $$

(B.5)

Notice that

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{1}} \partial_{t} q(x,t_{n-1}+2\tau \theta)\mathrm{d}\theta = \triangle_{t} q(x,t_{n}), \end{array} $$

(B.5) becomes

$$ \begin{array}{@{}rcl@{}} -h_{x}^{-\alpha}\delta_{x}^{\alpha} \triangle_{t} q(x,t_{n}) = \partial_{x}^{\alpha} \triangle_{t} q(x,t_{n}) + \mathcal{O}({h_{x}^{2}}), \end{array} $$

which implies (B.3). □

Next, we derive the local truncation errors $R_{ij}^{n}$ in (3.8) and $\triangle _{t} R_{ij}^{n}$ in (3.9). Denote

$$ \begin{array}{@{}rcl@{}} & (R^{(1)})_{ij}^{n} = \frac{\tau^{2}}{4} {{\int}_{0}^{1}}\left[\partial_{ttt}u(x_{i},y_{j},t_{n}+s\tau)+ \partial_{ttt}u(x_{i},y_{j},t_{n}-s\tau)\right](1-s)^{2}ds,\\ & (R^{(2)})_{ij}^{n} = \frac{\tau^{2}}{2} {{\int}_{0}^{1}}\left[\partial_{tt}u(x_{i},y_{j},t_{n}+s\tau)+ \partial_{tt}u(x_{i},y_{j},t_{n}-s\tau)\right](1-s)ds. \end{array} $$

Using (B.1) and (B.2), we have

$$ \begin{array}{@{}rcl@{}} &&\triangle_{t} U_{ij}^{n} - \partial_{t} u(x_{i},y_{j},t_{n}) = (R^{(1)})_{ij}^{n}, \end{array} $$

(B.6)

$$ \begin{array}{@{}rcl@{}} &&|U_{ij}^{n}|^{2} U_{ij}^{\bar{n}} - |u(x_{i},y_{j},t_{n})|u(x_{i},y_{j},t_{n}) = |U_{ij}^{n}|^{2} \cdot (R^{(2)})_{ij}^{n}, \end{array} $$

(B.7)

Using (B.2) in Lemmas B.7 and 2.2, we have

$$ \begin{array}{@{}rcl@{}} &&\delta_{x}^{\alpha} U_{ij}^{\bar{n}} - \partial_{x}^{\alpha} u(x_{i},y_{j},t_{n})= \partial_{x}^{\alpha}(R^{(2)})_{ij}^{n} + \mathcal{O}({h_{x}^{2}}), \end{array} $$

(B.8)

$$ \begin{array}{@{}rcl@{}} &&\delta_{y}^{\beta} U_{ij}^{\bar{n}} - \partial_{y}^{\beta} u(x_{i},y_{j},t_{n}) = \partial_{y}^{\beta}(R^{(2)})_{ij}^{n} + \mathcal{O}({h_{y}^{2}}). \end{array} $$

(B.9)

Subtracting (3.5) from (3.13) and noticing (B.6)–(B.9), we have

$$ R_{ij}^{n} = (R^{(1)})_{ij}^{n}-(\nu+\textbf{i}\eta)((\partial_{x}^{\alpha}+\partial_{y}^{\beta})(R^{(2)})_{ij}^{n}) + (\kappa+ \textbf{i} \zeta)|U_{ij}^{n}|^{2} \cdot (R^{(2)})_{ij}^{n} -\gamma (R^{(2)})_{ij}^{n} + \mathcal{O}({h_{x}^{2}} + {h_{y}^{2}}). $$

Thus, the local truncation errors $R_{ij}^{n}$ in (3.8) holds. Furthermore, we have

$$ \begin{array}{@{}rcl@{}} \triangle_{t}(R^{(1)})_{ij}^{n} &= & \frac{1}{2\tau}((R^{(1)})_{ij}^{n+1}-(R^{(1)})_{ij}^{n-1})\\ &= & \frac{\tau}{8}{{\int}_{0}^{1}} \Big[\partial_{ttt} u(x_{i},y_{j},t_{n+1}+s\tau) + \partial_{ttt} u(x_{i},y_{j},t_{n+1}-s\tau) \Big](1-s)^{2}ds\\ & & -\frac{\tau}{8}{{\int}_{0}^{1}} \Big[\partial_{ttt} u(x_{i},y_{j},t_{n-1}+s\tau) + \partial_{ttt} u(x_{i},y_{j},t_{n-1}-s\tau) \Big](1-s)^{2}ds\\ &= &\frac{\tau}{8} {{\int}_{0}^{1}} \Big[\partial_{ttt} u(x_{i},y_{j},t_{n+1}+s\tau) - \partial_{ttt} u(x_{i},y_{j},t_{n-1}+s\tau)\Big](1-s)^{2}ds\\ & & + \frac{\tau}{8} {{\int}_{0}^{1}} \Big[\partial_{ttt} u(x_{i},y_{j},t_{n+1}-s\tau) -\partial_{ttt} u(x_{i},y_{j},t_{n-1}-s\tau)\Big](1-s)^{2}ds\\ &= & \frac{\tau^{2}}{4}{{\int}_{0}^{1}} {{\int}_{0}^{1}} \Big[\partial_{tttt} u(x_{i},y_{j},t_{n-1} + (2\mu+s)\tau) + \partial_{tttt} u(x_{i},y_{j},t_{n-1} + (2\mu - s)\tau)\Big] (1-s)^{2}d\mu ds. \end{array} $$

(B.10)

Therefore,

$$ | \triangle_{t} (R^{(1)})_{ij}^{n}| \leqslant c\tau^{2}, $$

where c is a positive constant. Similarly, we have

$$ | \triangle_{t} (R^{(2)})_{ij}^{n}| \leqslant c\tau^{2},\quad |\triangle_{t}(U^{n} (R^{2})_{ij}^{n})| \leqslant c\tau^{2}. $$

Combining (B.3) in Lemma B.8 and (B.10), we have

$$ | \triangle_{t} (\partial_{x}^{\alpha}(R^{(2)})_{ij}^{n})| = \mathcal{O}(\tau^{2}+{h_{x}^{2}}),\quad | \triangle_{t} (\partial_{y}^{\beta}(R^{2})_{ij}^{n})| = \mathcal{O}(\tau^{2}+{h_{y}^{2}}). $$

Thus, the local truncation errors $\triangle _{t} R_{ij}^{n}$ in (3.9) holds.

Appendix: C

With the help of Lemma 3.4 and (3.42), the seven terms in (3.41) can be bounded, respectively, as follows

$$ \begin{array}{@{}rcl@{}} \|J_{1}\|&=& \|\triangle_{t}(U^{k})^{*}e^{\bar{k}}U^{\hat{k}}+\triangle_{t}e^{k} (U^{\bar{k}})^{*}U^{\hat{k}}+\triangle_{t}U^{\bar{k}}(U^{\bar{k}})^{*}e^{\bar{k}}\| \\ &\leqslant& \|u_{t}\|_{{\infty}}\cdot\|u\|_{{\infty}}\cdot\|e^{\bar{k}}\|+\|u\|^{2}_{{\infty}}\cdot\|\triangle_{t}e^{k}\| +\|u_{t}\|_{{\infty}}\cdot\|u\|_{{\infty}}\cdot\|e^{\bar{k}}\|\\ &\leqslant& c_{7}(\|e^{k+1}\|+\|e^{k-1}\|+\|R^{k}\|+\|p^{k}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+1}\|+\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-1}\|), \end{array} $$

$$ \begin{array}{@{}rcl@{}} \|J_{2}\|&=& \|\triangle_{t}U^{k} U^{\hat{n}}(e^{\bar{k}})^{*}+\triangle_{t}(e^{k})^{*} U^{\bar{k}}U^{\hat{k}}+\triangle_{t}U^{\bar{k}}U^{\bar{k}}(e^{\bar{k}})^{*}\| \\ &\leqslant& \|u_{t}\|_{{\infty}}\cdot\|u\|_{{\infty}}\cdot\|e^{\bar{k}}\|+\|u\|^{2}_{{\infty}}\cdot\|\triangle_{t}e^{k}\| +\|u_{t}\|_{{\infty}}\cdot\|u\|_{{\infty}}\cdot\|e^{\bar{k}}\| \\ &\leqslant& c_{7}(\|e^{k+1}\|+\|e^{k-1}\|+\|R^{k}\|+\|p^{k}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+1}\|+\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-1}\|), \end{array} $$

$$ \begin{array}{@{}rcl@{}} \|J_{3}\|&=& \|\triangle_{t}U^{k} (U^{k})^{*}e^{\hat{k}}+\triangle_{t}(U^{k})^{*} U^{\bar{k}}e^{\hat{k}} +U^{\bar{k}}(U^{\bar{k}})^{*}\triangle_{t}e^{\bar{k}}\| \\ &\leqslant& \|u_{t}\|_{{\infty}}\cdot\|u\|_{{\infty}}\cdot\|e^{\hat{k}}\| +\|u_{t}\|_{{\infty}}\cdot\|u\|_{{\infty}}\cdot\|e^{\hat{k}}\| +\|u\|^{2}_{{\infty}}\cdot\|\triangle_{t}e^{\bar{k}}\| \\ &\leqslant& c_{7}(\|e^{k+2}\|+\|e^{k}\|+\|e^{k-2}\| +\|R^{k+1}\|+\|R^{k-1}\| \\ && +\|p^{k+1}\|+\|p^{k-1}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+2}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-2}\|), \end{array} $$

$$ \begin{array}{@{}rcl@{}} \|J_{4}\|&=& \|\triangle_{t}U^{k} (e^{\bar{k}})^{*}e^{\hat{k}}+\triangle_{t}(e^{k})^{*} U^{\bar{k}}e^{\hat{k}} +U^{\bar{k}}(e^{\bar{k}})^{*}\triangle_{t}e^{\bar{k}}\|\\ &\leqslant& \|u_{t}\|_{{\infty}}\cdot\|e^{\bar{k}} \|\cdot\|e^{\hat{k}}\| +\|u\|_{{\infty}}\cdot\|e^{\hat{k}}\|\cdot\|\triangle_{t}e^{k}\| +\|u\|_{{\infty}}\cdot\|e^{\bar{k}}\|\cdot\|\triangle_{t}e^{\bar{k}}\| \\ &\leqslant& c_{7}(\|e^{\hat{k}}\|+\|\triangle_{t}e^{k}\|+\|\triangle_{t}e^{\bar{k}}\|) \\ &\leqslant& c_{7}(\|e^{k+2}\|+\|e^{k+1}\|+\|e^{k}\|+\|e^{k-1}\|+\|e^{k-2}\| \\ && +\|R^{k+1}\|+\|R^{k}\|+\|R^{k-1}\|+\|p^{k+1}\|+\|p^{k}\|+\|p^{k-1}\| \\ && +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+2}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+1}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k}\|\\ && +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-1}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-2}\|), \end{array} $$

$$ \begin{array}{@{}rcl@{}} \|J_{5}\|&=& \|\triangle_{t}(U^{k})^{*}e^{\bar{k}}e^{\hat{k}}+\triangle_{t}e^{k} (U^{\bar{k}})^{*}e^{\hat{k}} +(U^{\bar{k}})^{*}e^{\bar{k}}\triangle_{t}e^{\bar{k}}\|\\ &\leqslant& \|u_{t}\|_{{\infty}}\cdot\|e^{\bar{k}}\|\cdot\|e^{\hat{k}}\| +\|u\|_{{\infty}}\cdot\|e^{\hat{k}}\|\cdot\|\triangle_{t}e^{k}\| +\|u\|_{{\infty}}\cdot\|e^{\bar{k}}\|\cdot\|\triangle_{t}e^{\bar{k}}\| \\ &\leqslant& c_{7}(\|e^{\hat{k}}\|+\|\triangle_{t}e^{k}\|+\|\triangle_{t}e^{\bar{k}}\|) \\ &\leqslant& c_{7}(\|e^{k+2}\|+\|e^{k+1}\|+\|e^{k}\|+\|e^{k-1}\|+\|e^{k-2}\| \\ && +\|R^{k+1}\|+\|R^{k}\|+\|R^{k-1}\| +\|p^{k+1}\|+\|p^{k}\|+\|p^{k-1}\| \\ && +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+2}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+1}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k}\|\\ && +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-1}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-2}\|), \end{array} $$

$$ \begin{array}{@{}rcl@{}} \|J_{6}\|&=& \|\triangle_{t}e^{k} (e^{\bar{k}})^{*}U^{\hat{k}}+\triangle_{t}(e^{k})^{*} e^{\bar{k}}U^{\hat{k}} +\triangle_{t}U^{\bar{k}}e^{\bar{k}}(e^{\bar{k}})^{*}\| \\ &\leqslant& 2\|u\|_{{\infty}}\cdot\|e^{\bar{k}}\|\cdot\|\triangle_{t}e^{{k}}\| +\frac{1}{2}\|u_{t}\|_{{\infty}}\cdot\|e^{\bar{k}}\|\cdot(\|e^{k+1}\|+\|e^{k-1}\|) \\ &\leqslant& c_{7}(\|e^{k+1}\|+\|e^{k-1}\|+\|\triangle_{t}e^{k}\|) \\ &\leqslant& c_{7}(\|e^{k+1}\|+\|e^{k-1}\|+\|R^{k}\|+\|p^{k}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+1}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-1}\|), \end{array} $$

$$ \begin{array}{@{}rcl@{}} \|J_{7}\|&=& \|\triangle_{t}e^{k} (e^{\bar{k}})^{*}e^{\hat{k}}+\triangle_{t}(e^{k})^{*} e^{\bar{k}}e^{\hat{k}} +e^{\bar{k}}(e^{\bar{k}})^{*}\triangle_{t}e^{\bar{k}}\| \\ &\leqslant& 2\|e^{\bar{k}}\|\cdot\|e^{\hat{k}}\|\cdot\|\triangle_{t}e^{k}\| +\|e^{\bar{k}}\|^{2}\cdot\|\triangle_{t}e^{\bar{k}}\| \leqslant c_{7}(\|\triangle_{t}e^{k}\|+\|\triangle_{t}e^{\bar{k}}\|) \\ &\leqslant& c_{7}(\|e^{k+2}\|+\|e^{k+1}\|+\|e^{k}\|+\|e^{k-1}\|+\|e^{k-2}\|+\|R^{k+1}\|+\|R^{k}\|+\|R^{k-1}\| \\ && +\|p^{k+1}\|+\|p^{k}\|+\|p^{k-1}\|+\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+2}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k+1}\| \\ && +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-1}\| +\|(\delta_{x}^{\alpha}+\delta_{y}^{\beta})e^{k-2}\|). \end{array} $$