A Crank–Nicolson Compact Difference Method for Time-Fractional Damped Plate Vibration Equations

Abstract

This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. We discretize the space derivative via compact difference and approximate the time-integer-order derivative and fraction-order derivative via central difference and L1 interpolation, respectively, to obtain the compact difference formats with fourth-order space precision and $3-\alpha$($1<\alpha <2$)-order time precision. We apply the energy method to analyze the stability and convergence of this difference format. We provide numerical cases, which not only validate the convergence order and feasibility of the given difference format, but also simulate the influence of the damping coefficient on the amplitude of plate vibration.

1. Introduction

Viscoelastic materials possess both the mechanical properties of an elastic solid and a Newtonian fluid and are effective for mitigating vibration by energy dissipation [1,2]. The memory of viscoelastic damping materials and the fractional power frequency dependence of the viscoelastic damping factor can be described by time-fractional derivatives with good accuracy [3,4,5]. The fourth-order differential equations with time-fractional derivative can better describe and solve damped vibration problems in a viscoelastic medium, such as problems dealing with vibration reduction in bridges, aircrafts and spacecrafts, noise reduction, etc. The time-fractional damped plate vibration equations are difficult to solve analytically because they are fourth-order differential equations with time-fractional derivatives. Thus, the vibration models with fraction-order derivatives as the damping factor and their numerical solutions have gained increasing attention from researchers around the world [6,7,8], similar to various other types of studies of fractional problems [9,10,11].

The compact difference method, which can further accelerate the convergence of the general difference methods following a similar idea as the efficient numerical methods developed in [12] accelerate most existing approaches, is a high-accuracy numerical method that uses relatively fewer nodes arranged in a compact way. It handles boundary conditions efficiently and is often employed to solve second-order differential equations. Refs. [13,14] studied the compact difference method for the wave equations. Refs. [15,16,17,18,19] constructed the compact difference formats for the time fractional sub-diffusion equation, the fraction sub-diffusion equations and the time fractional convection-diffusion equation of groundwater pollution problems. For the fourth-order nonlinear elliptic boundary value problems, ref. [20] converted fourth-order differential equations into second-order differential equation systems by introducing intermediate variables and constructed the compact difference format with fourth-order space precision. For the type differential equations such as two-dimensional fourth-order hyperbolic equations, the fourt- order fractional diffusion-wave system and fourth-order temporal multi-term fractional wave equations, refs. [21,22,23,24,25,26,27] established the compact difference formats by introducing second-order derivatives as intermediate variables.

To date, the authors have found no studies that use the compact difference method to numerically simulate time-fractional damped plate vibration equations. Therefore, this paper seeks to solve time-fractional damped plate vibration equations by introducing the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. The space derivative is discretized by fourth-order compact difference, the time-integer-order derivative is discretized by the Crank–Nicolson (C-N) difference format, and the time-fractional derivative is approximated by L1 interpolation in the same way as [28]. The compact difference method for solving time-fractional damped plate vibration equations is established. The energy method is applied to analyze the stability and convergence of this difference format. Numerical cases are provided to validate the convergence order and feasibility of the given difference format. This paper not only discusses the compact difference method for the damped plate vibration equations with time-fractional derivatives but also simulates the influence of damping coefficients on the amplitude of vibration. This paper provides a novel effective numerical method for solving damped plate vibration problems, which enlarges the toolkit for simulating damped plate vibration problems and contributes to a more thorough and complete system of theories for numerically solving time-fractional fourth-order differential equations.

This paper is structured as follows: Section 2 establishes the C-N compact difference format for time-fractional damped plate vibration equations. Section 3 and Section 4 analyze the stability and convergence of the given difference format using the energy method. Section 5 provides the numerical cases to verify the convergence order and feasibility of the given difference format and simulates the influence of damping coefficient on the amplitude of vibration.

2. Compact Difference Scheme

In this section, the following fractional damped plate vibration equations are considered.

$$\left\{\begin{array}{cc}(a){\mu }_0^c{D}_t^{\alpha }u+{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+a{▵}^{2}u=f(x,y,t),\hfill & (x,y,t)\in \Omega \times (0,T],\hfill \\ {(b)u|}_{t=0}={\phi }_1(x,y),{\displaystyle \frac{\partial u}{\partial t}}{|}_{t=0}={\phi }_2(x,y),\hfill & (x,y)\in \Omega ,\hfill \\ {(c)u|}_{x=0}={\psi }_1{(y,t),u|}_{x=b}={\psi }_2(y,t),\hfill \\ {u|}_{y=0}={\psi }_3{(x,t),u|}_{y=c}={\psi }_4(x,t),\hfill & t\in (0,T],\hfill \\ {(d)▵u|}_{x=0}=g_1{(y,t),▵u|}_{x=b}=g_2(y,t),\hfill \\ {▵u|}_{y=0}=g_3{(x,t),▵u|}_{y=c}=g_4(x,t),\hfill & t\in (0,T],\hfill \end{array}\right.$$

(1)

where ${}_0^c{D}_t^{\alpha }u(t)=\frac{1}{\Gamma (2-\alpha )}{\int }_0^t\frac{f^{″}(s)}{{(t-s)}^{\alpha -1}}ds,(1<\alpha <2)$, $\Omega =(0,b)\times (0,c)\subset {\mathbf{R}}^2$, T is the total amount of time, $▵u=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}$, ${▵}^{2}u=\frac{{\partial }^{4}u}{\partial x^4}+2\frac{{\partial }^{4}u}{\partial x^2\partial y^2}+\frac{{\partial }^{4}u}{\partial y^4}$, $\mu (\mu >0)$ is the damping coefficient, and $a=\frac{D}{\rho l}$, D, $\rho$ and l are the bending stiffness, sheet density and sheet thickness, respectively. For simplicity, we assume that a and $\mu$ are both constants. The right-hand term $f(x,y,t)$ is a known smooth function, and ${\phi }_1(x,y)$, ${\phi }_2(x,y)$, ${\psi }_i(y,t)$, $g_i(y,t)$$(i=1,2)$ and ${\psi }_i(x,t)$, $g_i(x,t)$$(i=3,4)$ are all known functions.

By introducing two intermediate functions, $v=-a▵u$, $w={\displaystyle \frac{\partial u}{\partial t}}$, (1) becomes

$$\left\{\begin{array}{cc}(a){\mu }_0^c{D}_t^{\alpha }u+{\displaystyle \frac{\partial w}{\partial t}}-▵v=f(x,y,t),\hfill & (x,y,t)\in \Omega \times (0,T],\hfill \\ (b)a▵w+{\displaystyle \frac{\partial v}{\partial t}}=0,\hfill & (x,y,t)\in \Omega \times (0,T],\hfill \\ {(c)v|}_{t=0}=-a▵{\phi }_1{(x,y),w|}_{t=0}={\phi }_2(x,y),\hfill & (x,y)\in \Omega ,\hfill \\ {(d)w|}_{x=0}={\displaystyle \frac{\partial {\psi }_1}{\partial t}}{(y,t),w|}_{x=b}={\displaystyle \frac{\partial {\psi }_2}{\partial t}}(y,t),\hfill \\ {w|}_{y=0}={\displaystyle \frac{\partial {\psi }_3}{\partial t}}{(x,t),w|}_{y=c}={\displaystyle \frac{\partial {\psi }_4}{\partial t}}(x,t),\hfill & t\in (0,T],\hfill \\ {(e)v|}_{x=0}=-a{g}_1{(y,t),v|}_{x=b}=-a{g}_2(y,t),\hfill \\ {v|}_{y=0}=-a{g}_3{(x,t),v|}_{y=c}=-a{g}_4(x,t),\hfill & t\in (0,T].\hfill \end{array}\right.$$

(2)

For establishing the C-N compact difference scheme of Problem (1), region $\Omega$ is divided along the x and y directions, respectively. Let $h_x=\frac{b}{M_x}$, $h_y=\frac{c}{M_y}$ be the spatial step, $h=max\{h_x,h_y\}$ and let $\tau =\frac{T}{N}$ be the time step; then $x_i=i{h}_x$$(i=0,1,\cdots M_x)$, $y_j=j{h}_y$$(j=0,1,\cdots M_y)$, $t_n=n\tau$$(n=0,1,\cdots N)$. ${\widehat{\Omega }}_h=\{(x_i,y_j)|0\le i\le M_x,0\le j\le M_y\}$, ${\widehat{\Omega }}_{\tau }=\{t_n|0\le n\le N\}$, $u_{ij},0\le i\le M_x,0\le j\le M_y$ is the grid function on ${\widehat{\Omega }}_h,u_{ij}^n,0\le i\le M_x,$ and $0\le j\le M_y,0\le n\le N$ is the grid function on ${\widehat{\Omega }}_h\times {\widehat{\Omega }}_{\tau }$.

Introduce the following notations:

$$\begin{array}{c}\begin{array}{cc}\hfill & u_{ij}^{n+\frac{1}{2}}=\frac{1}{2}(u_{ij}^n+u_{ij}^{n+1}),{\delta }_t{u}_{ij}^{n+\frac{1}{2}}=\frac{1}{\tau }(u_{ij}^{n+1}-u_{ij}^n),\hfill \\ \hfill & {\delta }_x^2{u}_{ij}^n=\frac{1}{h_x^2}(u_{i+1,j}^n-2u_{ij}^n+u_{i-1,j}^n),{\delta }_y^2{u}_{ij}^n=\frac{1}{h_y^2}(u_{i,j+1}^n-2u_{ij}^n+u_{i,j-1}^n).\hfill \end{array}\hfill \end{array}$$

Let $A_x$ and $A_y$ denote compact difference operators in the x and y directions, respectively, and [29]

$$A_x{u}_{ij}=\left\{\begin{array}{cccc}& \frac{1}{12}(u_{i-1,j}+10u_{ij}+u_{i+1,j}),\hfill & & 1\le i\le M_x-1,0\le j\le M_y,\hfill \\ & u_{ij},\hfill & & i=0,M_x,0\le j\le M_y,\hfill \end{array}\right.$$

$$A_y{u}_{ij}=\left\{\begin{array}{cccc}& \frac{1}{12}(u_{i,j-1}+10u_{ij}+u_{i,j+1}),\hfill & & 1\le j\le M_y-1,0\le i\le M_x,\hfill \\ & u_{ij},\hfill & & j=0,M_y,0\le i\le M_x.\hfill \end{array}\right.$$

Let $A_h=A_x{A}_y$, $B_h=A_y{\delta }_x^2+A_x{\delta }_y^2$.

Let $S_h=\{u|u=u_{ij},0\le i\le M_x,0\le j\le M_y\},$ $S_{h0}=\{u|u\in S_h,u_{0,j}=u_{M_x,j}=u_{i,0}=u_{i,M_y}=0,0\le i\le M_x,0\le j\le M_y\}$. For any two functions $u,v\in S_h$, we introduce the following inner product and norms:

$$\begin{array}{cc}\hfill & \hfill (u,v)=h_x{h}_y\sum _{i=1}^{M_x-1}\sum _{j=1}^{M_y-1}{u}_{ij}{v}_{ij},\parallel u\parallel ={(u,u)}^{\frac{1}{2}}{,\parallel u\parallel }_{\infty }=\underset{\begin{array}{c}1\le i\le M_x-1\\ 1\le j\le M_y-1\end{array}}{max}|u_{ij}|.\hfill \end{array}$$

Next, we shall give three Lemmas. One is the L1 discrete format of the fractional derivative, and the other two are error estimates and symmetry for compact operators.

Lemma 1

([29]). If $u(t)\in C^3[t_0,t_n]$, $1<\alpha <2$, let $w(t)=u^{\prime }(t)$, there is

$$\begin{array}{c}\hfill {}_0^c{D}_t^{\alpha }u(t_n)=\frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}[b_0^{(\alpha )}w(t_n)-\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})w(t_k)-b_{n-1}^{(\alpha )}w(t_0)]+R(w(t_n)),\end{array}$$

where

$$\begin{array}{cc}\hfill & b_k^{(\alpha )}={(k+1)}^{2-\alpha }-k^{2-\alpha },\hfill \\ \hfill & |R(w(t_n))|\le \frac{1}{2\Gamma (2-\alpha )}[\frac{1}{4}+\frac{\alpha -1}{(2-\alpha )(3-\alpha )}]\underset{t_0\le t\le t_n}{max}|u^{‴}(t)|{\tau }^{3-\alpha }.\hfill \end{array}$$

Lemma 2

([30]). If $g(x)\in C^6[c-h,c+h]$, there is

$$\begin{array}{c}\hfill \frac{1}{12}[g^{″}(c-h)+10g^{″}(c)+g^{″}(c+h)]=\frac{1}{h^2}[g(c+h)-2g(c)+g(c-h)]+\frac{h^4}{240}{g}^{(6)}(\xi ),\end{array}$$

where $\xi \in (c-h,c+h)$.

Now, we discretize the system of Equation (2).

Firstly, at node $(x_i,y_j,t_{n-\frac{1}{2}})$, $1\le i\le M_x-1$, $1\le j\le M_y-1$, $1\le n\le N$, Equation (2a) can be written as

$$\begin{array}{cc}\hfill & {\mu }_0^c{D}_t^{\alpha }u(x_i,y_j,t_{n-\frac{1}{2}})+\frac{\partial w}{\partial t}(x_i,y_j,t_{n-\frac{1}{2}})-\frac{{\partial }^{2}v}{\partial x^2}(x_i,y_j,t_{n-\frac{1}{2}})\hfill \\ \hfill & -\frac{{\partial }^{2}v}{\partial y^2}(x_i,y_j,t_{n-\frac{1}{2}})=f(x_i,y_j,t_{n-\frac{1}{2}}).\hfill \end{array}$$

(3)

According to Lemma 1, for $u(x,y,t)\in C^3[t_0,t_n]$, the fractional derivative is approximated by L1 interpolation

$$\begin{array}{cc}\hfill {}_0^c{D}_t^{\alpha }u(x_i,y_j,t_{n-\frac{1}{2}})=& \hfill \frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}[b_0^{(\alpha )}{w}_{ij}^{n-\frac{1}{2}}-\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})w_{ij}^{k-\frac{1}{2}}\hfill \\ \hfill & \hfill -b_{n-1}^{(\alpha )}{\phi }_{2,ij}]+R_{ij,n-\frac{1}{2}}^{(1)},\hfill \end{array}$$

(4)

where

$$|R_{ij,n-\frac{1}{2}}^{(1)}|\le \frac{1}{\Gamma (2-\alpha )}[\frac{1}{4}+\frac{\alpha -1}{(2-\alpha )(3-\alpha )}]\underset{t_0\le t\le t_n}{max}|u^{‴}(x,y,t)|{\tau }^{3-\alpha }.$$

For the integer derivative terms, by Taylor’s expansion, there are

$$\frac{\partial w}{\partial t}(x_i,y_j,t_{n-\frac{1}{2}})={\delta }_t{w}_{ij}^{n-\frac{1}{2}}-\frac{{\tau }^2}{24}\frac{{\partial }^{3}w}{\partial t^3}(x_i,y_j,t_{n-\frac{1}{2}})+O({\tau }^4),$$

$$\begin{array}{cc}\hfill \frac{{\partial }^{2}v}{\partial x^2}(x_i,y_j,t_{n-\frac{1}{2}})& \hfill =\frac{1}{2}[\frac{{\partial }^{2}v}{\partial x^2}(x_i,y_j,t_{n-1})+\frac{{\partial }^{2}v}{\partial x^2}(x_i,y_j,t_n)]\hfill \\ \hfill & \hfill -\frac{{\tau }^2}{8}\frac{{\partial }^{4}v}{\partial x^2\partial t^2}(x_i,y_j,t_{n-\frac{1}{2}})+O({\tau }^4),\hfill \\ \hfill \frac{{\partial }^{2}v}{\partial y^2}(x_i,y_j,t_{n-\frac{1}{2}})& \hfill =\frac{1}{2}[\frac{{\partial }^{2}v}{\partial y^2}(x_i,y_j,t_{n-1})+\frac{{\partial }^{2}v}{\partial y^2}(x_i,y_j,t_n)]\hfill \\ \hfill & \hfill -\frac{{\tau }^2}{8}\frac{{\partial }^{4}v}{\partial y^2\partial t^2}(x_i,y_j,t_{n-\frac{1}{2}})+O({\tau }^4).\hfill \end{array}$$

Substituting the above three equations and (4) into (3), we obtain

$$\begin{array}{cc}\hfill & \mu \frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}[b_0^{(\alpha )}{w}_{ij}^{n-\frac{1}{2}}-\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})w_{ij}^{k-\frac{1}{2}}-b_{n-1}^{(\alpha )}{\phi }_{2,ij}]+{\delta }_t{w}_{ij}^{n-\frac{1}{2}}\hfill \\ \hfill & -\frac{1}{2}[\frac{{\partial }^{2}v}{\partial x^2}(x_i,y_j,t_{n-1})+\frac{{\partial }^{2}v}{\partial x^2}(x_i,y_j,t_n)]-\frac{1}{2}[\frac{{\partial }^{2}v}{\partial y^2}(x_i,y_j,t_{n-1})+\frac{{\partial }^{2}v}{\partial y^2}(x_i,y_j,t_n)]\hfill \\ \hfill & =f_{ij}^{n-\frac{1}{2}}+{\tau }^2{g}_{ij}^{n-\frac{1}{2}}+O({\tau }^{3-\alpha }),\hfill \end{array}$$

(5)

where

$$g_{ij}^{n-\frac{1}{2}}=\frac{1}{24}\frac{{\partial }^{3}w}{\partial t^3}(x_i,y_j,t_{n-\frac{1}{2}})+\frac{1}{8}\frac{{\partial }^{4}v}{\partial x^2\partial t^2}(x_i,y_j,t_{n-\frac{1}{2}})+\frac{1}{8}\frac{{\partial }^{4}v}{\partial x^2\partial t^2}(x_i,y_j,t_{n-\frac{1}{2}}).$$

Applying the compact operators $A_x{A}_y$ on both sides of Formula (5), we obtain

$$\begin{array}{cc}\hfill & A_h\mu \frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}[b_0^{(\alpha )}{w}_{ij}^{n-\frac{1}{2}}-\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})w_{ij}^{k-\frac{1}{2}}-b_{n-1}^{(\alpha )}{\phi }_{2,ij}]+A_h{\delta }_t{w}_{ij}^{n-\frac{1}{2}}\hfill \\ \hfill & -\frac{1}{2}{A}_h[\frac{{\partial }^{2}v}{\partial x^2}(x_i,y_j,t_{n-1})+\frac{{\partial }^{2}v}{\partial x^2}(x_i,y_j,t_n)]-\frac{1}{2}{A}_h[\frac{{\partial }^{2}v}{\partial y^2}(x_i,y_j,t_{n-1})+\frac{{\partial }^{2}v}{\partial y^2}(x_i,y_j,t_n)]\hfill \\ \hfill & =A_h{f}_{ij}^{n-\frac{1}{2}}+{\tau }^2{A}_h{g}_{ij}^{n-\frac{1}{2}}+O({\tau }^{3-\alpha }).\hfill \end{array}$$

(6)

According to Lemma 2, using the commutativity of operator $A_x{A}_y$, we obtain

$$\begin{array}{cc}\hfill & \mu A_h\frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}[b_0^{(\alpha )}{w}_{ij}^{n-\frac{1}{2}}-\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})w_{ij}^{n-\frac{1}{2}}-b_{n-1}^{(\alpha )}{\phi }_{2,ij}]\hfill \\ \hfill & +A_h{\delta }_t{w}_{ij}^{n-\frac{1}{2}}+B_h{v}_{ij}^{n-\frac{1}{2}}=A_h{f}_{ij}^{n-\frac{1}{2}}+R_{ij,n-\frac{1}{2}}^{(2)},\hfill \end{array}$$

(7)

where

$$\begin{array}{cc}\hfill |R_{ij,n-\frac{1}{2}}^{(2)}|\le & A_h\frac{-\mu }{\Gamma (2-\alpha )}[\frac{1}{4}+\frac{\alpha -1}{(2-\alpha )(3-\alpha )}]\underset{t_0\le t\le t_n}{max}|u^{‴}(x,y,t)|{\tau }^{3-\alpha }+{\tau }^2{A}_h{g}_{ij}^{n-\frac{1}{2}}\hfill \\ \hfill & +\frac{h_x^4}{480}{A}_y[\frac{{\partial }^{6}v}{\partial x^6}(x_i,y_j,t_{n-1})+\frac{{\partial }^{6}v}{\partial x^6}(x_i,y_j,t_n)]+\frac{h_y^4}{480}{A}_x[\frac{{\partial }^{6}v}{\partial y^6}(x_i,y_j,t_{n-1})\hfill \\ \hfill & +\frac{{\partial }^{6}v}{\partial y^6}(x_i,y_j,t_n)]+O({\tau }^4+h_x^6+h_y^6).\hfill \end{array}$$

Assuming that w and v are sufficiently smooth, there exists a constant $C_1$ such that

$$\begin{array}{c}\hfill |R_{ij,n-\frac{1}{2}}^{(2)}|\le C_1({\tau }^{3-\alpha }+h_x^4+h_y^4).\end{array}$$

By omitting $R_{ij,n-\frac{1}{2}}^{(2)}$ in (7) and replacing $w_{ij}^{n-\frac{1}{2}}$ and $v_{ij}^{n-\frac{1}{2}}$ with approximate solutions $W_{ij}^{n-\frac{1}{2}}$ and $V_{ij}^{n-\frac{1}{2}}$, the differential discretization scheme of Equation (2a) can be obtained

$$\begin{array}{cc}\hfill & \mu A_h\frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}[b_0^{(\alpha )}{W}_{ij}^{n-\frac{1}{2}}-\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})W_{ij}^{n-\frac{1}{2}}-b_{n-1}^{(\alpha )}{\phi }_{2,ij}]\hfill \\ \hfill & +A_h{\delta }_t{W}_{ij}^{n-\frac{1}{2}}+B_h{V}_{ij}^{n-\frac{1}{2}}=A_h{f}_{ij}^{n-\frac{1}{2}}.\hfill \end{array}$$

(8)

Secondly, at node $(x_i,y_j,t_{n-\frac{1}{2}})$, $1\le i\le M_x-1$, $1\le j\le M_y-1$, $1\le n\le N$, Equation (2b) can be written as

$$\begin{array}{c}\hfill a▵w(x_i,y_j,t_{n-\frac{1}{2}})+\frac{\partial v}{\partial t}(x_i,y_j,t_{n-\frac{1}{2}})=0.\end{array}$$

(9)

Applying the compact operators $A_x{A}_y$ on both sides of Formula (9), we obtain

$$\begin{array}{c}\hfill a{A}_h\frac{{\partial }^{2}w}{\partial x^2}(x_i,y_j,t_{n-\frac{1}{2}})+a{A}_h\frac{{\partial }^{2}w}{\partial y^2}(x_i,y_j,t_{n-\frac{1}{2}})+A_h\frac{\partial v}{\partial t}(x_i,y_j,t_{n-\frac{1}{2}})=0.\end{array}$$

(10)

Similarly to (7), (10) can be written as follows:

$$\begin{array}{c}\hfill a{B}_h{w}_{ij}^{n-\frac{1}{2}}+A_h{\delta }_t{v}_{ij}^{n-\frac{1}{2}}=R_{ij,n-\frac{1}{2}}^{(3)},\end{array}$$

(11)

where

$$\begin{array}{cc}\hfill & R_{ij,n-\frac{1}{2}}^{(3)}={\tau }^2{A}_h[\frac{a}{8}\frac{{\partial }^{4}w}{\partial x^2\partial t^2}(x_i,y_j,t_{n-\frac{1}{2}})+\frac{a}{8}\frac{{\partial }^{4}w}{\partial y^2\partial t^2}(x_i,y_j,t_{n-\frac{1}{2}})\hfill \\ \hfill & +\frac{1}{24}\frac{{\partial }^{3}v}{\partial t^3}(x_i,y_j,t_{n-\frac{1}{2}})]+\frac{h_x^4}{480}{A}_y[\frac{{\partial }^{6}v}{\partial x^6}(x_i,y_j,t_{n-1})+\frac{{\partial }^{6}v}{\partial x^6}(x_i,y_j,t_n)]\hfill \\ \hfill & +\frac{h_y^4}{480}{A}_x[\frac{{\partial }^{6}v}{\partial y^6}(x_i,y_j,t_{n-1})+\frac{{\partial }^{6}v}{\partial y^6}(x_i,y_j,t_n)]+O({\tau }^4+h_x^6+h_y^6).\hfill \end{array}$$

Assuming that w and v are sufficiently smooth, there exists a constant $C_2$ such that

$$\begin{array}{c}\hfill |R_{ij,n-\frac{1}{2}}^{(3)}|\le C_2({\tau }^2+h_x^4+h_y^4).\end{array}$$

By omitting $R_{ij,n-\frac{1}{2}}^{(3)}$ in (11) and replacing $w_{ij}^{n-\frac{1}{2}}$ and $v_{ij}^{n-\frac{1}{2}}$ with approximate solutions $W_{ij}^{n-\frac{1}{2}}$ and $V_{ij}^{n-\frac{1}{2}}$, the differential discretization scheme of Equation (2b) can be obtained:

$$\begin{array}{c}\hfill a{B}_h{W}_{ij}^{n-\frac{1}{2}}+A_h{\delta }_t{V}_{ij}^{n-\frac{1}{2}}=0.\end{array}$$

(12)

The initial value condition (2c) can be directly discretized as

$$\begin{array}{c}\hfill W_{ij}^0=w_{ij}^0={\phi }_2(x_i,y_j),V_{ij}^0=v_{ij}^0=-a\Delta {\phi }_1(x_i,y_j).\end{array}$$

(13)

Considering the boundary value conditions (2d), (2e), we obtain

$$\begin{array}{cc}\hfill & W_{0,j}^n=\frac{\partial {\psi }_1}{\partial t}(y_j,t_n),W_{M_x,j}^n=\frac{\partial {\psi }_2}{\partial t}(y_j,t_n),W_{i,0}^n=\frac{\partial {\psi }_3}{\partial t}(x_i,t_n),W_{i,M_y}^n=\frac{\partial {\psi }_4}{\partial t}(x_i,t_n),\hfill \\ \hfill & V_{0,j}^n=-a{g}_1(y_j,t_n),V_{M_x,j}^n=-a{g}_2(y_j,t_n),V_{i,0}^n=-a{g}_3(x_i,t_n),V_{i,M_y}^n=-a{g}_4(x_i,t_n).\hfill \\ \hfill & \hfill 1\le i\le M_x-1,1\le i\le M_y-1,0\le n\le N.\hfill \end{array}$$

(14)

By combining Equations (8), (12), (13) and (14), the C-N compact difference scheme of the system of Equation (2) can be obtained as follows:

$$\left\{\begin{array}{c}(a)\mu A_h{\displaystyle \frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}}[b_0^{(\alpha )}{W}_{ij}^{n-\frac{1}{2}}-{\displaystyle \sum _{k=1}^{n-1}}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})W_{ij}^{n-\frac{1}{2}}-b_{n-1}^{(\alpha )}{\phi }_{2,ij}]+A_h{\delta }_t{W}_{ij}^{n-\frac{1}{2}}\hfill \\ +B_h{V}_{ij}^{n-\frac{1}{2}}=A_h{f}_{ij}^{n-\frac{1}{2}},\hfill \\ (b)a{B}_h{W}_{ij}^{n-\frac{1}{2}}+A_h{\delta }_t{V}_{ij}^{n-\frac{1}{2}}=0,\hfill \\ (c)W_{ij}^0={\phi }_2(x_i,y_j),V_{ij}^0=-a\Delta {\phi }_1(x_i,y_j),\hfill \\ (d)W_{0,j}^n={\displaystyle \frac{\partial {\psi }_1}{\partial t}}(y_j,t_n),W_{M_x,j}^n={\displaystyle \frac{\partial {\psi }_2}{\partial t}}(y_j,t_n),\hfill \\ W_{i,0}^n={\displaystyle \frac{\partial {\psi }_3}{\partial t}}(x_i,t_n),W_{i,M_y}^n={\displaystyle \frac{\partial {\psi }_4}{\partial t}}(x_i,t_n),\hfill \\ (e)V_{0,j}^n=-a{g}_1(y_j,t_n),V_{M_x,j}^n=-a{g}_2(y_j,t_n),\hfill \\ V_{i,0}^n=-a{g}_3(x_i,t_n),V_{i,M_y}^n=-a{g}_4(x_i,t_n).\hfill \end{array}\right.$$

(15)

Discrete the equation $w={\displaystyle \frac{\partial u}{\partial t}}$; similarly to (12), we get

$$\begin{array}{c}\hfill {\delta }_t{U}_{ij}^{n-\frac{1}{2}}=W_{ij}^{n-\frac{1}{2}}.\end{array}$$

(16)

It can be seen from the above derivation process that the truncation error of the differential formats (15) and (16) is $O({\tau }^{3-\alpha }+h_x^4+h_y^4)$.

3. Stability Analysis

In this part, the energy method is used to analyze the stability of the C-N compact difference scheme (15).

Theorem 1.

Set ${\psi }_i=g_i=0(i=1,2,3,4)$, where $\{w^n,v^n\}$ are the solutions of the following difference scheme,

$$\begin{array}{cc}\hfill & \mu A_h\frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}[b_0^{(\alpha )}{w}_{ij}^{n-\frac{1}{2}}-\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})w_{ij}^{n-\frac{1}{2}}-b_{n-1}^{(\alpha )}{\phi }_{2,ij}]+A_h{\delta }_t{w}_{ij}^{n-\frac{1}{2}}\hfill \\ \hfill & +B_h{v}_{ij}^{n-\frac{1}{2}}=P_{ij}^{n-\frac{1}{2}},1\le i\le M_x-1,1\le i\le M_y-1,1\le n\le N,\hfill \end{array}$$

(17)

$$\begin{array}{c}a{B}_h{w}_{ij}^{n-\frac{1}{2}}+A_h{\delta }_t{v}_{ij}^{n-\frac{1}{2}}=Q_{ij}^{n-\frac{1}{2}},1\le i\le M_x-1,1\le i\le M_y-1,1\le n\le N,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & w_{ij}^0=w_{ij}^0={\phi }_{2,ij},v_{ij}^0=v_{ij}^0=-\rho \Delta {\phi }_{1,ij},1\le i\le M_x-1,1\le i\le M_y-1,\hfill \\ \hfill & w_{0,j}^n=w_{M_x,j}^n=w_{i,0}^n=w_{i,M_y}^n=0,v_{0,j}^n=v_{M_x,j}^n=v_{i,0}^n=v_{i,M_y}^n=0,0\le n\le N,\hfill \end{array}$$

(19)

then

$$\begin{array}{cc}\hfill & a\parallel A_h{w}^n{\parallel }^2+\parallel A_h{v}^n{\parallel }^2\le exp(n\tau )[a\parallel A_h{w}^0{\parallel }^2+\parallel A_h{v}^0{\parallel }^2+\frac{a\mu t_n^{2-\alpha }}{\Gamma (3-\alpha )}{\parallel A_h{\phi }_2\parallel }^2\hfill \\ \hfill & +\frac{\tau }{\mu }\Gamma (2-\alpha )t_n^{\alpha -1}\sum _{k=1}^n\parallel P^{k-\frac{1}{2}}{\parallel }^2+\tau \sum _{k=1}^n\parallel Q^{k-\frac{1}{2}}{\parallel }^2].\hfill \end{array}$$

Proof of Theorem 1. 

Let $\epsilon =\Gamma (3-\alpha ){\tau }^{\alpha -1}$. Taking the inner product of Equation (17) with $A_h{w}^{n-\frac{1}{2}}$, we obtain

$$\begin{array}{cc}\hfill & \frac{\mu }{\epsilon }(A_h{w}^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})-\frac{\mu }{\epsilon }[\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})(A_h{w}^{k-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})]\hfill \\ \hfill & +b_{n-1}^{(\alpha )}(A_h{\phi }_2,A_h{w}^{n-\frac{1}{2}})]+(A_h{\delta }_t{w}^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})\hfill \\ \hfill & -(B_h{v}^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})=(P^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}}).\hfill \end{array}$$

(20)

Taking the inner product of the Equation (18) with $A_h{v}^{n-\frac{1}{2}}$, we obtain

$$\begin{array}{c}\hfill a(B_h{w}^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}})+(A_h{\delta }_t{v}^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}})=(Q^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}}).\end{array}$$

(21)

Multiplying both sides of (20) by a, adding them to (21) and using the commutativity of operator $A_x{A}_y$, we obtain

$$\begin{array}{cc}\hfill & \frac{\mu a}{\epsilon }(A_h{w}^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})-\frac{\mu a}{\epsilon }[\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})(A_h{w}^{k-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})]\hfill \\ \hfill & +b_{n-1}^{(\alpha )}(A_h{\phi }_2,A_h{w}^{n-\frac{1}{2}})]+a(A_h{\delta }_t{w}^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})+(A_h{\delta }_t{v}^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}})\hfill \\ \hfill & =a(P^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})+(Q^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}}).\hfill \end{array}$$

(22)

Due to

$$\begin{array}{c}\hfill (A_h{\delta }_t{v}^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}})=\frac{1}{2\tau }(\parallel A_h{v}^n{\parallel }^2-\parallel A_h{v}^{n-1}{\parallel }^2),\\ \hfill (A_h{\delta }_t{w}^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})=\frac{1}{2\tau }(\parallel A_h{w}^n{\parallel }^2-\parallel A_h{w}^{n-1}{\parallel }^2),\end{array}$$

Equation (22) can become

$$\begin{array}{cc}\hfill & \frac{\mu a}{\epsilon }\parallel A_h{w}^{n-\frac{1}{2}}{\parallel }^2+\frac{a}{2\tau }(\parallel A_h{w}^n{\parallel }^2-\parallel A_h{w}^{n-1}{\parallel }^2)+\frac{1}{2\tau }(\parallel A_h{v}^n{\parallel }^2-\parallel A_h{v}^{n-1}{\parallel }^2)\hfill \\ \hfill & =\frac{\mu a}{\epsilon }[\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})(A_h{w}^{k-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})]+b_{n-1}^{(\alpha )}(A_h{\phi }_2,A_h{w}^{n-\frac{1}{2}})]\hfill \\ \hfill & +a(P^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})+(Q^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}}).\hfill \end{array}$$

Applying the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{cc}\hfill & \frac{\mu a}{\epsilon }\parallel A_h{w}^{n-\frac{1}{2}}{\parallel }^2+\frac{a}{2\tau }(\parallel A_h{w}^n{\parallel }^2-\parallel A_h{w}^{n-1}{\parallel }^2)+\frac{1}{2\tau }(\parallel A_h{v}^n{\parallel }^2-\parallel A_h{v}^{n-1}{\parallel }^2)\hfill \\ \hfill & \le \frac{\mu a}{2\epsilon }\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})(\parallel A_h{w}^{k-\frac{1}{2}}{\parallel }^2+\parallel A_h{w}^{n-\frac{1}{2}}{\parallel }^2)+\frac{\mu a}{2\epsilon }{b}_{n-1}^{(\alpha )}(\parallel A_h{\phi }_2{\parallel }^2\hfill \\ \hfill & +\parallel A_h{w}^{n-\frac{1}{2}}{\parallel }^2)+a(P^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})+(Q^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}}).\hfill \end{array}$$

Multiplying both sides of the above equation by $2\tau$, we obtain

$$\begin{array}{cc}\hfill & \frac{a\mu \tau }{\epsilon }\parallel A_h{w}^{n-\frac{1}{2}}{\parallel }^2+a(\parallel A_h{w}^n{\parallel }^2-\parallel A_h{w}^{n-1}{\parallel }^2)+(\parallel A_h{v}^n{\parallel }^2-\parallel A_h{v}^{n-1}{\parallel }^2)\hfill \\ \hfill & \le \frac{a\mu \tau }{\epsilon }[\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})\parallel A_h{w}^{k-\frac{1}{2}}{\parallel }^2+b_{n-1}^{(\alpha )}\parallel A_h{\phi }_2{\parallel }^2]\hfill \\ \hfill & +2\tau a(P^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})+2\tau (Q^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}}).\hfill \end{array}$$

(23)

Set

$$\begin{array}{c}\hfill F^n=a\parallel A_h{w}^n{\parallel }^2+\parallel A_h{v}^n{\parallel }^2+\frac{a\mu \tau }{\epsilon }\sum _{k=1}^n{b}_{n-k}^{(\alpha )}{\parallel A_h{w}^{k-\frac{1}{2}}\parallel }^2,\end{array}$$

Equation (23) can become

$$\begin{array}{c}\hfill F^n\le F^{n-1}+\frac{a\mu \tau }{\epsilon }{b}_{n-1}^{(\alpha )}{\parallel A_h{\phi }_2\parallel }^2+2\tau a(P^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})+2\tau (Q^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}}).\end{array}$$

Recursively, we obtain

$$\begin{array}{cc}\hfill F^n\le & \hfill F^0+\frac{a\mu \tau }{\epsilon }\sum _{k=0}^{n-1}{b}_k^{(\alpha )}{\parallel A_h{\phi }_2\parallel }^2+2\tau a\sum _{k=1}^n(P^{n-\frac{1}{2}},A_h{w}^{n-\frac{1}{2}})\hfill \\ \hfill & \hfill +2\tau \sum _{k=1}^n(Q^{n-\frac{1}{2}},A_h{v}^{n-\frac{1}{2}}).\hfill \end{array}$$

Applying the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{cc}\hfill & \hfill F^n\le F^0+\frac{a\mu \tau }{\epsilon }\sum _{k=0}^{n-1}{b}_k^{(\alpha )}\parallel A_h{\phi }_2{\parallel }^2+\tau \sum _{k=1}^n(\frac{\epsilon }{\mu b_{n-k}^{(\alpha )}}\parallel P^{k-\frac{1}{2}}{\parallel }^2+\frac{a\mu b_{n-k}^{(\alpha )}}{\epsilon }\parallel A_h{w}^{k-\frac{1}{2}}{\parallel }^2)\hfill \\ \hfill & \hfill +\tau \sum _{k=1}^n\parallel Q^{k-\frac{1}{2}}{\parallel }^2+\tau \sum _{k=1}^n{\parallel A_h{v}^k\parallel }^2.\hfill \end{array}$$

(24)

It can be obtained from the nature and definition of $b_k^{(\alpha )}$ that

$$\begin{array}{c}\hfill \frac{\epsilon }{b_{n-k}^{(\alpha )}}\le \Gamma (2-\alpha )t_n^{\alpha -1},\end{array}$$

(25)

$$\begin{array}{c}\hfill \tau \sum _{k=0}^{n-1}\frac{b_k^{(\alpha )}}{\epsilon }=\frac{t_n^{2-\alpha }}{\Gamma (3-\alpha )}.\end{array}$$

(26)

Substituting (25) and (26) into (24), we obtain

$$\begin{array}{cc}\hfill & \hfill a\parallel A_h{w}^n{\parallel }^2+\parallel A_h{v}^n{\parallel }^2\le a\parallel A_h{w}^0{\parallel }^2+\parallel A_h{v}^0{\parallel }^2+\frac{a\mu t_n^{2-\alpha }}{\Gamma (3-\alpha )}{\parallel A_h{\phi }_2\parallel }^2\hfill \\ \hfill & \hfill +\frac{\tau }{\mu }\Gamma (2-\alpha )t_n^{\alpha -1}\sum _{k=1}^n\parallel P^{k-\frac{1}{2}}{\parallel }^2+\tau \sum _{k=1}^n\parallel Q^{k-\frac{1}{2}}{\parallel }^2+\tau \sum _{k=0}^n{\parallel A_h{v}^k\parallel }^2.\hfill \end{array}$$

(27)

Let

$$\begin{array}{c}\hfill H^n=a\parallel A_h{w}^n{\parallel }^2+{\parallel A_h{v}^n\parallel }^2;\end{array}$$

then from (27), we obtain

$$\begin{array}{cc}\hfill H^n\le & \hfill H^0+\frac{a\mu t_n^{2-\alpha }}{\Gamma (3-\alpha )}\parallel A_h{\phi }_2{\parallel }^2+\frac{\tau }{\mu }\Gamma (2-\alpha )t_n^{\alpha -1}\sum _{k=1}^n{\parallel P^{k-\frac{1}{2}}\parallel }^2\hfill \\ \hfill & \hfill +\tau \sum _{k=0}^n{H}^k+\tau \sum _{k=1}^n{\parallel Q^{k-\frac{1}{2}}\parallel }^2.\hfill \end{array}$$

It can be obtained by the discrete Gronwall inequality that

$$\begin{array}{cc}\hfill H^n\le & \hfill exp(n\tau )[H^0+\frac{a\mu t_n^{2-\alpha }}{\Gamma (3-\alpha )}\parallel A_h{\phi }_2{\parallel }^2+\tau \sum _{k=1}^n\parallel Q^{k-\frac{1}{2}}{\parallel }^2\hfill \\ \hfill & \hfill +\frac{\tau }{\mu }\Gamma (2-\alpha )t_n^{\alpha -1}\sum _{k=1}^n\parallel P^{k-\frac{1}{2}}{\parallel }^2].\hfill \end{array}$$

□

From Theorem 1, we know that the difference scheme (15) is unconditionally stable.

4. Convergence Analysis

In this section, the energy method is used to analyze the convergence of the C-N compact difference scheme (15). For $1\le i\le M_x-1,1\le j\le M_y-1,1\le n\le N$, let $e_{ij}^n=w_{ij}^n-W_{ij}^n$, ${\eta }_{ij}^n=v_{ij}^n-V_{ij}^n$, where $w_{ij}^n,v_{ij}^n$ are the solutions of the system of Equation (2), and $W_{ij}^n$ and $V_{ij}^n$ are the solutions of the C-N compact difference format (15). Subtracting (7) and (11) from (8) and (12) respectively, we obtain the error equations system from (13) and (14)

$$\left\{\begin{array}{cc}\hfill \hfill & \mu A_h{\displaystyle \frac{{\tau }^{1-\alpha }}{\Gamma (3-\alpha )}}[b_0^{(\alpha )}{e}_{ij}^{n-\frac{1}{2}}-\sum _{k=1}^{n-1}(b_{n-k-1}^{(\alpha )}-b_{n-k}^{(\alpha )})e_{ij}^{n-\frac{1}{2}}+A_h{\delta }_t{e}_{ij}^{n-\frac{1}{2}}\hfill \\ \hfill \hfill & +B_h{\eta }_{ij}^{n-\frac{1}{2}}=R_{ij,n-\frac{1}{2}}^{(4)},1\le i\le M_x-1,1\le j\le M_y-1,1\le n\le N,\hfill \\ \hfill \hfill & a{B}_h{e}_{ij}^{n-\frac{1}{2}}+A_h{\delta }_t{\eta }_{ij}^{n-\frac{1}{2}}=R_{ij,n-\frac{1}{2}}^{(5)},1\le i\le M_x-1,1\le j\le M_y-1,1\le n\le N,\hfill \\ \hfill \hfill & e_{ij}^0=0,{\eta }_{ij}^0=0,0\le i\le M_x,0\le i\le M_y,\hfill \\ \hfill \hfill & e_{0,j}^n=e_{M_x,j}^n=e_{i,0}^n=e_{i,M_y}^n=0,{\eta }_{0,j}^n={\eta }_{M_x,j}^n={\eta }_{i,0}^n={\eta }_{i,M_y}^n=0,0\le n\le N.\hfill \end{array}\right.$$

(28)

Theorem 2.

Let $\{W^n,V^n\}$ be the solutions of the C-N compact difference scheme (15) and let $\{w^n,v^n\}$ be the solutions of the initial boundary value problem (2); then, we get the error estimation

$$\begin{array}{c}\hfill a\parallel A_h{e}^n{\parallel }^2+{\parallel A_h{\eta }^n\parallel }^2\le exp(T)[\frac{\Gamma (2-\alpha )}{\mu }{T}^{\alpha }{C}_3^2+T{C}_4^2]{({\tau }^{3-\alpha }+h^4)}^2.\end{array}$$

Proof of Theorem 2. 

Similar to the stability proof, according to Theorem 1 and the system of error Equation (28), we obtain

$$\begin{array}{cc}\hfill a\parallel A_h{e}^n{\parallel }^2+{\parallel A_h{\xi }^n\parallel }^2& \le exp(\frac{\tau }{\mu }\Gamma (2-\alpha )t_n^{\alpha -1}\sum _{k=1}^n\parallel R_{ij,n-\frac{1}{2}}^{(4)}{\parallel }^2+\tau \sum _{k=1}^n\parallel R_{ij,n-\frac{1}{2}}^{(5)}{\parallel }^2]\hfill \\ \hfill & \le exp(n\tau )[\frac{\Gamma (2-\alpha )}{\mu }{t}_n^{\alpha -1}n\tau C_3^2{({\tau }^{3-\alpha }+h^4)}^2+n\tau C_4^2({\tau }^2+h^4)]\hfill \\ \hfill & \le exp(T)[\frac{\Gamma (2-\alpha )}{\mu }{T}^{\alpha }{C}_3^2{({\tau }^{3-\alpha }+h^4)}^2+T{C}_4^2({\tau }^2+h^4)].\hfill \end{array}$$

□

5. Numerical Examples

In this part, three numerical examples are given to verify the convergence order and validity of the C-N compact difference schemes (15) and (16) for Problem (1). The first and second examples verify the time and space convergence of the error estimates when $\alpha$ is different; the third example verifies the influence of the damping coefficient $\mu$ on the vibration amplitude of the damping plate.

Example 1.

In the problem (1), let $\Omega =(0,1)\times (0,1)$, $T=1$, $\mu =1,a=1$,

$$\begin{array}{c}\hfill f(x,y,t)=[\frac{24}{\Gamma (5-\alpha )}{t}^{4-\alpha }+12t^2+4{\pi }^4{t}^4]sin\pi xsin\pi y,\end{array}$$

the corresponding exact solution is $u(x,y,t)=t^{4}sin\pi xsin\pi y$.

Now solve this problem using the C-N compact difference schemes (15) and (16) proposed in Section. Set $h=\frac{1}{M_x}=\frac{1}{M_y}$ as the spatial step and $\tau =\frac{1}{N}$ as the time step. Let $h=\frac{1}{500}$ and $\tau$ take different values to verify the time-direction convergence order of the C-N compact difference schemes. Let $\tau =\frac{1}{100000}$ and h take different values to verify the spatial-direction convergence order of the C-N compact difference schemes.

When $\alpha =1.3,1.5,1.8$, the computational errors and convergence orders are shown in

Table 1. The computational errors and spatial convergence orders of U.

h	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{4}$	3.106 × ${10}^{-3}$	−	1.553 × ${10}^{-3}$	−
$\frac{1}{8}$	1.094 × ${10}^{-4}$	4.028	9.522 × ${10}^{-5}$	4.028
$\frac{1}{16}$	1.185 × ${10}^{-5}$	4.007	5.924 × ${10}^{-6}$	4.007
$\frac{1}{32}$	7.936 × ${10}^{-7}$	4.002	3.698 × ${10}^{-7}$	4.002

,

Table 2. The computational errors and time convergence orders of U.

$\mathit{\tau }$	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{32}$	3.379 × ${10}^{-5}$	−	1.690 × ${10}^{-5}$	−
$\frac{1}{64}$	9.995 × ${10}^{-6}$	1.757	4.998 × ${10}^{-6}$	1.757
$\frac{1}{128}$	2.986 × ${10}^{-6}$	1.743	1.493 × ${10}^{-6}$	1.743
$\frac{1}{256}$	8.957 × ${10}^{-7}$	1.737	4.479 × ${10}^{-7}$	1.737

,

Table 3. The computational errors and spatial convergence orders of U.

h	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{4}$	3.094 × ${10}^{-3}$	−	1.547 × ${10}^{-3}$	−
$\frac{1}{8}$	1.897 × ${10}^{-4}$	4.028	9.485 × ${10}^{-5}$	4.028
$\frac{1}{16}$	1.180 × ${10}^{-5}$	4.007	5.901 × ${10}^{-6}$	4.007
$\frac{1}{32}$	7.372 × ${10}^{-7}$	4.001	3.686 × ${10}^{-7}$	4.001

,

Table 4. The computational errors and time convergence orders of U.

$\mathit{\tau }$	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{16}$	2.733 × ${10}^{-4}$	−	1.367 × ${10}^{-4}$	−
$\frac{1}{32}$	8.938 × ${10}^{-5}$	1.613	4.469 × ${10}^{-5}$	1.613
$\frac{1}{64}$	3.050 × ${10}^{-5}$	1.551	1.525 × ${10}^{-5}$	1.551
$\frac{1}{128}$	1.056 × ${10}^{-5}$	1.530	5.281 × ${10}^{-6}$	1.530

,

Table 5. The computational errors and spatial convergence orders of U.

h	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{4}$	3.027 × ${10}^{-3}$	−	1.536 × ${10}^{-3}$	−
$\frac{1}{8}$	1.884 × ${10}^{-4}$	4.027	9.419 × ${10}^{-5}$	4.027
$\frac{1}{16}$	1.1740 × ${10}^{-5}$	4.004	5.871 × ${10}^{-6}$	4.004
$\frac{1}{32}$	7.550 × ${10}^{-7}$	3.959	3.775 × ${10}^{-7}$	3.959

and

Table 6. The computational errors and time convergence orders of U.

$\mathit{\tau }$	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{16}$	8.699 × ${10}^{-4}$	−	4.350 × ${10}^{-4}$	−
$\frac{1}{32}$	3.751 × ${10}^{-4}$	1.214	1.875 × ${10}^{-4}$	1.214
$\frac{1}{64}$	1.620 × ${10}^{-4}$	1.211	8.101 × ${10}^{-5}$	1.211
$\frac{1}{128}$	7.013 × ${10}^{-5}$	1.208	3.506 × ${10}^{-5}$	1.208

, respectively.

When $\alpha =1.4$, the images of the numerical solution U and the exact solution u at $t=1$ with the mesh h = 1/64, $\tau =1/100$ are shown in the Figure 1 and Figure 2.

Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6 show the computational errors and convergence orders of the numerical solution U when $\alpha$ takes different values. It can be seen that in the sense of the maximum norm and the $L^2$ norm, the order of time convergence is $3-\alpha$, and the order of space convergence is 4, which demonstrates the effectiveness of the C-N compact difference schemes established in Section 2. Figure 1 and Figure 2 show the graphs of the numerical solution U and the exact solution u, which show good fitting results.

Example 2.

In the problem (1), let $\Omega =(0,\pi )\times (0,\pi )$, $T=1$, $\mu =1,a=1$,

$$\begin{array}{c}\hfill f(x,y,t)=[\frac{6}{\Gamma (4-\alpha )}{t}^{3-\alpha }+6t+4t^3]sinxsiny,\end{array}$$

the corresponding exact solution is $u(x,y,t)=t^{3}sinxsiny$.

Similar to Example 1, let $h=\frac{\pi }{500}$ and $\tau$ take different values to verify the time-direction convergence order of the schemes. Let $\tau =\frac{1}{100000}$ and h take different values to verify the spatial-direction convergence order of the schemes.

When $\alpha =1.2,1.4,1.7$, the computational errors and convergence orders are shown in

Table 7. The computational errors and spatial convergence orders of U.

h	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{\pi }{4}$	4.792 × ${10}^{-4}$	−	7.537 × ${10}^{-4}$	−
$\frac{\pi }{8}$	2.946 × ${10}^{-5}$	4.024	4.628 × ${10}^{-5}$	4.024
$\frac{\pi }{16}$	1.839 × ${10}^{-6}$	4.004	2.889 × ${10}^{-6}$	4.002
$\frac{\pi }{32}$	1.202 × ${10}^{-7}$	3.935	1.888 × ${10}^{-7}$	3.935

,

Table 8. The computational errors and time convergence orders of U.

$\mathit{\tau }$	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{16}$	1.938 × ${10}^{-3}$	−	3.045 × ${10}^{-3}$	−
$\frac{1}{32}$	5.365 × ${10}^{-4}$	1.853	8.429 × ${10}^{-4}$	1.853
$\frac{1}{64}$	1.489 × ${10}^{-4}$	1.850	2.339 × ${10}^{-4}$	1.850
$\frac{1}{128}$	4.143 × ${10}^{-5}$	1.845	6.508 × ${10}^{-5}$	1.845

,

Table 9. The computational errors and spatial convergence orders of U.

h	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{\pi }{4}$	4.441 × ${10}^{-4}$	−	6.976 × ${10}^{-4}$	−
$\frac{\pi }{8}$	2.730 × ${10}^{-5}$	4.024	4.289 × ${10}^{-5}$	4.024
$\frac{\pi }{16}$	1.702 × ${10}^{-6}$	4.004	2.673 × ${10}^{-6}$	4.004
$\frac{\pi }{32}$	1.089 × ${10}^{-7}$	3.965	1.711 × ${10}^{-7}$	3.965

,

Table 10. The computational errors and time convergence orders of U.

$\mathit{\tau }$	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{64}$	4.055 × ${10}^{-4}$	−	6.370 × ${10}^{-4}$	−
$\frac{1}{128}$	1.310 × ${10}^{-4}$	1.631	2.057 × ${10}^{-4}$	1.630
$\frac{1}{256}$	4.249 × ${10}^{-5}$	1.624	6.674 × ${10}^{-5}$	1.624
$\frac{1}{512}$	1.383 × ${10}^{-5}$	1.619	2.173 × ${10}^{-5}$	1.619

,

Table 11. The computational errors and spatial convergence orders of U.

h	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{\pi }{2}$	6.352 × ${10}^{-3}$	−	9.978 × ${10}^{-3}$	−
$\frac{\pi }{4}$	3.811 × ${10}^{-4}$	4.059	5.987 × ${10}^{-4}$	4.059
$\frac{\pi }{8}$	2.730 × ${10}^{-5}$	4.023	4.289 × ${10}^{-5}$	3.805
$\frac{\pi }{16}$	1.599 × ${10}^{-6}$	4.094	2.512 × ${10}^{-6}$	4.094

and

Table 12. The computational errors and time convergence orders of U.

$\mathit{\tau }$	${\mathit{l}}^{\mathit{\infty }}-\mathit{Error}$	$\mathit{Order}$	${\mathit{l}}^2-\mathit{Error}$	$\mathit{Order}$
$\frac{1}{16}$	1.256 × ${10}^{-2}$	−	1.972 × ${10}^{-2}$	−
$\frac{1}{32}$	5.054 × ${10}^{-3}$	1.313	7.939 × ${10}^{-3}$	1.313
$\frac{1}{64}$	2.039 × ${10}^{-3}$	1.310	3.202 × ${10}^{-3}$	1.310
$\frac{1}{128}$	8.239 × ${10}^{-4}$	1.307	1.294 × ${10}^{-3}$	1.307

, respectively.

When $\alpha =1.4$, the images of the numerical solution U and the exact solution u at $t=1$ with the mesh $h=1/64,\tau =1/100$ are shown in Figure 3 and Figure 4.

Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12 show the computational errors and convergence orders of the numerical solution U when $\alpha$ takes different values. It can be seen that in the sense of the maximum norm and the $L^2$ norm, the order of time convergence is $3-\alpha$, and the order of space convergence is 4, which demonstrates the effectiveness of the C-N compact difference schemes established in Section 2. Figure 3 and Figure 4 show the graphs of the numerical solution U and the exact solution u, which show good fitting results.

Example 3.

In the problem (1), let $\Omega =(0,1)\times (0,1),T=3$, $a=2.78$, $u(x,y,0)={\phi }_1(x,y)=sin\pi xsin\pi y$, ${\displaystyle \frac{\partial u}{\partial t}}(x,y,0)={\phi }_2(x,y)=0$, $f(x,y,t)=0$.

We simulate the influence of the damping coefficient $\mu$ on the vibration amplitude of the damped plate using the C-N compact difference schemes (15) and (16) proposed in Section 2. Let $\alpha$ = 1.5; when $\mu$ is 1, 2, 3 and 5, the images of the center point of the thin plate are shown in the Figure 5, Figure 6, Figure 7 and Figure 8, respectively, with the spatial step $h=\frac{1}{M_x}=\frac{1}{M_y}$ and the time step $\tau =\frac{3}{N}$.

From Figure 5, Figure 6, Figure 7 and Figure 8, it is shown that using the fractional derivative equation to simulate the vibration of the damped plate and using the C-N compact difference method to solve this model can well characterize the way the amplitude decreases more quickly with the increase in the damping coefficient.

6. Conclusions

This paper seeks to solve time-fractional damped plate vibration equations. By introducing the second-order space derivative and the first-order time derivative as intermediate variables, the fourth-order differential equations are converted into second-order differential equation systems, and the C-N compact difference formats for solving time-fractional damped plate vibration problems are established. Numerical cases are given to validate the convergence order and feasibility of the given difference formats, as well as to simulate the influence of the damping coefficient on the vibration amplitude. This research topic will be further pursued by investigating algorithms that are more effective for solving time-fractional damped plate vibration equations and further discussing the simulation performance with different values of $\alpha$, which is the order of the fraction-order derivative, for solving time-fractional damped plate vibration equations.