Numerical Schemes for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Model in a Double-Layered Nanoscale Thin Film

Abstract

This article proposes a time fractional dual-phase-lagging (DPL) heat conduction model in a double-layered nanoscale thin film with the temperature-jump boundary condition and a thermal lagging effect interfacial condition between layers. The model is proved to be well-posed. A finite difference scheme with second-order spatial convergence accuracy in maximum norm is then presented for solving the fractional DPL model. Unconditional stability and convergence of the scheme are proved by using the discrete energy method. A numerical example without exact solution is given to verify the accuracy of the scheme. Finally, we show the applicability of the time fractional DPL model by predicting the temperature rise in a double-layered nanoscale thin film, where a gold layer is on a chromium padding layer exposed to an ultrashort-pulsed laser heating.

1 Introduction

When dealing with micro/nano scale heat conduction, such as thin film exposed to ultrashort-pulsed laser heating or thermal analysis in nanoscale metal-oxide-semiconductor field-effect-transistor (MOSFET) in the semiconductor industry, the dual-phase-lagging (DPL) model,

$$\begin{aligned} \rho c_{\rho } (T_t + \tau _q T_{tt})(x,t) = \kappa (T + \tau _{T} T_t)_{xx} (x,t) + f(x,t), \end{aligned}$$

(1)

is one of excellent candidates for solving micro/nanoscale heat transfer problems [1,2,3,4,5,6,7,8,9]. The DPL model was derived based on the non-Fourier’s law

$$\begin{aligned} q(x,t) + \tau _q q_t(x,t) = -\kappa (T+\tau _{T}T_t)_x(x,t), \end{aligned}$$

(2)

coupled with the energy equation

$$\begin{aligned} \rho c_{\rho } T_t(x,t) = -q_x(x,t) + S(x,t). \end{aligned}$$

(3)

Here, $\kappa$ is the conductivity, $\tau _{q}$, $\tau _T$ are the phase lags of the heat flux q and temperature gradient $\nabla T$, respectively, $\rho$ and $c_{\rho }$ denote the density and the specific heat of the material, respectively, and $f(x,t) = S(x,t) + \tau _q S_t(x,t)$, where S(x, t) is a source term.

Fractional calculus has been successfully used to modulate several models in heat conduction and other media and has gained importance in heat conduction and thermoelastic problems [10]. Sherief et al. [11] have proposed the fractional non-Fourier law as $q+\tau _{q}$$_{0}^{C}D_{t}^{\alpha }q=-\kappa T_{x}$, $0<\alpha <1,$ where $_{0}^{C}D_{t}^{\alpha }$ is the Caputo-type time-fractional derivative such that $_{0}^{C}D_{t}^{\alpha }u(t)=\frac{1}{\varGamma (1-\alpha )} \displaystyle \int _{0}^{t} \frac{u^{\prime }(s)}{(t-s)^{\alpha }}ds,~ 0< \alpha < 1$, $\varGamma (\cdot )$ is the gamma function [12]. Results show a good agreement with experimental data when using fractional derivatives for description of viscoelastic materials [11]. Youssef [13] has assumed another form for the nonlocal non-Fourier law as $q+\tau _{q}q_{t}=-\kappa I^{\alpha -1}T_{x},~0<\alpha \le 2,$ where $I^{\alpha }$ is the Riemann-Louisville fractional integral such that $I^{\alpha }u(t)=\frac{1}{\varGamma (n-\alpha )}\frac{d^{n}}{dt^{n}}\int \limits _{0}^{t}\frac{ u(s)}{(t-s)^{\alpha -n+1}}ds,$$n-1\le \alpha <n,$ as given in [14]. When $\alpha >1$, it indicates the strong conductivity [14]. Yu et al. [15] has applied the fractional-order generalized DPL model for nanoscale heat transfer in electro-magneto-thermoelastic media. It has been reported that there is larger heat conductivity for nanostructured carbon materials [16, 17]. Thus, the fractional-order generalized DPL model that includes the concept of non-locality [18] can be an excellent candidate for such a nanoscale heat transfer. Recently, we [19] have studied a time-fractional DPL heat conduction equation

$$\begin{aligned} \rho c_{\rho } \left( T_t + \frac{(\tau _q)^{\alpha }}{\varGamma (1+\alpha )} {}^C_{0}D^{\alpha +1}_{t} T\right) (x,t) = \kappa \Bigg ( T + \frac{(\tau _T)^{\alpha }}{\varGamma (1+\alpha )} \cdot {}^C_{0} D^{\alpha }_{t} T\Bigg )_{xx}(x,t)+ f(x,t).\nonumber \\ \end{aligned}$$

(4)

Here, $\kappa = \rho c_{\rho } |v| \frac{\ell _f}{3}$ where $\ell _f$ is the phonon mean free path, |v| is the heat carrier group velocity and $|v|=\frac{\ell _{f}}{\tau _{q}}$, $f(x,t) = S(x,t) + \frac{(\tau _q)^{\alpha }}{\varGamma (1+\alpha )} {}^C_{0} D^{\alpha }_t S(x,t)$ and the Caputo fractional derivatives [20] are defined by

$$\begin{aligned} {}^C_{0} D^{\alpha }_t T(x,t)= & {} \frac{1}{\varGamma (1-\alpha )} \int ^{t}_{0} \frac{T_{s}(x,s)}{(t-s)^{\alpha }} ds, \quad t>0,\quad 0<\alpha<1, \\ {}^C_{0} D^{\alpha +1}_t T(x,t)= & {} \frac{1}{\varGamma (1-\alpha )} \int ^{t}_{0} \frac{T_{ss}(x,s)}{(t-s)^{\alpha }} ds, \quad t>0,\quad 0<\alpha <1. \end{aligned}$$

We chose the Caputo fractional definition for two reasons. One is that the Caputo derivative allows the utilization of physically interpretable initial conditions [20]. The other is that L1 approximation for the Caputo derivative is well-developed, which we will use for the development of the numerical scheme in this study.

By introducing these non-dimensional parameters:

$$\begin{aligned} u = \frac{T-T_0}{T_0},\quad t_{\star } =[ \varGamma (1+\alpha )]^{\frac{1}{\alpha }} \frac{t}{\tau _q},\quad x_{\star } = \frac{x}{L_c},\quad K_n = \frac{\ell _f}{L_c},\quad B =\frac{\tau _T}{\tau _q}, \end{aligned}$$

where $L_c$ is the characteristic length, and $K_n$ is the Knudsen number, we obtained a fractional DPL model in dimensionless form as follows (without asterisk):

$$\begin{aligned} u_t(x,t)+{}^{C}_{0}D^{\alpha +1}_{t}u(x,t) =\frac{K_n^2}{3[\varGamma (1+\alpha )]^{1/\alpha }}\bigg ( u+B^{\alpha } \cdot {}^{C}_{0}D^{\alpha }_{t} u\bigg ) _{xx}(x,t)+F(x,t) \end{aligned}$$

(5)

with the temperature-jump boundary conditions

$$\begin{aligned} -\gamma K_nu_x(0,t)+u(0,t)= \phi _1(t), \quad \gamma K_nu_x(L,t)+u(L,t)=\phi _2(t), \end{aligned}$$

(6)

and the initial condition

$$\begin{aligned} u(x,0)=\varphi (x),\quad u_t(x,0)=\psi (x). \end{aligned}$$

(7)

In [19], we further proposed an accurate and unconditionally stable finite difference scheme for solving the above fractional DPL model. By changing $K_n$ and fractional order $\alpha$, we applied the fractional DPL model to simulate a simple nanoscale semiconductor silicon device. Results indicate that our model can be an excellent candidate for analyzing the temperature instability appearing in electronic [21].

It is noted that layered structures have appeared in many engineering systems such as biological tissues, micro-electronic devices, thin films, reactor walls, thermoelectric power conversion, thermal coating, metal oxide semiconductors, and thermal processing of DNA origami nanostructures [4, 22,23,24,25]. In particular, the multi-layered metal thin-films, for example, gold-coated metal mirrors, are often used in high-power infrared-laser systems to avoid thermal damage at the front surface of a single layer film caused by the high-power laser energy [26]. Furthermore, to achieve high thermoelectric efficiency, a low thermal conductivity is required. Low thermal conductivity is often realized by nano-structuring with the introduction of a high density of materials [24]. All semiconductor devices possess metal contacts and hence the study of heat transport through metal-semiconductor interfaces is a technologically relevant problem [27]. Thus, analyzing heat transfer in layered structures is of crucial importance for the design and operation of nano-devices and the optimization of thermal processing of nano-materials.

For this purpose, we extend our study to the multilayered structure case and propose a time fractional DPL heat conduction model in a double-layered nanoscale thin film, as shown in Fig. 1. We then develop a second-order finite difference scheme for solving the fractional DPL model. The rest of the article is organized as follows: In Sect. 2, we propose a fractional DPL heat conduction model for nanoscale heat conduction in a double-layered thin film with the temperature-jump boundary condition and a thermal lagging effect interfacial condition between layers. In Sect. 3, we obtain an energy estimate for ensuring the mathematical model to be well-posed. Some useful notations and lemmas are presented in Sect. 4. After that, we construct a finite difference scheme for solving the mathematical model. In Sect. 5, the unconditional stability and convergence of the scheme are rigorously analyzed. In Sect. 6, we provide a numerical example to support the theoretical analysis and then apply the numerical scheme to the thermal analysis for a gold thin layer on a chromium padding layer irradiated by an ultrashort-pulsed laser. Finally, we summarize the major results of this work in Sect. 7.

Fig. 1

Double-layered structure (above) and the mesh for numerical schemes (below)

2 Time Fractional DPL Heat Conduction Model

Consider the time fractional DPL heat conduction equations in a double-layered nanoscale thin film as follows:

$$\begin{aligned}&\rho _1c^1_{\rho } \bigg (T_t +\frac{(\tau _{q,1})^{\alpha }}{\varGamma (1+\alpha )} {}^C_{0} D^{\alpha +1}_t T\bigg )(x,t) =\kappa _1\bigg (T+\frac{(\tau _{T,1})^{\beta }}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_{t} T\bigg )_{xx}(x,t)\nonumber \\&\quad +f_1(x,t), ~~ x\in (0,l'_x),~ t\in (0,L'_t], \end{aligned}$$

(8)

$$\begin{aligned}&\rho _2c^2_{\rho } \bigg (T_t+\frac{(\tau _{q,2})^{\alpha }}{\varGamma (1+\alpha )} {}^C_{0}D^{\alpha +1}_t T \bigg )(x,t) =\kappa _2\bigg (T+\frac{(\tau _{T,2})^{\beta }}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_t T\bigg )_{xx}(x,t)\nonumber \\&\quad +f_2(x,t), ~~x\in (l'_x,L'_x),~t\in (0,L'_t], \end{aligned}$$

(9)

where $0<\alpha ,\beta <1.$ Introducing several non-dimensional parameters:

$$\begin{aligned} u=\frac{T-T_0}{T_0},~~ t_{\star }=\frac{t}{\tau _{q,1}+\tau _{q,2}},~~ x_{\star }=\frac{x}{L_{c}},~~ K_{\mathrm{n},i}=\frac{\ell _{f,i}}{L_{c}},~~B_i=\frac{\tau _{T,i}}{\tau _{q,1}+\tau _{q,2}},\quad i=1,2 \end{aligned}$$

where $\ell _{f,i}$ and $L_{c}$ are the phonon free path length and the characteristic length in different layer nanoscale thin films, respectively.

Let $\eta _1=\frac{\tau _{q,1}}{\tau _{q,1}+\tau _{q,2}}$ and $\eta _2=\frac{\tau _{q,2}}{\tau _{q,1}+\tau _{q,2}}$. We can express the governing equations (8) and (9) in a dimensionless form by omitting the subscripts of the non-dimensional parameters $x_{\star }$ and $t_{\star }$ as follows,

$$\begin{aligned}&u_t(x,t)+\frac{\eta _{1}^{\alpha }}{\varGamma (1+\alpha )} {}^C_{0} D^{\alpha +1}_tu(x,t) =\frac{K^2_{\mathrm{n,1}}}{3\eta _{1}}\bigg (u+\frac{B_1^{\beta }}{\varGamma (1+\beta )} {}^C_{0} D^{\beta }_t u\bigg )_{xx}(x,t)\nonumber \\&\quad +F_1(x,t),~x\in (0,l_x), ~t\in (0,L_t], \end{aligned}$$

(10)

$$\begin{aligned}&u_t(x,t)+\frac{\eta _{2}^{\alpha }}{\varGamma (1+\alpha )} {}^C_{0}D^{\alpha +1}_tu(x,t) =\frac{ K^2_{\mathrm{n,2}}}{3\eta _{2}}\bigg (u+\frac{B_2^{\beta }}{\varGamma (1+\beta )}{}^C_{0}D^{\beta }_t u\bigg )_{xx}(x,t)\nonumber \\&\quad +F_2(x,t), ~x\in (l_x,L_x), ~t\in (0,L_t], \end{aligned}$$

(11)

with the source terms

$$\begin{aligned}&F_1(x,t)=\frac{\tau _{q,1}+\tau _{q,2}}{\rho _1c_1T_0}f_1(x,t), \quad x \in (0,l_x), \quad t \in (0,L_t], \\&F_2(x,t)=\frac{\tau _{q,1}+\tau _{q,2}}{\rho _2c_2T_0}f_2(x,t), \quad x \in (l_x,L_x), \quad t \in (0,L_t]. \end{aligned}$$

Based on the assumption of the perfect thermal contact between double layers, we propose the interfacial conditions at $x=l_x$ below

$$\begin{aligned}&u(l_x-0,t)=u(l_x+0,t),\quad t\in [0,L_t], \end{aligned}$$

(12)

$$\begin{aligned}&\frac{ K^2_{\mathrm{n,1}}}{\eta _1}\bigg (u+\frac{B^{\beta }_1}{\varGamma (1+\beta )}{}^C_{0} D^{\beta }_t u\bigg )_x(l_x-0,t) \nonumber \\&\quad =\frac{ K^2_{\mathrm{n,2}}}{\eta _2}\bigg (u+\frac{B^{\beta }_2}{\varGamma (1+\beta )}{}^C_{0} D^{\beta }_t u\bigg )_x(l_x+0,t),\quad t\in [0,L_t], \end{aligned}$$

(13)

which ensures that the normalized fractional DPL model is well-posed. In order to catch the effects of boundary phonon scattering inside a nano-size geometry, the temperature-jump boundary conditions [19, 28] are introduced as follows,

$$\begin{aligned}&-\gamma _1 K_{\mathrm{n,1}}u_x(0,t)+u(0,t)= \phi _1(t), \quad t\in (0,L_t], \end{aligned}$$

(14)

$$\begin{aligned}&\gamma _2 K_{\mathrm{n,2}}u_x(L_x,t)+u(L_x,t)=\phi _2(t), \quad t\in (0,L_t], \end{aligned}$$

(15)

together with the initial conditions

$$\begin{aligned} u(x,0)=\varphi (x),\quad u_t(x,0)=\psi (x),\quad x\in [0,L_x]. \end{aligned}$$

(16)

It should be pointed out that from Eqs. (8) to (9), we can see that when $\alpha =\beta =0$, the fractional DPL heat conduction equation reduces to the traditional heat conduction equation, and on the other hand, when $\alpha =\beta =1$, it reduces to the DPL heat conduction equation with integer order derivatives. To see how the temperature changes from the traditional heat conduction equation to the DPL heat conduction equation with integer order derivatives, we consider only the case when $0\le \alpha ,\beta \le 1.$ This may be considered as some type of sub-DPL heat conduction behavior.

3 Well Posedness of the Fractional DPL Model

In our recent work [19], an energy estimation for the time fractional DPL in a single nanoscale thin film with the fractional order $\alpha =\beta$, i.e.,

$$\begin{aligned}&u_t(x,t) + \frac{\eta ^{\alpha }}{\varGamma (1+\alpha )} {}^C_{0} D^{\alpha +1}_t u(x,t) = \frac{K^2_{\mathrm{n}}}{3\eta } \left( u+ \frac{B^{\alpha }}{\varGamma (1+\alpha )} {}^C_{0}D^{\alpha }_t u\right) _{xx} (x,t) \nonumber \\&\quad +F(x,t),\quad x\in (0,L_x),\quad t\in (0,L_t], \end{aligned}$$

(17)

was deduced by introducing an intermediate variable

$$\begin{aligned} v(x,t) = u(x,t) + \frac{B^{\alpha }}{\varGamma (1+\alpha )} \cdot {}^C_{0}D^{\alpha }_{t} u(x,t). \end{aligned}$$

(18)

This transform brings in an inevitable restriction on the initial condition $u_{t}(x,0)=0$ when exchanging the first-order differential operator $D_t$ and the Caputo fractional operator ${}^C_{0}D^{\alpha }_t$. The commutativity of the two operators is needed in the proof of Theorem 2.3 in [19]. To detour this restrictive condition, in this section, we do not introduce an intermediate variable like (18), but flexibly utilize the corresponding properties of the Caputo fractional derivative ${}^C_{0}D^{\alpha }_t~(0<\alpha <1)$ to verify that the time fractional DPL system in a double-layered nanoscale thin films (10)–(16) is well-posed. This will ensure that the presented energy function can be effectively controlled by the initial values and the source terms.

We now present several useful lemmas with respect to the Caputo fractional derivative operator ${}^C_{0}D^{\alpha }_t~(0<\alpha <1)$, which will be used for obtaining an energy estimation of the governing model (10)–(16). We firstly introduce the Riemann-Liouville (RL) fractional integral of order $\alpha$ [20],

$$\begin{aligned} {}^{RL}_{~~0} D^{-\alpha }_{t} y(t) =\frac{1}{\varGamma (\alpha )} \int ^t_0 \frac{y(\xi )d\xi }{(t-\xi )^{1-\alpha }}, \quad t>0,\quad \alpha >0, \end{aligned}$$

(19)

and the RL fractional derivative of order $\alpha ~,$

$$\begin{aligned} {}^{RL}_{~~0} D^{\alpha }_{t} y(t) =\frac{1}{\varGamma (1-\alpha )} \frac{d}{dt} \int ^t_0 \frac{y(\xi )d\xi }{(t-\xi )^{\alpha }},\quad t>0,\quad 0<\alpha <1. \end{aligned}$$

(20)

Lemma 1

For any function $y(t)\in C^1([0,T]),$ if $~0<\alpha <1$, it holds that

$$\begin{aligned}&y'(t)\cdot {}^C_{0} D^{\alpha }_{t} y(t)=\frac{1}{2} \frac{d}{dt} {}^{RL}_{~~0} D^{-\alpha }_t \big ({}^C_{0} D^{\alpha }_{t} y(t)\big )^2 \nonumber \\&\quad +\frac{1-\alpha }{2\varGamma (\alpha )} \int ^{t}_{0} \bigg (\bigg [ \int ^{\xi }_{0} \frac{\big ({}^C_{0} D^{\alpha }_{\eta } y(\eta ) \big )^\prime }{(t-\eta )^{1-\alpha }}\mathrm{d}\eta \bigg ] \frac{1}{(t-\xi )^{\alpha }}\mathrm{d}\xi \bigg )^2,\quad t\in (0,T],\quad 0<\alpha <1. \end{aligned}$$

(21)

Proof

For simplicity, denote

$$\begin{aligned} G[y](\alpha ,t)= \frac{\alpha }{2\varGamma (1-\alpha )} \int ^{t}_0 \frac{d \xi }{(t-\xi )^{1-\alpha }} \bigg (\int ^{\xi }_{0} \frac{y_{\eta }(\eta )}{(t-\eta )^{\alpha }}\mathrm{d}\eta \bigg )^2. \end{aligned}$$

Alikhanov [29] proved that if $~0<\alpha <1$,

$$\begin{aligned} y(t)\cdot {}^C_{0} D^{\alpha }_{t} y(t)=\frac{1}{2} {}^C_{0} D^{\alpha }_t y^2(t) + G[y](\alpha ,t). \end{aligned}$$

(22)

We further notice the fact ${}^C_{0}D^{\alpha }_t y(t)|_{t=0}=0$ and use the relation (22). The detailed process is shown as follows,

$$\begin{aligned} y'(t)\cdot {}^C_{0} D^{\alpha }_{t} y(t)&={}^{RL}_{~~0}D^{1-\alpha }_{t} \big ( {}^{RL}_{~~0} D^{\alpha -1}_{t} y'(t)\big ) \cdot {}^C_{0} D^{\alpha }_{t} y(t) \\&={}^{RL}_{~~0} D^{1-\alpha }_{t} ({}^C_{0} D^{\alpha }_{t} y(t) ) \cdot {}^C_{0} D^{\alpha }_{t} y(t) \\&= {}^C_{0} D^{1-\alpha }_t \big ( {}^C_{0} D^{\alpha }_{t} y(t) \big ) \cdot {}^C_{0} D^{\alpha }_{t} y(t) \\&= \frac{1}{2} {}^C_{0} D^{1-\alpha }_t \Big (\big ({}^C_{0}D^{\alpha }_{t}y(t)\big )^2\Big ) +G\big [{}^C_{0}D^{\alpha }_ty\big ](1-\alpha ,t) \\&= \frac{1}{2} {}^{RL}_{~~0} D^{1-\alpha }_t \Big (\big ({}^C_{0}D^{\alpha }_{t}y(t)\big )^2\Big ) +G\big [{}^C_{0}D^{\alpha }_ty\big ](1-\alpha ,t)\\&=\frac{1}{2} \frac{d}{dt} {}^{RL}_{~~0} D^{-\alpha }_t\Big (\big ({}^C_{0} D^{\alpha }_{t}y(t)\big )^2\Big ) +G\big [{}^C_{0} D^{\alpha }_ty\big ](1-\alpha ,t). \end{aligned}$$

Hence, the conclusion holds. $\square$

Lemma 2

Let the function $y(t)\in C^1([0,T]).$ If $~0<\alpha <1,$ then it holds that

$$\begin{aligned} \int ^t_0 {}^{C}_{0}D^{\alpha }_{\tau } y(\tau )d\tau ={}^{RL}_{~~0}D^{\alpha -1}_{t}y(t) - \frac{t^{1-\alpha }}{\varGamma (2-\alpha )} y(0). \end{aligned}$$

Proof

According to the property of the semigroup of the RL fractional integral, we obtain

$$\begin{aligned} \int ^t_0 {}^C_{0} D^{\alpha }_s y(s) ds&={}^{RL}_{~~0} D^{-1}_t \big ( {}^{RL}_{~~0} D^{\alpha -1}_t y'(t)\big ) ={}^{RL}_{~~0} D^{\alpha -1}_t \big ({}^{RL}_{~~0} D^{-1}_t y'(t)\big ) \nonumber \\&= {}^{RL}_{~~0}D^{\alpha -1}_t y(t) - {}^{RL}_{~~0}D^{\alpha -1}_{t}y(0)\nonumber \\&= {}^{RL}_{~~0} D^{\alpha -1}_t y(t) - \frac{t^{1-\alpha }}{\varGamma (2-\alpha )} y(0). \end{aligned}$$

(23)

$\square$

For the solution u(x, t) of the normalized fractional DPL equations in a double-layered nanoscale thin film (10)–(16), we have an energy estimation as shown in the following theorem. For simplicity, we first define an energy function

$$\begin{aligned} E_1(t)&=\frac{1}{\gamma _1 K_{\mathrm{n,1}}} \Big (u^2(0,t) +\frac{B^{\beta }_1}{\varGamma (1+\beta )} {}^{RL}_{~~0} D^{-\beta }_t\big ({}^C_{0} D^{\beta }_t u(0,t) \big )^2\Big ) \\&\quad +\int ^{l_x}_0 \Big (u^2_x(x,t) + \frac{B^{\beta }_1}{\varGamma (1+\beta )} {}^{RL}_{~~0} D^{-\beta }_t\big ({}^C_{0}D^{\beta }_t u_x(x,t) \big )^2 \Big )dx, \quad 0< t \le L_t.\\ E_2(t)&=\frac{1}{\gamma _2 K_{\mathrm{n,2}}} \Big ( u^2(L_x,t) + \frac{B^{\beta }_2}{\varGamma (1+\beta )} {}^{RL}_{~~0}D^{-\beta }_t \big ({}^C_{0} D^{\beta }_tu(L_x,t)\big )^2\Big )\\&\quad +\int ^{L_x}_{l_x} \Big (u^2_x(x,t) + \frac{B^{\beta }_2}{\varGamma (1+\beta )}{}^{RL}_{~~0} D^{-\beta }_t\big ({}^C_{0}D^{\beta }_t u_x(x,t) \big )^2 \Big )dx, \quad 0< t \le L_t\\ E(t)&=\frac{ K^2_{\mathrm{n,1}}}{6\eta _1}E_1(t)+\frac{ K^2_{\mathrm{n,2}}}{6\eta _2}E_2(t), \quad 0< t \le L_t. \end{aligned}$$

Theorem 1

Let u(x, t) be the solution of the normalized time fractional DPL model (10)–(16), subject to the homogeneous boundary conditions, i.e., $\phi _1(t)\equiv 0$ and $\phi _2(t)\equiv 0$. Then, it holds that

$$\begin{aligned} E(t)&\le E(0) + \frac{t^{1-\alpha }}{2\varGamma (2-\alpha )} \cdot \bigg ( \frac{\eta ^{\alpha }_1}{{\varGamma (1+\alpha )}} \int ^{l_x}_{0} \psi _2^2(x)dx+ \frac{\eta ^{\alpha }_2}{\varGamma (1+\alpha )} \int ^{L_x}_{l_x}\psi ^2_2(x) dx \bigg ) \nonumber \\&\quad + \frac{1}{4} \int ^t_0 \Big (\int ^{l_x}_0 F^2_1(x,s)dx + \int ^{L_x}_{l_x} F^2_2(x,s) dx\Big ) ds,\quad 0< t \le L_t. \end{aligned}$$

(24)

Proof

Multiplying (10) by $u_t(x,t)$ and integrating the result with respect to the variable x from 0 to $l_{x}$, we have

$$\begin{aligned}&\int ^{l_{x}}_{0}u^2_t(x,t)dx+ \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )}\int ^{l_{x}}_0u_t(x,t) \cdot {}^C_{0}D^{\alpha +1}_tu(x,t)dx\nonumber \\&\quad =\frac{K^2_{\mathrm{n,1}}}{3\eta _1}\int ^{l_{x}}_0u_t(x,t)\Big (u +\frac{B_1^{\beta }}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_t u\Big )_{xx}(x,t)dx\nonumber \\&\qquad +\int ^{l_{x}}_0u_t(x,t)\cdot F_1(x,t)dx, \quad 0< t\le L_t. \end{aligned}$$

(25)

By virtue of the relation (22), we obtain the inequality

$$\begin{aligned} \int ^{l_{x}}_0u_t(x,t)\cdot {}^C_{0}D^{\alpha +1}_tu(x,t)dx \ge \frac{1}{2}{}^C_{0}D^{\alpha }_{t} \Big (\int ^{l_{x}}_0u^2_t(x,t)dx\Big ). \end{aligned}$$

(26)

Applying the integration by parts, then noticing the boundary condition (14) and Lemma 1, it holds that

$$\begin{aligned}&\int ^{l_{x}}_0u_t(x,t)\cdot \Big (u+\frac{B_1^{\beta }}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_t u \Big )_{xx}(x,t)dx\nonumber \\&\quad = \bigg (u_t \cdot \Big (u+\frac{B_1^{\beta }}{\varGamma (1+\beta )}{}^C_{0}D^{\beta }_t u \Big )_x\bigg )(x,t)\bigg |^{l_x-0}_{x=0}\nonumber \\&\qquad -\int ^{l_x}_{0} u_{xt}(x,t)\cdot \Big (u+\frac{B_1^{\beta }}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_t u \Big )_{x}(x,t) dx \nonumber \\&\quad \le u_t(l_x-0,t) \cdot \Big (u+\frac{B_1^{\beta }}{\varGamma (1+\beta )}{}^C_{0}D^{\beta }_t u \Big )_x(l_x-0,t) -\frac{1}{2}\frac{d}{dt} E_1(t). \end{aligned}$$

(27)

Substituting (26)–(27) into (25) and using the Cauchy–Schwarz inequality with a suitable parameter yield

$$\begin{aligned}&\frac{\eta _1^{\alpha }}{2\varGamma (1+\alpha )} {}^C_{0} D^{\alpha }_{t}\Big (\int ^{l_x}_0u^2_t(x,t)dx\Big ) + \frac{ K^2_{\mathrm{n,1}}}{6\eta _1}\frac{d}{dt} E_1(t) \nonumber \\&\quad \le \frac{ K^2_{\mathrm{n,1}}}{3\eta _1} u_t(l_x-0,t) \cdot \Big (u+\frac{B_1^{\beta }}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_t u \Big )_x(l_x-0,t) \nonumber \\&\qquad +\frac{1}{4} \int ^{l_x}_0 F_1^2(x,t)dx,\quad 0<t\le L_t. \end{aligned}$$

(28)

Similarly, multiplying (11) by $u_t(x,t)$ and integrating the result with respect to the variable x from $l_x$ to $L_x$, we get

$$\begin{aligned}&\int ^{L_x}_{l_x}u^2_t(x,t)dx+ \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )}\int ^{L_x}_{l_x}u_t(x,t) \cdot {}^C_{0}D^{\alpha +1}_tu(x,t)dx\nonumber \\&\quad =\frac{ K_{\mathrm{n,2}}^2}{3\eta _2}\int ^{L_x}_{l_x}u_t(x,t)\Big (u+\frac{B_2^{\beta }}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_t u\Big )_{xx}(x,t)dx\nonumber \\&\qquad +\int ^{L_x}_{l_x}u_t(x,t)\cdot F_2(x,t)dx,\quad 0< t\le L_t. \end{aligned}$$

(29)

Using the similar argument as (26) and (27) gives

$$\begin{aligned}&\frac{\eta _2^{\alpha }}{2\varGamma (1+\alpha )} {}^C_{0} D^{\alpha }_t\Big (\int ^{L_x}_{l_x}u^2_t(x,t)dx\Big ) + \frac{ K^2_{\mathrm{n,2}}}{6\eta _2} \frac{d}{dt}E_2(t)\nonumber \\&\quad \le -\frac{ K_{\mathrm{n,2}}^2}{3\eta _2} u_t(l_x+0,t)\cdot \Big (u+\frac{B_2^{\beta }}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_t u \Big )_x(l_x+0,t)\nonumber \\&\qquad +\frac{1}{4} \int ^{L_x}_{l_x} F_2^2(x,t)dx, \quad 0<t\le L_t. \end{aligned}$$

(30)

Adding (28) and (30) and using the interfacial conditions (12) and (13) yield

$$\begin{aligned}&{}^C_{0} D^{\alpha }_{t} \Big (\frac{\eta ^{\alpha }_1}{2\varGamma (1+\alpha )}\int ^{l_x}_0 u^2_t(x,t)dx +\frac{\eta ^{\alpha }_2}{2\varGamma (1+\alpha )}\int ^{L_x}_{l_x} u^2_t(x,t)dx \Big ) \nonumber \\&\quad +\frac{d E(t)}{dt} \le \frac{1}{4} \left[ \int ^{l_x}_0F^2_1(x,t) dx + \int ^{L_x}_{l_x}F^2_2(x,t) dx \right] . \end{aligned}$$

(31)

Replacing the variable t with s on both sides of (31), integrating the result with respect to the variable s from 0 to t, and then using Lemma 2, we arrive at

$$\begin{aligned}&{}^{RL}_{~~0}D^{\alpha -1}_{t} \Big ( \frac{\eta ^{\alpha }_1}{{2\varGamma (1+\alpha )}} \int ^{l_x}_{0} u^2_t (x,t)dx+ \frac{\eta ^{\alpha }_2}{2\varGamma (1+\alpha )} \int ^{L_x}_{l_x}u^2_t(x,t) dx \Big ) +E(t) \nonumber \\&\quad \le \frac{t^{1-\alpha }}{\varGamma (2-\alpha )} \Big ( \frac{\eta ^{\alpha }_1}{{2\varGamma (1+\alpha )}} \int ^{l_x}_{0} \psi _2^2(x)dx+ \frac{\eta ^{\alpha }_2}{2\varGamma (1+\alpha )} \int ^{L_x}_{l_x}\psi ^2_2(x) dx \Big ) \nonumber \\&\qquad +E(0) + \frac{1}{4} \int ^t_0 \Big [\int ^{l_x}_0 F^2_1(x,s)dx + \int ^{L_x}_{l_x} F^2_2(x,s) dx\Big ] ds. \end{aligned}$$

(32)

Hence, the conclusion holds. $\square$

4 Numerical Method for the Fractional DPL Model

4.1 Discrete Approximations

Divide the interval $[0,L_t]$ into N-subintervals with $\tau = \frac{L_t}{N}$ and $t_k = k\tau ,~0\le k\le N.$ Suppose that $u=(u^0,u^1,\cdots ,u^N)$ be a grid function defined on $\varOmega _{\tau }\equiv \{t_k~|~0\le k \le N\}$. For simplicity, we denote a difference quotient operator and an average operator in time,

$$\begin{aligned} \delta _t u^{k-\frac{1}{2}} = \frac{1}{\tau } (u^k-u^{k-1}),\quad u^{k-\frac{1}{2}} =\frac{1}{2} (u^k+u^{k-1}), \end{aligned}$$

and some fractional numerical differentiation operators (called L1 formula [30]) for the Caputo fractional derivatives as follows,

$$\begin{aligned}&\delta _{\tau }^{\beta } u^k = \frac{\tau ^{-\beta }}{\varGamma (2-\beta )} \Big [ a^{(\beta )}_0 u^k - \sum ^{k-1}_{\ell =1} (a^{(\beta )}_{k-\ell -1} -a^{(\beta )}_{k-\ell }) u^{\ell } -a^{(\beta )}_{k-1} u^0 \Big ], \end{aligned}$$

(33)

$$\begin{aligned}&\delta ^{\beta }_{\tau } u^{k-\frac{1}{2}} =\frac{\tau ^{-\beta }}{\varGamma (2-\beta )} \Big [a^{(\beta )}_0 u^{k-\frac{1}{2}} -\sum ^{k-1}_{\ell =1}(a^{(\beta )}_{k-\ell -1} -a^{(\beta )}_{k-\ell })u^{\ell -\frac{1}{2}} -a^{(\beta )}_{k-1} u^0 \Big ], \end{aligned}$$

(34)

$$\begin{aligned}&\varDelta ^{\alpha }_{\tau } (\delta _t u^{k-\frac{1}{2}},q)= \frac{\tau ^{-\alpha }}{\varGamma (2-\alpha )}\Big [ a^{(\alpha )}_{0} \delta _t u^{k-\frac{1}{2}}-\sum ^{k-1}_{\ell =1}(a^{(\alpha )}_{k-\ell -1}-a^{(\alpha )}_{k-\ell }) \delta _t u^{\ell -\frac{1}{2}}-a_{k-1}^{(\alpha )} q \Big ], \end{aligned}$$

(35)

where q is a given value, and the coefficients

$$\begin{aligned} a^{(\zeta )}_{\ell } = (\ell +1)^{1-\zeta }-\ell ^{1-\zeta },\quad \ell \ge 0,\quad 0<\zeta = \alpha ,\beta <1. \end{aligned}$$

(36)

From the above definition, it is obvious that $\{a^{(\zeta )}_{\ell }\}_{\ell =0}^{\infty }$ is a nonnegative and descending series, i.e.,

$$\begin{aligned} a^{(\zeta )}_{\ell } \ge 0\quad (\ell \ge 0); \quad a^{(\zeta )}_{\ell } - a^{(\zeta )}_{\ell -1} \le 0\quad (\ell \ge 1). \end{aligned}$$

We then list the truncation errors of the L1 formulas in the following lemma.

Lemma 3

[30] Suppose $f(t)\in C^2[0,t_n]$ and $0<\alpha , \beta <1$. We have

$$\begin{aligned} \frac{1}{2} \Big [ {}^C_{0} D^{\beta }_{t} f(t_k) +{}^C_{0} D^{\beta }_{t} f(t_{k-1}) \Big ] = \delta ^{\beta }_{\tau } f^{k-\frac{1}{2}} + O(\tau ^{2-\beta }), \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2} \Big [ {}^C_0 D^{\alpha +1}_t f(t_k) + {}^C_0 D^{\alpha +1}_t f(t_{k-1}) \Big ] = \varDelta _{\tau }^{\alpha } (\delta _{t} f^{k-\frac{1}{2}}, f'(0)) + O(\tau ^{2-\alpha }), \end{aligned}$$

where $f = (f^0,f^1,f^2, \cdots , f^N)$ and each component $f^k\equiv f(t_k),~0\le k\le N$.

For the numerical approximation of the first order temporal derivative, a lemma is prepared below.

Lemma 4

[31] Suppose $f(t) \in C^3[t_{k-1},t_{k}]$ and denote $t_{k-\frac{1}{2}} = \frac{1}{2}(t_{k-1}+t_k)$, then it holds

$$\begin{aligned} \frac{1}{2} \big [ f'(t_k)+ f'(t_{k-1})\big ]&=\frac{f(t_{k}) - f(t_{k-1})}{\tau } + \frac{\tau ^2}{16} \int ^1_0 \Big [ f'''\big (t_{k-\frac{1}{2}} +\frac{\tau }{2}\theta \big ) \\&\quad + f'''\big (t_{k-\frac{1}{2}}-\frac{\tau }{2}\theta \big ) \Big ] (1-\theta ^2) ds. \end{aligned}$$

Next, we introduce some discrete notations in space. Take two positive integers m and $M~(m<M).$ Let $h_1=\frac{l_x}{m},~\omega _l=\{i~|~0\le i\le m\},~ \varOmega _{h,l}=\{x_i~|~x_i=ih_1,~i\in \omega _l\}$ and $h_2=\frac{L_x-l_x}{M-m},~\omega _r=\{i~|~m\le i\le M\}, ~\varOmega _{h,r}=\{x_i~|~x_i=x_m+(i-m)h_2,~i\in \omega _r\},$ as shown in Fig. 1. Let ${\mathscr {U}}_h=\{u~|~u=(u_0,u_1,\cdots , u_M) \}$ be the grid function space defined on $\varOmega _h=\varOmega _{h,l} \cup \varOmega _{h,r}$. For any $u, v\in {\mathscr {U}}_h$, we have

$$\begin{aligned}&\delta _xu_{i-\frac{1}{2}}=\frac{1}{h_1}(u_i-u_{i-1}),\quad i\in \omega _{l}\backslash \{0\}; \quad \delta _xu_{i-\frac{1}{2}}=\frac{1}{h_2}(u_i-u_{i-1}),\quad i\in \omega _{r}\backslash \{m\};\\&(u,v)_l=h_1\bigg ( \frac{1}{2}u_0v_0+\sum ^{m-1}_{i=1}u_iv_i+\frac{1}{2}u_mv_m \bigg ), \quad \Vert u\Vert _l=\sqrt{(u,u)_l};\\&(u,v)_r=h_2\bigg ( \frac{1}{2}u_mv_m+\sum ^{M-1}_{i=m+1}u_iv_i+\frac{1}{2}u_Mv_M \bigg ), \quad \Vert u\Vert _r=\sqrt{(u,u)_r};\\&(\delta _xu,\delta _xv)_l=h_1\sum ^{m}_{i=1}\big (\delta _xu_{i-\frac{1}{2}}\big )\delta _xv_{i-\frac{1}{2}}, \quad \Vert \delta _xu\Vert _l=\sqrt{(\delta _x u,\delta _xu)_l};\\&(\delta _xu,\delta _xv)_r=h_2\sum ^{M}_{i=m+1}\big (\delta _xu_{i-\frac{1}{2}}\big )\delta _xv_{i-\frac{1}{2}}, \quad \Vert \delta _xu\Vert _r=\sqrt{(\delta _x u,\delta _xu)_r};\\&\Vert u\Vert _{l,\infty } = \max _{0\le i \le m} |u_i|, \quad \Vert u\Vert _{r,\infty } = \max _{m\le i \le M} |u_i|. \end{aligned}$$

For any $u\in {\mathscr {U}}_h$, we denote

$$\begin{aligned} \delta ^2_{x} u_i=\left\{ \begin{array}{ll} \frac{2}{h_1} (\delta _{x} u_{\frac{1}{2}} - \frac{1}{\gamma _1 K_{\mathrm{n,1}}}u_0), \quad i=0, \\ \frac{1}{h_1}(\delta _x u_{i+\frac{1}{2}}- \delta _x u_{i-\frac{1}{2}}),\quad 1\le i\le m-1, \end{array} \right. \end{aligned}$$

(37)

and

$$\begin{aligned} \delta ^2_{x} u_i=\left\{ \begin{array}{ll} \frac{1}{h_2}(\delta _x u_{i+\frac{1}{2}}- \delta _x u_{i-\frac{1}{2}}),\quad m+1\le i\le M-1,\\ \frac{2}{h_2}(-\frac{1}{\gamma _2 K_{\mathrm{n,2}}}u_M-\delta _x u_{M-\frac{1}{2}}),\quad i=M. \end{array} \right. \end{aligned}$$

(38)

Lemma 5

Let c and h be two given constants and $h>0.$

• (I)    Suppose $g(x)\in C^3[c,c+h],$ then it holds that

  $$\begin{aligned} g''(c) = \frac{2}{h} \left[ \frac{g(c+h)-g(c)}{h} -g'(c) \right] - h\int ^1_0 g'''(c+\theta h)(1-\theta )^2 d\theta ; \end{aligned}$$

• (II) Suppose $g(x)\in C^3[c-h,c],$ then it holds that

  $$\begin{aligned} g''(c) = \frac{2}{h} \left[ g'(c) - \frac{g(c)-g(c-h)}{h} \right] +h \int ^1_0 g'''(c-\theta h)(1-\theta )^2d\theta ; \end{aligned}$$

• (III)    Suppose $g(x)\in C^4[c-h,c+h],$ then it holds that

  $$\begin{aligned} g''(c)= & {} \frac{1}{h^2} \left[ g(c+h) - 2g(c) + g(c-h)\right] \\&- \frac{h^2}{6} \int ^1_0 [g^{(4)}(c-\theta h) + g^{(4)}(c+\theta h)](1-\theta )^3d\theta . \end{aligned}$$

Proof

The conclusions (I)–(III) are easy to achieve by applying the Taylor expansion with the integral remainder term for the smooth function g(x). $\square$

4.2 Finite Difference Scheme

From now on, we will construct a discretization scheme for the problem (10)–(16). The derivation of the scheme adopts the tactic that the semi-discretization of the spatial derivative is prior to the approximation of the temporal derivative. Let u(x, t) be the solution of (10)–(16). Denote the discrete functions by

$$\begin{aligned} U_i(t)= & {} u(x_i,t), ~i \in \omega _l \cup \omega _r; ~(F_1)_i(t) = F_1(x_i,t),~i \in \omega _l ; ~(F_2)_i(t) \\= & {} F_2(x_i,t), ~ i \in \omega _r; ~0\le t\le L_t;\\ U^k_i= & {} u(x_i,t_k),~i \in \omega _l \cup \omega _r; ~(F_1)^k_i = F_1(x_i,t_k),~ i \in \omega _l; ~(F_2)^k_i \\= & {} F_2(x_i,t_k),~ i \in \omega _r;~0\le k\le N; \end{aligned}$$

and $\varphi _i = \varphi (x_i),~~ \psi _i = \psi (x_i),~~i \in \omega _l \cup \omega _r.$

For the sake of brevity in writing the scheme, we firstly introduce several functions with respect to the variable t, i.e.,

$$\begin{aligned} (G_1)_i(t)= & {} \left\{ \begin{array}{ll} \frac{2 K_{\mathrm{n,1}}}{3\gamma _1\eta _1h_1}\Big (\phi _1(t) +\frac{B^{\beta }_1}{\varGamma (1+\beta )} {}^C_{0} D^{\beta }_{t} \phi _1(t) \Big )+ (F_1)_0(t),\quad i=0,\\ (F_1)_i(t),~~ 1\le i\le m-1,\\ \frac{h_1}{h_1+h_2} (F_1)_m(t), \quad i=m; \end{array} \right. \end{aligned}$$

(39)

$$\begin{aligned} (G_2)_i(t)= & {} \left\{ \begin{array}{ll} \frac{h_2}{h_1+h_2}(F_2)_m(t), \quad i=m, \\ (F_2)(t), \quad m+1\le i\le M-1, \\ \frac{2K_{\mathrm{n,2}}}{3\gamma _2\eta _2h_2}\Big (\phi _2(t) +\frac{B^{\beta }_2}{\varGamma (1+\beta )} {}^C_{0} D^{\beta }_{t} \phi _2(t) \Big )+ (F_2)_M(t),\quad i = M; \end{array} \right. \end{aligned}$$

(40)

(41)

and

(42)

Let us begin by considering (10) at $(x_i,t) \in \{\varOmega _{h,l}\backslash {x_m} \}\times (0,L_t]$. According to Lemma  5 and the left boundary condition (14), a semi-discretization scheme reads

$$\begin{aligned}&\frac{d}{dt} U_0(t)+ \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} {}^{C}_0D^{\alpha +1}_{t}U_0(t) =\frac{K_{\mathrm{n,1}}^2}{3\eta _1}\delta ^2_x \Big (U_0(t)+\frac{B_1^{\beta }}{\varGamma (1+\beta )}\cdot {}^C_{0}D^{\beta }_t U_0(t)\Big )\nonumber \\&\quad +(G_1)_0(t) + h_1\cdot r_0(t), \quad i=0, \end{aligned}$$

(43)

$$\begin{aligned}&\frac{d}{dt} U_i(t)+ \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} {}^{C}_0D^{\alpha +1}_{t}U_i(t) =\frac{K_{\mathrm{n,1}}^2}{3\eta _1}\delta ^2_x \Big (U_i(t)+\frac{B_1^{\beta }}{\varGamma (1+\beta )}\cdot {}^C_{0}D^{\beta }_t U_i(t)\Big )\nonumber \\&\quad +(G_1)_i(t) + h^2_1 \cdot r_i(t), \quad 1\le i\le m-1. \end{aligned}$$

(44)

We then move on to consider (11) at $(x_i,t)\in \{\varOmega _{h,r}\backslash {x_m}\} \times (0,L_t]$. A similar technique with (43) and (44) yields the following semi-discretization scheme:

$$\begin{aligned}&\frac{d}{dt} U_i(t)+ \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} {}^{C}_0D^{\alpha +1}_{t}U_i(t) \nonumber \\&\quad = \frac{K_{\mathrm{n,2}}^2}{3\eta _2}\delta ^2_x \Big (U_i(t)+\frac{B_2^{\beta }}{\varGamma (1+\beta )}\cdot {}^C_{0}D^{\beta }_tU_i(t)\Big )+(G_2)_i(t) \nonumber \\&\qquad + h^2_2\cdot s_i(t),\quad m+1\le i\le M-1, \end{aligned}$$

(45)

$$\begin{aligned}&\frac{d}{dt} U_M(t)+ \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} {}^{C}_0D^{\alpha +1}_{t}U_M(t) \nonumber \\&\quad = \frac{K_{\mathrm{n,2}}^2}{3\eta _2}\delta ^2_x \Big (U_M(t)+\frac{B_2^{\beta }}{\varGamma (1+\beta )}\cdot {}^C_{0}D^{\beta }_tU_M(t)\Big )+(G_2)_M(t) + h_2 \cdot s_M(t), \quad i=M. \end{aligned}$$

(46)

Next, a semi-discretization scheme on the interface is considered. Using Lemma 5 for (10) at $(x_m-0,t)$ and for (11) at $(x_m+0,t)$, respectively, and noticing the interfacial condition (12), we can obtain

$$\begin{aligned}&\frac{d}{dt} U_m(t)+ \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} {}^{C}_0D^{\alpha +1}_{t}U_m(t) =\frac{ K^2_{\mathrm{n,1}}}{3\eta _1} \frac{2}{h_1}\bigg [ \Big (u + \frac{B^{\beta }_1}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_tu \Big )_x(x_m-0,t) \nonumber \\&\quad - \delta _x\Big (U_{m-\frac{1}{2}}(t) +\frac{B^{\beta }_{1}}{\varGamma (1+\beta )}{}^C_0 D^{\beta }_{t} U_{m-\frac{1}{2}}(t)\Big ) \bigg ] + (F_1)_m(t) + h_1 \cdot r_m(t), \end{aligned}$$

(47)

$$\begin{aligned}&\frac{d}{dt} U_m(t)+ \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} {}^{C}_0D^{\alpha +1}_{t}U_m(t) =\frac{ K^2_{\mathrm{n,2}}}{3\eta _2}\frac{2}{h_2} \bigg [ - \Big (u + \frac{B^{\beta }_2}{\varGamma (1+\beta )} {}^C_{0}D^{\beta }_tu \Big )_x(x_m+0,t) \nonumber \\&\quad + \delta _x\Big (U_{m+\frac{1}{2}}(t) +\frac{B^{\beta }_{2}}{\varGamma (1+\beta )}{}^C_{0}D^{\beta }_t U_{m+\frac{1}{2}}(t)\Big ) \bigg ] + (F_2)_m(t) + h_2 \cdot s_m(t). \end{aligned}$$

(48)

Multiplying (47) by $\frac{h_1}{h_1+h_2}$ and (48) by $\frac{h_2}{h_1+h_2}$, respectively, and then adding the results and noticing the interfacial condition (13), we deduce

$$\begin{aligned}&\frac{h_1}{h_1+h_2} \Big [ \frac{d}{dt} U_m(t)+ \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )}{}^{C}_0D^{\alpha +1}_{t}U_m(t) \Big ]\nonumber \\&\qquad + \frac{h_2}{h_1+h_2} \Big [ \frac{d}{dt} U_m(t)+ \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )}{}^{C}_0D^{\alpha +1}_{t}U_m(t) \Big ] \nonumber \\&\quad =\frac{2}{3(h_1+h_2)}\bigg [ \frac{ K^2_{\mathrm{n,2}}}{\eta _2} \delta _x\Big (U_{m+\frac{1}{2}}(t) + \frac{B^{\beta }_2}{\varGamma (1+\beta )} {}^C_0 D^{\beta }_t U_{m+\frac{1}{2}}(t)\Big ) \nonumber \\&\qquad - \frac{ K^2_{\mathrm{n,1}}}{\eta _1} \delta _x\Big ( U_{m-\frac{1}{2}}(t) + \frac{B^{\beta }_1}{\varGamma (1+\beta )} \cdot {}^C_0 D^{\beta }_t U_{m-\frac{1}{2}}(t) \Big )\bigg ]\nonumber \\&\qquad +(G_1)_m(t) + (G_2)_m(t) + \frac{h_1}{h_1+h_2}h_1 \cdot r_m(t) +\frac{h_2}{h_1+h_2} h_2 \cdot s_m(t). \end{aligned}$$

(49)

Further, considering (43)–(46) and (49) at $t=t_{k-1}$ and $t = t_k$, respectively, averaging the results and then using Lemmas 3 and 4, we have

$$\begin{aligned}&\delta _tU^{k-\frac{1}{2}}_i +\frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \varDelta _{\tau }^{\alpha } \big (\delta _tU^{k-\frac{1}{2}}_i, \psi _i \big )= \frac{ K_{\mathrm{n,1}}^2}{3\eta _1}\delta _x^2\Big (U^{k-\frac{1}{2}}_i +\frac{B^{\beta }_1}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }U^{k-\frac{1}{2}}_i\Big ) +(G_1)^{k-\frac{1}{2}}_i \nonumber \\&\qquad +(R^t)^{k-\frac{1}{2}}_i+(R^x)^{k-\frac{1}{2}}_i, \quad i \in \omega _l\backslash \{m\},\quad 1\le k\le N, \end{aligned}$$

(50)

$$\begin{aligned}&\frac{h_1}{h_1+h_2} \Big (\delta _tU^{k-\frac{1}{2}}_m + \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \varDelta ^{\alpha }_{\tau } \big (\delta _tU_m^{k-\frac{1}{2}},\psi _m\big ) \Big )\nonumber \\&\qquad +\frac{h_2}{h_1+h_2}\Big (\delta _tU^{k-\frac{1}{2}}_m + \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} \varDelta ^{\alpha }_{\tau } \big (\delta _tU_m^{k-\frac{1}{2}},\psi _m\big ) \Big ) \nonumber \\&\quad =\frac{2}{3(h_1+h_2)}\bigg [ \frac{ K_{\mathrm{n,2}}^2}{\eta _2} \delta _x\Big (U^{k-\frac{1}{2}}_{m+\frac{1}{2}} + \frac{B^{\beta }_2}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }U^{k-\frac{1}{2}}_{m+\frac{1}{2}} \Big ) \nonumber \\&\qquad -\frac{ K_{\mathrm{n,1}}^2}{\eta _1} \delta _x\Big ( U^{k-\frac{1}{2}}_{m-\frac{1}{2}}+ \frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau }U^{k-\frac{1}{2}}_{m-\frac{1}{2}} \Big ) \bigg ]\nonumber \\&\qquad +(G_1)^{k-\frac{1}{2}}_m + (G_2)^{k-\frac{1}{2}}_m + \frac{h_1}{h_1+h_2}[(R^t)^{k-\frac{1}{2}}_m+ (R^x)^{k-\frac{1}{2}}_m]\nonumber \\&\qquad +\frac{h_2}{h_1+h_2}[(S^t)^{k-\frac{1}{2}}_m + (S^x)^{k-\frac{1}{2}}_m],\quad 1\le k\le N, \end{aligned}$$

(51)

$$\begin{aligned}&\qquad \delta _tU^{k-\frac{1}{2}}_i +\frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} \varDelta _{\tau }^{\alpha } \big (\delta _tU^{k-\frac{1}{2}}_i, \psi _i \big )\nonumber \\&\quad =\frac{ K_{\mathrm{n,2}}^2}{3\eta _2}\delta _x^2\Big (U^{k-\frac{1}{2}}_i +\frac{B^{\beta }_2}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }U^{k -\frac{1}{2}}_i\Big )+(G_2)^{k-\frac{1}{2}}_i\nonumber \\&\qquad +(S^t)^{k-\frac{1}{2}}_i+(S^x)^{k-\frac{1}{2}}_i,\quad i \in \omega _r\backslash \{m\},\quad 1\le k\le N. \end{aligned}$$

(52)

Here the truncation errors satisfy, for $1\le k\le N,$

$$\begin{aligned}&\big |(R^t)^{k-\frac{1}{2}}_i\big | \le c_0 \tau ^{\min \{2-\alpha ,2-\beta \}},~~i\in \omega _l; \quad \big |(S^t)^{k-\frac{1}{2}}_i\big | \le c_0 \tau ^{\min \{2-\alpha ,2-\beta \}},~~i\in \omega _r ; \end{aligned}$$

(53)

$$\begin{aligned}&\big |(R^x)^{k-\frac{1}{2}}_i\big |= h^2_1 \Big |\frac{r_i(t_k) + r_i(t_{k-1})}{2}\Big | \le c_0 h^2_1,~~i\in \omega _l \backslash \{0,m\}; \end{aligned}$$

(54)

$$\begin{aligned}&\big |(R^x)^{k-\frac{1}{2}}_i\big |= h_1 \Big |\frac{r_i(t_k) + r_i(t_{k-1})}{2}\Big | \le c_0 h_1,\quad i\in \{0,m\}; \end{aligned}$$

(55)

$$\begin{aligned}&\big |(S^x)^{k-\frac{1}{2}}_i\big |= h^2_2 \Big |\frac{s_i(t_k) + s_i(t_{k-1})}{2}\Big | \le c_0 h^2_2,~~i\in \omega _r \backslash \{m,M\}; \end{aligned}$$

(56)

$$\begin{aligned}&\big |(S^x)^{k-\frac{1}{2}}_i\big |= h_2 \Big |\frac{s_i(t_k) + s_i(t_{k-1})}{2}\Big | \le c_0 h_2,\quad i\in \{m,M\}; \end{aligned}$$

(57)

and, for $1\le k\le N-1,$

$$\begin{aligned}&\Big | \frac{(R^x)_i^{k+\frac{1}{2}}- (R^x)_i^{k-\frac{1}{2}} }{\tau }\Big | \nonumber \\&\quad = \frac{h_1}{2} \Big | \frac{r_i(t_{k+1})-r_i(t_k)}{\tau } + \frac{r_i(t_{k})-r_i(t_{k-1})}{\tau }\Big | \le c_0 h_1,\quad i \in \{0,m\}; \end{aligned}$$

(58)

$$\begin{aligned}&\Big | \frac{(S^x)_i^{k+\frac{1}{2}}- (S^x)_i^{k-\frac{1}{2}} }{\tau }\Big | \nonumber \\&\quad = \frac{h_2}{2} \Big | \frac{s_i(t_{k+1})-s_i(t_k)}{\tau } + \frac{s_i(t_{k})-s_i(t_{k-1})}{\tau }\Big | \le c_0 h_2,\quad i \in \{m,M\}, \end{aligned}$$

(59)

where $c_0$ is a positive constant number independent of $\tau , h_1$ and $h_2$.

Noticing the initial condition,

$$\begin{aligned} U^0_i = \varphi (x_i),\quad i \in \omega _l \cup \omega _r, \end{aligned}$$

(60)

and dropping small terms in (50)–(52), respectively, we obtain a finite difference scheme for the problem (10)–(16) as follows

$$\begin{aligned}&\delta _tu^{k-\frac{1}{2}}_i +\frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \varDelta _{\tau }^{\alpha } \big (\delta _tu^{k-\frac{1}{2}}_i, \psi _i \big )=\frac{K^2_{\mathrm{{n,1}}}}{3\eta _1}\delta _x^2\Big (u^{k-\frac{1}{2}}_i +\frac{B^{\beta }_1}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_i\Big )+(G_1)^{k-\frac{1}{2}}_i, \nonumber \\&\qquad i \in \omega _l\backslash \{m\},\quad 1\le k\le N, \end{aligned}$$

(61)

$$\begin{aligned}&\frac{h_1}{h_1+h_2} \Big (\delta _tu^{k-\frac{1}{2}}_m + \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \varDelta ^{\alpha }_{\tau } \big (\delta _tu_m^{k-\frac{1}{2}},\psi _m\big ) \Big )\nonumber \\&\qquad +\frac{h_2}{h_1+h_2}\Big (\delta _tu^{k-\frac{1}{2}}_m + \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} \varDelta ^{\alpha }_{\tau } \big (\delta _tu_m^{k-\frac{1}{2}},\psi _m\big ) \Big ) \nonumber \\&\quad =\frac{2}{3(h_1+h_2)}\bigg [ \frac{ K^2_{\mathrm{{n,2}}}}{\eta _2} \delta _x\Big (u^{k-\frac{1}{2}}_{m+\frac{1}{2}} + \frac{B^{\beta }_2}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }u^{k -\frac{1}{2}}_{m+\frac{1}{2}} \Big ) \nonumber \\&\qquad -\frac{ K^2_{\mathrm{{n,1}}}}{\eta _1} \delta _x\Big ( u^{k-\frac{1}{2}}_{m-\frac{1}{2}}+ \frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau }u^{k-\frac{1}{2}}_{m-\frac{1}{2}} \Big ) \bigg ]\nonumber \\&\qquad +(G_1)^{k-\frac{1}{2}}_m + (G_2)^{k-\frac{1}{2}}_m,\quad 1\le k\le N, \end{aligned}$$

(62)

$$\begin{aligned}&\delta _tu^{k-\frac{1}{2}}_i +\frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} \varDelta _{\tau }^{\alpha } \big (\delta _tu^{k-\frac{1}{2}}_i, \psi _i \big ) =\frac{ K^2_{\mathrm{{n,2}}}}{3\eta _2}\delta _x^2\Big (u^{k-\frac{1}{2}}_i +\frac{B^{\beta }_2}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }u^{k-\frac{1}{2}}_i\Big ) +(G_2)^{k-\frac{1}{2}}_i, \nonumber \\&\quad \quad i \in \omega _r\backslash \{m\},\quad 1\le k\le N, \end{aligned}$$

(63)

$$\begin{aligned}&u^0_i=\varphi _i,\quad i \in \omega _l \cup \omega _r. \end{aligned}$$

(64)

Let $u^{n}=(u^n_0,u^n_1,\cdots , u^n_m, \cdots , u^n_{M-1}, u^n_{M})^T,$$v^{n}=(v^n_0,v^n_1,\cdots , v^n_m, \cdots , v^n_{M-1}, v^n_{M})^T,$$n=0,1,\cdots ,N.$ The scheme (61)–(64) can be written in the following matrix form:

$$\begin{aligned} {\left\{ \begin{array}{ll} Av^n = \sum ^{n-1}_{j=1} A_{n-j+1} v^j +B u^{n-1} + b^n,\\ u^n = u^{n-1} + \tau v^n, \end{array}\right. } \quad n=1,2, \cdots , N. \end{aligned}$$

(65)

The coefficient matrix A is strictly diagonally dominant. Hence, the presented scheme (61)–(64) has a unique solution and can be easily solved using the Thomas algorithm.

5 Stability and Convergence of the Scheme

In this section, we will analyze the stability and convergence of the presented scheme (61)–(64) for the normalized time fractional DPL equations in a double-layered nanoscale thin film (10)–(16). Prior to this, some useful lemmas will be listed in preparation for the theoretical analysis.

Lemma 6

Let $\varDelta _{\tau }^{\alpha }(\delta _t y^{k-\frac{1}{2}},q)$ be defined in (35). Then we have

$$\begin{aligned}&\big (\varDelta _{\tau }^{\alpha }(\delta _ty^{k-\frac{1}{2}},q)\big )\cdot \delta _t y^{k-\frac{1}{2}} \\&\quad \ge \frac{1}{\tau } \bigg [ \frac{\tau ^{1-\alpha }}{2\varGamma (2-\alpha )} \sum ^{k}_{\ell =1} a^{(\alpha )}_{k-\ell } (\delta _t y^{\ell -\frac{1}{2}})^2 -\frac{\tau ^{1-\alpha }}{2\varGamma (2-\alpha )} \sum ^{k-1}_{\ell =1} a^{(\alpha )}_{k-1-\ell } (\delta _t y^{\ell -\frac{1}{2}})^2 \bigg ] \\&\qquad + \Big (\frac{1}{2} -\epsilon \Big ) \frac{\tau ^{-\alpha }}{\varGamma (2-\alpha )} a^{(\alpha )}_{k-1} (\delta _t y^{k-\frac{1}{2}})^2 -\frac{1}{4\epsilon } \frac{\tau ^{-\alpha }}{\varGamma (2-\alpha )} a^{(\alpha )}_{k-1} q^2, \end{aligned}$$

where $0<\epsilon \le \frac{1}{2}$.

Proof

An immediate application of the Cauchy–Schwarz inequality for the fractional numerical differentiation operator $\varDelta _{\tau }^{\alpha }(\delta _t y^{k-\frac{1}{2}},q)$ leads to the following estimate

$$\begin{aligned}&\big (\varDelta _{\tau }^{\alpha }(\delta _ty^{k-\frac{1}{2}},q)\big )\cdot \delta _t y^{k-\frac{1}{2}} \\&\quad = \frac{\tau ^{-\alpha }}{\varGamma (2-\alpha )} \bigg [ a^{(\alpha )}_0 \delta _{t} y^{k-\frac{1}{2}} - \sum ^{k-1}_{\ell =1} \big (a^{(\alpha )}_{k-\ell -1}-a_{k-\ell }^{(\alpha )}\big ) \delta _{t} y^{\ell -\frac{1}{2}} -a^{(\alpha )}_{k-1} q \bigg ] \cdot \delta _t y^{k-\frac{1}{2}} \\&\quad \ge \frac{\tau ^{-\alpha }}{\varGamma (2-\alpha )} \bigg [ a^{(\alpha )}_0 (\delta _{t} y^{k-\frac{1}{2}})^2 - \sum ^{k-1}_{\ell =1} (a^{(\alpha )}_{k-\ell -1}-a_{k-\ell }^{(\alpha )}) \frac{(\delta _{t} y^{k-\frac{1}{2}})^2 + (\delta _{t} y^{\ell -\frac{1}{2}})^2}{2} \\&\qquad - a^{(\alpha )}_{k-1} \Big (\epsilon (\delta _t y^{k-\frac{1}{2}})^2 + \frac{1}{4\epsilon } q^2 \Big ) \bigg ] \\&\quad = \frac{1}{\tau } \bigg [ \frac{\tau ^{1-\alpha }}{2\varGamma (2-\alpha )} \sum ^{k}_{\ell =1} a^{(\alpha )}_{k-\ell } (\delta _t y^{\ell -\frac{1}{2}})^2 -\frac{\tau ^{1-\alpha }}{2\varGamma (2-\alpha )} \sum ^{k-1}_{\ell =1} a^{(\alpha )}_{k-1-\ell } (\delta _t y^{\ell -\frac{1}{2}})^2 \bigg ] \\&\qquad + \Big (\frac{1}{2} -\epsilon \Big ) \frac{\tau ^{-\alpha }}{\varGamma (2-\alpha )} a^{(\alpha )}_{k-1} (\delta _t y^{k-\frac{1}{2}})^2 -\frac{1}{4\epsilon } \frac{\tau ^{-\alpha }}{\varGamma (2-\alpha )} a^{(\alpha )}_{k-1} q^2, \end{aligned}$$

where $\epsilon$ is a positive constant and $\epsilon \le \frac{1}{2}$. Hence, the conclusion holds. $\square$

Lemma 7

Let $\delta _{\tau }^{\beta } y^{k-\frac{1}{2}}$ be defined in (34). Then we have

$$\begin{aligned} \sum ^{N}_{k=1} (\delta _{\tau }^{\beta } y^{k-\frac{1}{2}}) \cdot \delta _{t} y^{k-\frac{1}{2}} \ge 0. \end{aligned}$$

Proof

According to the definition of the fractional numerical differentiation operator $\delta _{\tau }^{\beta } y^{k-\frac{1}{2}}$, we point out [19]

$$\begin{aligned} \delta _{\tau }^{\beta } y^{k-\frac{1}{2}}= & {} \frac{\delta _{\tau }^{\beta } y^k + \delta ^{\beta }_{\tau }y^{k-1}}{2} =\frac{1}{2} \bigg [ \frac{\tau ^{1-\beta }}{\varGamma (2-\beta )} \sum ^{k}_{\ell =1} a^{(\beta )}_{k-\ell } \delta _t y^{\ell -\frac{1}{2}}\\&+\,\frac{\tau ^{1-\beta }}{\varGamma (2-\beta )} \sum ^{k-1}_{\ell =1} a^{(\beta )}_{k-1-\ell } \delta _t y^{\ell -\frac{1}{2}} \bigg ]. \end{aligned}$$

Then, from [32], it holds that

$$\begin{aligned} \sum ^{N}_{k=1} (\delta ^{\beta }_{\tau } y^{k-\frac{1}{2}}) \cdot \delta _t y^{k-\frac{1}{2}}&=\frac{1}{2} \frac{\tau ^{1-\beta }}{\varGamma (2-\beta )} \bigg [ \sum ^{N}_{k =1} \sum ^{k}_{\ell =1} a^{(\beta )}_{k-\ell } \delta _t y^{\ell -\frac{1}{2}} \cdot \delta _t y^{k-\frac{1}{2}} \\&\quad +\sum ^{N}_{k =1} \sum ^{k-1}_{\ell =1} a^{(\beta )}_{k-1-\ell } \delta _t y^{\ell -\frac{1}{2}} \cdot \delta _t y^{k-\frac{1}{2}}\bigg ] \ge 0. \end{aligned}$$

Hence, the conclusion holds. $\square$

Lemma 8

Suppose that $u\in {\mathscr {U}}_{h}$, then for any $\epsilon >0,$ it holds that

$$\begin{aligned}&\Vert u\Vert ^2_{l,\infty }\le (1+\epsilon )u^2_0 + \big (1+\frac{1}{\epsilon } \big )l_x\Vert \delta _xu\Vert ^2_{l}, \\&\Vert u\Vert ^2_{r,\infty }\le (1+\epsilon )u^2_M + \big (1+\frac{1}{\epsilon } \big )(L_x-l_x)\Vert \delta _xu\Vert ^2_{r}, \end{aligned}$$

where $\epsilon$ is an any positive constant.

Proof

Notice

$$\begin{aligned}&u_i = u_0 + \sum ^{i}_{j=1} (u_j-u_{j-1})=u_0 + h_1 \sum ^{i}_{j=1} \delta _xu_{j-\frac{1}{2}}, \quad 0\le i \le m, \end{aligned}$$

(66)

$$\begin{aligned}&u_i = u_M - \sum ^{M}_{j=i+1} (u_j-u_{j-1})=u_M - h_2 \sum ^{M}_{j=i+1} \delta _xu_{j-\frac{1}{2}}, \quad m\le i \le M. \end{aligned}$$

(67)

Squaring both sides of (66) and using the Cauchy–Schwarz inequality, we have

$$\begin{aligned} u^2_i&= \bigg ( u_0 + h_1 \sum ^{i}_{j=1} \delta _xu_{j-\frac{1}{2}} \bigg )^2 \nonumber \\&\le (1+\epsilon ) u^2_0 + \Big (1+\frac{1}{\epsilon }\Big ) \Big (h_1\sum ^{i}_{j=1} \delta _x u_{j-\frac{1}{2}}\Big )^2 \nonumber \\&\le (1+\epsilon ) u^2_0 + \Big (1+\frac{1}{\epsilon }\Big ) l_x \Vert \delta _x u\Vert ^2_l,\quad 0\le i\le m. \end{aligned}$$

(68)

Similarly, it follows from (67) that

$$\begin{aligned} u^2_i \le (1+\epsilon ) u^2_M + \Big ( 1+ \frac{1}{\epsilon } \Big )(L_x-l_x) \Vert \delta _xu\Vert ^2_r,\quad m\le i\le M. \end{aligned}$$

(69)

Hence, the conclusion holds. $\square$

As sufficient preparatory works, the above lemmas assist us proving the following discrete energy estimation of the difference scheme (61)–(64). Assume $\{u^k_i~|~0\le i\le M,~0\le k\le N\}$ is the solution of the scheme (61)–(64). For simplicity, we firstly define a discrete energy function, i.e.,

$$\begin{aligned} E^k&=\frac{\tau ^{1-\alpha }}{2\varGamma (2-\alpha )} \sum ^k_{\ell =1} a^{(\alpha )}_{k-\ell } \bigg [ \frac{\eta ^{\alpha }_1}{\varGamma (1+\alpha )}\Vert \delta _t u^{\ell -\frac{1}{2}}\Vert ^2_l +\frac{\eta ^{\alpha }_2}{\varGamma (1+\alpha )} \Vert \delta _t u^{\ell -\frac{1}{2}}\Vert ^2_r\bigg ] \nonumber \\&\quad +\frac{ K^2_{\mathrm{n,1}}}{6\eta _1}\Big [\Vert \delta _x u^k\Vert ^2_l + \frac{1}{\gamma _1K_{\mathrm{n,1}}}(u^k_0)^2\Big ] +\frac{ K^2_{\mathrm{n,2}}}{6\eta _2}\Big [\Vert \delta _x u^k\Vert ^2_r + \frac{1}{\gamma _2K_{\mathrm{n,2}}}(u^k_M)^2\Big ],\quad 1\le k\le N. \end{aligned}$$

(70)

Theorem 2

The solution $\{u^k_i~|~0\le i\le M,~0\le k\le N\}$ of the scheme (61)–(64) satisfies

$$\begin{aligned} E^k&\le E^0 + \frac{1}{2}\frac{t^{1-\alpha }_k}{\varGamma (2-\alpha )} \Big [ \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \Vert \psi \Vert _l^2 + \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} \Vert \psi \Vert _r^2\Big ] \nonumber \\&\quad + \frac{\tau }{4} \sum ^k_{s=1}(\Vert (G)_1^{s-\frac{1}{2}}\Vert ^2_l + \Vert (G)_2^{s-\frac{1}{2}}\Vert ^2_r), \quad 1\le k\le N. \end{aligned}$$

(71)

Proof

Multiplying (61) by $\frac{h_1}{2}\delta _t u^{k-\frac{1}{2}}_{0}$ for $i=0$ and $h_1 \delta _t u^{k-\frac{1}{2}}_{i}$ for $1\le i\le m-1$, (62) by $\frac{h_1+h_2}{2}\delta _t u^{k-\frac{1}{2}}_{m}$, (63) by $h_2\delta _t u^{k-\frac{1}{2}}_{i}$ for $m+1\le i\le M-1$ and $\frac{h_2}{2}\delta _t u^{k-\frac{1}{2}}_{M}$ for $i=M$, respectively, and then summing up the results, we obtain

$$\begin{aligned}&\bigg (\delta _t u^{k-\frac{1}{2}}+\frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )}\varDelta _{\tau }^{\alpha }(\delta _t u^{k-\frac{1}{2}},\psi ),\delta _t u^{k-\frac{1}{2}}\bigg )_l\nonumber \\&\qquad + \bigg (\delta _t u^{k-\frac{1}{2}}+\frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )}\varDelta _{\tau }^{\alpha }(\delta _t u^{k-\frac{1}{2}},\psi ),\delta _t u^{k-\frac{1}{2}}\bigg )_r \nonumber \\&\quad = \frac{ K^2_{\mathrm{n,1}}}{3\eta _1} \bigg [ \frac{h_1}{2} \delta ^2_x \Big (u^{k-\frac{1}{2}}_0 +\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau }u^{k-\frac{1}{2}}_0\Big ) \cdot \delta _t u^{k-\frac{1}{2}}_0 \nonumber \\&\qquad + h_1 \sum ^{m-1}_{i=1} \delta _x^2\Big (u^{k-\frac{1}{2}}_i +\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_i\Big )\cdot \delta _tu^{k-\frac{1}{2}}_i \nonumber \\&\qquad -\delta _x\Big (u^{k-\frac{1}{2}}_{m-\frac{1}{2}}+\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_{m-\frac{1}{2}}\Big )\cdot \delta _t u^{k-\frac{1}{2}}_{m} \bigg ] \nonumber \\&\qquad + \frac{ K^2_{\mathrm{n,2}}}{3\eta _2} \bigg [ \delta _x\Big (u^{k-\frac{1}{2}}_{m+\frac{1}{2}} + \frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_{m+\frac{1}{2}}\Big )\cdot \delta _t u^{k-\frac{1}{2}}_m \nonumber \\&\qquad +h_2 \sum ^{M-1}_{i=m+1} \delta _x^2\Big (u^{k-\frac{1}{2}}_i +\frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_i\Big )\cdot \delta _tu^{k-\frac{1}{2}}_i\nonumber \\&\qquad +\frac{h_2}{2} \delta _x^2\Big (u^{k-\frac{1}{2}}_M+\frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_M\Big )\cdot \delta _t u^{k-\frac{1}{2}}_M \bigg ] \nonumber \\&\qquad + ((G_1)^{k-\frac{1}{2}},\delta _t u^{k-\frac{1}{2}})_l+((G_2)^{k-\frac{1}{2}},\delta _t u^{k-\frac{1}{2}})_r, \quad 1\le k \le N. \end{aligned}$$

(72)

For the first term on the right hand side of (72), by virtue of the definition (37), the summation by parts and the Cauchy–Schwarz inequality, we have

$$\begin{aligned}&\frac{K^2_{\mathrm{n,1}}}{3\eta _1} \bigg [ \frac{h_1}{2} \delta ^2_x \Big (u^{k-\frac{1}{2}}_0 +\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau }u^{k-\frac{1}{2}}_0\Big ) \cdot \delta _t u^{k-\frac{1}{2}}_0 \nonumber \\&\qquad + h_1 \sum ^{m-1}_{i=1} \delta _x^2\Big (u^{k-\frac{1}{2}}_i+\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_i\Big )\cdot \delta _tu^{k-\frac{1}{2}}_i \nonumber \\&\qquad -\delta _x\Big (u^{k-\frac{1}{2}}_{m-\frac{1}{2}}+\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_{m-\frac{1}{2}}\Big )\cdot \delta _t u^{k-\frac{1}{2}}_{m} \bigg ] \nonumber \\&\quad = - \frac{K^2_{\mathrm{n,1}}}{3\eta _1} \bigg [ \bigg ( \delta _x \Big (u^{k-\frac{1}{2}} +\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau }u^{k-\frac{1}{2}}\Big ), \delta _x(\delta _t u^{k-\frac{1}{2}}) \bigg )_l \nonumber \\&\qquad + \frac{1}{\gamma _1 K_{\mathrm{n,1}}} \Big (u^{k-\frac{1}{2}}_0 +\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau }u^{k-\frac{1}{2}}_0\Big ) \cdot \delta _t u^{k-\frac{1}{2}}_0 \bigg ] \nonumber \\&\quad \le -\frac{1}{\tau } \bigg [ \frac{K^2_{\mathrm{n,1}}}{6\eta _1} \Big ( \Vert \delta _x u^k\Vert ^2_l +\frac{1}{\gamma _1 K_{\mathrm{n,1}}} (u^k_0)^2\Big ) -\frac{K^2_{\mathrm{n,1}}}{6\eta _1} \Big ( \Vert \delta _x u^{k-1}\Vert ^2_l +\frac{1}{\gamma _1 K_{\mathrm{n,1}}} (u^{k-1}_0)^2\Big ) \bigg ] \nonumber \\&\qquad - \frac{K^2_{\mathrm{n,1}}}{3\eta _1}\frac{B^{\beta }_1}{\varGamma (1+\beta )} \bigg [ \big (\delta ^{\beta }_{\tau } (\delta _x u^{k-\frac{1}{2}}), \delta _t(\delta _x u^{k-\frac{1}{2}})\big )_l + \frac{1}{\gamma _1 K_{\mathrm{n,1}}} (\delta ^{\beta }_{\tau }u^{k-\frac{1}{2}}_0) \cdot \delta _t u^{k-\frac{1}{2}}_0 \bigg ]. \end{aligned}$$

(73)

Similarly, for the second term on the right hand side of (72), we get

$$\begin{aligned}&\frac{ K^2_{\mathrm{n,2}}}{3\eta _2} \bigg [ \delta _x\Big (u^{k-\frac{1}{2}}_{m+\frac{1}{2}} + \frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_{m+\frac{1}{2}}\Big )\cdot \delta _t u^{k-\frac{1}{2}}_m \nonumber \\&\qquad +h_2 \sum ^{M-1}_{i=m+1} \delta _x^2\Big (u^{k-\frac{1}{2}}_i+\frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_i\Big )\cdot \delta _tu^{k-\frac{1}{2}}_i \nonumber \\&\qquad +\frac{h_2}{2} \delta _x^2\Big (u^{k-\frac{1}{2}}_M+\frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } u^{k-\frac{1}{2}}_M\Big )\cdot \delta _t u^{k-\frac{1}{2}}_M \bigg ] \nonumber \\&\quad \le -\frac{1}{\tau } \bigg [ \frac{K^2_{\mathrm{n,2}}}{6\eta _2} \Big ( \Vert \delta _x u^k\Vert ^2_r +\frac{1}{\gamma _2 K_{\mathrm{n,2}}} (u^k_M)^2\Big ) -\frac{K^2_{\mathrm{n,2}}}{6\eta _2} \Big ( \Vert \delta _x u^{k-1}\Vert ^2_r +\frac{1}{\gamma _2K_{\mathrm{n,2}}} (u^{k-1}_M)^2\Big ) \bigg ] \nonumber \\&\qquad - \frac{K^2_{\mathrm{n,2}}}{3\eta _2}\frac{B^{\beta }_2}{\varGamma (1+\beta )} \bigg [ \big (\delta ^{\beta }_{\tau } (\delta _x u^{k-\frac{1}{2}}), \delta _t(\delta _x u^{k-\frac{1}{2}})\big )_r + \frac{1}{\gamma _2 K_{\mathrm{n,2}}} (\delta ^{\beta }_{\tau }u^{k-\frac{1}{2}}_M) \cdot \delta _t u^{k-\frac{1}{2}}_M \bigg ]. \end{aligned}$$

(74)

Using the Cauchy–Schwarz inequality again for the last two terms on the right hand side of (72), we have

$$\begin{aligned}&((G_1)^{k-\frac{1}{2}},\delta _t u^{k-\frac{1}{2}})_l+((G_2)^{k-\frac{1}{2}}, \delta _t u^{k-\frac{1}{2}})_r \le \Vert \delta _t u^{k-\frac{1}{2}}\Vert ^2_l + \Vert \delta _t u^{k-\frac{1}{2}}\Vert ^2_r \nonumber \\&\quad + \frac{1}{4} (\Vert (G)_1^{k-\frac{1}{2}}\Vert ^2_l + \Vert (G)_2^{k-\frac{1}{2}}\Vert ^2_r). \end{aligned}$$

(75)

Inserting (73)–(75) into (72), replacing the superscript k with s and summing up s from 1 to k on both sides of the result, then using Lemma  6 with $\epsilon = \frac{1}{2}$ and Lemma 7, we get the conclusion. $\square$

From Lemma 8, we have an estimate

$$\begin{aligned}&\frac{ K^2_{\mathrm{n,1}}}{6\eta _1}\Big [\Vert \delta _x u^k\Vert ^2_l + \frac{1}{\gamma _1K_{\mathrm{n,1}}}(u^k_0)^2\Big ] + \frac{ K^2_{\mathrm{n,2}}}{6\eta _2}\Big [\Vert \delta _x u^k\Vert ^2_r + \frac{1}{\gamma _2K_{\mathrm{n,2}}}(u^k_M)^2\Big ] \nonumber \\&\quad = c_1\bigg [(1+\frac{l_x}{\gamma _1K_{\mathrm{n,1}}})(u^k_0)^2+ (1+\frac{\gamma _1K_{\mathrm{n,1}}}{l_x})l_x \Vert \delta _x u^k\Vert ^2_l\bigg ] \nonumber \\&\qquad +c_2\bigg [(1+\frac{L_x-l_x}{\gamma _2K_{\mathrm{n,2}}})(u^k_M)^2+(1 +\frac{\gamma _2K_{\mathrm{n,2}}}{L_x-l_x})(L_x-l_x) \Vert \delta _x u^k\Vert ^2_r\bigg ] \nonumber \\&\quad \ge c_1 \Vert u^k\Vert ^2_{l,\infty } +c_2\Vert u^k\Vert ^2_{r,\infty }, \end{aligned}$$

(76)

where $c_1=\frac{ K^2_{\mathrm{n,1}}}{6\eta _1(\gamma _1K_{\mathrm{n,1}}+l_x)}$ and $c_2=\frac{ K^2_{\mathrm{n,2}}}{6\eta _2( \gamma _2K_{\mathrm{n,2}}+L_x-l_x)}$.

Theorem 2 and the estimate (76) indicate the following stability conclusion.

Corollary 1

(Stability) The scheme (61)–(64) is unconditionally stable with respect to the initial values and the source terms, i.e.,

$$\begin{aligned} c_1 \Vert u^k\Vert ^2_{l,\infty } +c_2\Vert u^k\Vert ^2_{r,\infty }&\le \frac{ K^2_{\mathrm{n,1}}}{6\eta _1}\Big [\Vert \delta _x u^0\Vert ^2_l + \frac{1}{\gamma _1K_{\mathrm{n,1}}}(u^0_0)^2\Big ]\\&\quad + \frac{ K^2_{\mathrm{n,2}}}{6\eta _2}\Big [\Vert \delta _x u^0\Vert ^2_r + \frac{1}{\gamma _2K_{\mathrm{n,2}}}(u^0_M)^2\Big ] \\&\quad +\frac{1}{2}\frac{t^{1-\alpha }_k}{\varGamma (2-\alpha )} \Big [ \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \Vert \psi \Vert _l^2 + \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} \Vert \psi \Vert _r^2\Big ]\\&\quad + \frac{\tau }{4} \sum ^k_{s=1}(\Vert (G)_1^{s-\frac{1}{2}}\Vert ^2_l + \Vert (G)_2^{s-\frac{1}{2}}\Vert ^2_r),~~1\le k\le N. \end{aligned}$$

Assume u(x, t) is the solution of (10)–(16) and $\{u^k_i~|~0\le i\le M,~0\le k\le N\}$ is the solution of the scheme (61)–(64). We denote the errors

$$\begin{aligned} e^n_i=u(x_i,t_n)-u^n_i,\quad 0 \le i \le M,\quad 0 \le k \le N. \end{aligned}$$

(77)

Subtracting (61)–(64) from (50)– (52) and (60), respectively, we can easily get the following error equations

$$\begin{aligned}&\delta _te^{k-\frac{1}{2}}_i +\frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \varDelta _{\tau }^{\alpha } \big (\delta _te^{k-\frac{1}{2}}_i, 0\big )=\frac{K^2_{\mathrm{{n,1}}}}{3\eta _1}\delta _x^2\Big (e^{k-\frac{1}{2}}_i +\frac{B^{\beta }_1}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_i\Big )\nonumber \\&\qquad +(R^t)^{k-\frac{1}{2}}_i + (R^x)^{k-\frac{1}{2}}_i, \nonumber \\&\qquad i \in \omega _l\backslash \{m\},\quad 1\le k\le N, \end{aligned}$$

(78)

$$\begin{aligned}&\frac{h_1}{h_1+h_2} \Big (\delta _te^{k-\frac{1}{2}}_m + \frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \varDelta ^{\alpha }_{\tau } \big (\delta _te_m^{k-\frac{1}{2}},0\big ) \Big )\nonumber \\&\qquad +\frac{h_2}{h_1+h_2}\Big (\delta _te^{k-\frac{1}{2}}_m + \frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} \varDelta ^{\alpha }_{\tau } \big (\delta _te_m^{k-\frac{1}{2}},0\big ) \Big ) \nonumber \\&\quad =\frac{2}{3(h_1+h_2)}\bigg [ \frac{ K^2_{\mathrm{{n,2}}}}{\eta _2} \delta _x\Big (e^{k-\frac{1}{2}}_{m+\frac{1}{2}} + \frac{B^{\beta }_2}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_{m+\frac{1}{2}} \Big ) \nonumber \\&\qquad -\frac{ K^2_{\mathrm{{n,1}}}}{\eta _1} \delta _x\Big ( e^{k-\frac{1}{2}}_{m-\frac{1}{2}}+ \frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_{m-\frac{1}{2}} \Big ) \bigg ]\nonumber \\&\qquad +\frac{h_1}{h_1+h_2} \left( (R^t)^{k-\frac{1}{2}}_m + (R^x)^{k-\frac{1}{2}}_m\right) \nonumber \\&\qquad +\frac{h_2}{h_1+h_2} \left( (S^t)^{k-\frac{1}{2}}_m + (S^x)^{k-\frac{1}{2}}_m\right) ,\quad 1\le k\le N, \end{aligned}$$

(79)

$$\begin{aligned}&\delta _te^{k-\frac{1}{2}}_i +\frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )} \varDelta _{\tau }^{\alpha } \big (\delta _te^{k-\frac{1}{2}}_i, 0 \big )=\frac{ K^2_{\mathrm{{n,2}}}}{3\eta _2}\delta _x^2\left( e^{k-\frac{1}{2}}_i +\frac{B^{\beta }_2}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_i\right) \nonumber \\&\qquad +(S^t)^{k-\frac{1}{2}}_i + (S^x)^{k-\frac{1}{2}}_i, \nonumber \\&\qquad \qquad i \in \omega _r\backslash \{m\},\quad 1\le k\le N, \end{aligned}$$

(80)

$$\begin{aligned}&e^0_i=0,\quad i \in \omega _l \cup \omega _r. \end{aligned}$$

(81)

An immediate application of Theorem  2 deduces that the convergence order in space of the scheme (61)–(64) is $O(h_1^{3/2}+h_2^{3/2})$. In fact, the spatial convergence accuracy of the scheme in this paper is higher than the order of 3 / 2,  which can be confirmed in the subsequent simulations. In the next theorem, we will give the theoretical analysis of convergence accuracies. We omit some processes which are similar to those in Theorem 2.

Theorem 3

(Convergence) The finite difference scheme (61)–(64) for the fractional DPL model (10)–(16) admits

$$\begin{aligned} \Vert e^k \Vert _{\infty } \le c_3 (\tau ^{\min \{2-\alpha ,2-\beta \}}+h_1^2+h^2_2),\quad 0\le k \le N, \end{aligned}$$

(82)

where $c_3$ is a positive constant independent of $\tau$, $h_1$ and $h_2$.

Proof

Multiplying (78) by $\frac{h_1}{2}\delta _t e^{k-\frac{1}{2}}_{0}$ for $i=0$ and $h_1 \delta _t e^{k-\frac{1}{2}}_{i}$ for $1\le i\le m-1$, (79) by $\frac{h_1+h_2}{2}\delta _t e^{k-\frac{1}{2}}_{m}$, (80) by $h_2\delta _t e^{k-\frac{1}{2}}_{i}$ for $m+1\le i\le M-1$ and $\frac{h_2}{2}\delta _t e^{k-\frac{1}{2}}_{M}$ for $i=M$, respectively, and then summing up the results, we obtain

$$\begin{aligned}&\bigg (\delta _t e^{k-\frac{1}{2}}+\frac{\eta _1^{\alpha }}{\varGamma (1+\alpha )} \varDelta _{\tau }^{\alpha }(\delta _t e^{k-\frac{1}{2}},0),\delta _t e^{k-\frac{1}{2}}\bigg )_l\nonumber \\&\qquad + \bigg (\delta _t e^{k-\frac{1}{2}} +\frac{\eta _2^{\alpha }}{\varGamma (1+\alpha )}\varDelta _{\tau }^{\alpha } (\delta _t e^{k-\frac{1}{2}},0),\delta _t e^{k-\frac{1}{2}}\bigg )_r \nonumber \\&\quad = \frac{ K_{\mathrm{n,1}}^2}{3\eta _1} \bigg [ \frac{h_1}{2} \delta ^2_x \Big (e^{k-\frac{1}{2}}_0+\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta _{\tau }^{\beta } e^{k-\frac{1}{2}}_0\Big ) \cdot \delta _t e^{k-\frac{1}{2}}_0 \nonumber \\&\qquad + h_1 \sum ^{m-1}_{i=1} \delta _x^2\Big (e^{k-\frac{1}{2}}_i +\frac{B^{\beta }_1}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_i\Big ) \cdot \delta _t e^{k-\frac{1}{2}}_i \nonumber \\&\qquad -\delta _x\Big (e^{k-\frac{1}{2}}_{m-\frac{1}{2}}+\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } e^{k-\frac{1}{2}}_{m-\frac{1}{2}}\Big )\cdot \delta _t e^{k-\frac{1}{2}}_{m} \bigg ] \nonumber \\&\qquad + \frac{ K_{\mathrm{n,2}}^2}{3\eta _2} \bigg [ \delta _x\Big (e^{k-\frac{1}{2}}_{m+\frac{1}{2}} + \frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } e^{k-\frac{1}{2}}_{m+\frac{1}{2}}\Big )\cdot \delta _te^{k-\frac{1}{2}}_m \nonumber \\&\qquad +h_2 \sum ^{M-1}_{i=m+1} \delta _x^2\Big (e^{k-\frac{1}{2}}_i +\frac{B^{\beta }_2}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_i\Big )\cdot \delta _t e^{k-\frac{1}{2}}_i \nonumber \\&\qquad +\frac{h_2}{2} \delta _x^2\Big (e^{k-\frac{1}{2}}_M+\frac{B^{\beta }_2}{\varGamma (1+\beta )}\delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_M\Big )\cdot \delta _t e^{k-\frac{1}{2}}_M \bigg ] \nonumber \\&\qquad + ((R^t)^{k-\frac{1}{2}}+(R^x)^{k-\frac{1}{2}},\delta _t e^{k-\frac{1}{2}})_l+((S^t)^{k-\frac{1}{2}}+(S^x)^{k-\frac{1}{2}},\delta _t e^{k-\frac{1}{2}})_r, \quad 1\le k \le N. \end{aligned}$$

(83)

According to (73) and (74), for the first two terms on the right hand side of (83), we have

$$\begin{aligned}&\frac{K^2_{\mathrm{n,1}}}{3\eta _1} \bigg [ \frac{h_1}{2} \delta ^2_x \Big (e^{k-\frac{1}{2}}_0 +\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_0\Big )\delta _t e^{k-\frac{1}{2}}_0 \nonumber \\&\qquad + h_1 \sum ^{m-1}_{i=1} \delta _x^2\Big (e^{k-\frac{1}{2}}_i+\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } e^{k-\frac{1}{2}}_i\Big )\cdot \delta _te^{k-\frac{1}{2}}_i \nonumber \\&\qquad -\delta _x\Big (e^{k-\frac{1}{2}}_{m-\frac{1}{2}}+\frac{B^{\beta }_1}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } e^{k-\frac{1}{2}}_{m-\frac{1}{2}}\Big )\cdot \delta _t e^{k-\frac{1}{2}}_{m}\bigg ] \nonumber \\&\quad \le -\frac{1}{\tau } \bigg [ \frac{K^2_{\mathrm{n,1}}}{6\eta _1} \Big ( \Vert \delta _x e^k\Vert ^2_l +\frac{1}{\gamma _1 K_{\mathrm{n,1}}} (e^k_0)^2\Big ) -\frac{K^2_{\mathrm{n,1}}}{6\eta _1} \Big ( \Vert \delta _x e^{k-1}\Vert ^2_l +\frac{1}{\gamma _1 K_{\mathrm{n,1}}} (e^{k-1}_0)^2\Big ) \bigg ] \nonumber \\&\qquad - \frac{K^2_{\mathrm{n,1}}}{3\eta _1}\frac{B^{\beta }_1}{\varGamma (1+\beta )} \bigg [ \big (\delta ^{\beta }_{\tau } (\delta _x e^{k-\frac{1}{2}}), \delta _t(\delta _x e^{k-\frac{1}{2}})\big )_l + \frac{1}{\gamma _1 K_{\mathrm{n,1}}} (\delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_0) \cdot \delta _t e^{k-\frac{1}{2}}_0 \bigg ], \end{aligned}$$

(84)

Table 1 Maximum errors and spatial convergence orders for Example 1 ($\tau =\frac{1}{500}$)

Table 2 Maximum errors and temporal convergence orders for Example 1 ($h=\frac{1}{100}$)

Table 3 Properties of gold and chromium

Table 4 Maximum errors and spatial convergence orders for Example 2 ($\tau =\frac{1}{500}$)

Table 5 Maximum errors and temporal convergence orders for Example 2 ($h=\frac{{ L_x}}{200}$)

Fig. 2

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.1,0.1)$

Fig. 3

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.5,0.1)$

Fig. 4

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.9,0.1)$

Fig. 5

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.1,0.5)$

Fig. 6

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.5,0.5)$

Fig. 7

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.9,0.5)$

Fig. 8

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.1,0.9)$

and

$$\begin{aligned}&\frac{ K^2_{\mathrm{n,2}}}{3\eta _2} \bigg [ \delta _x\Big (e^{k-\frac{1}{2}}_{m+\frac{1}{2}} + \frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } e^{k-\frac{1}{2}}_{m+\frac{1}{2}}\Big )\cdot \delta _t e^{k-\frac{1}{2}}_m \nonumber \\&\qquad +h_2 \sum ^{M-1}_{i=m+1} \delta _x^2\Big (e^{k-\frac{1}{2}}_i+\frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } e^{k-\frac{1}{2}}_i\Big )\delta _te^{k-\frac{1}{2}}_i \nonumber \\&\qquad +\frac{h_2}{2} \delta _x^2\Big (e^{k-\frac{1}{2}}_M+\frac{B^{\beta }_2}{\varGamma (1+\beta )} \delta ^{\beta }_{\tau } e^{k-\frac{1}{2}}_M\Big )\cdot \delta _t e^{k-\frac{1}{2}}_M \bigg ] \nonumber \\&\quad \le -\frac{1}{\tau } \bigg [ \frac{K^2_{\mathrm{n,2}}}{6\eta _2} \Big ( \Vert \delta _x e^k\Vert ^2_r +\frac{1}{\gamma _2 K_{\mathrm{n,2}}} (e^k_M)^2\Big ) -\frac{K^2_{\mathrm{n,2}}}{6\eta _2} \Big ( \Vert \delta _x e^{k-1}\Vert ^2_r +\frac{1}{\gamma _2K_{\mathrm{n,2}}} (e^{k-1}_M)^2\Big ) \bigg ] \nonumber \\&\qquad - \frac{K^2_{\mathrm{n,2}}}{3\eta _2}\frac{B^{\beta }_2}{\varGamma (1+\beta )} \bigg [ \big (\delta ^{\beta }_{\tau } (\delta _x e^{k-\frac{1}{2}}), \delta _t(\delta _x e^{k-\frac{1}{2}})\big )_r + \frac{1}{\gamma _2 K_{\mathrm{n,2}}} (\delta ^{\beta }_{\tau }e^{k-\frac{1}{2}}_M) \cdot \delta _t e^{k-\frac{1}{2}}_M \bigg ]. \end{aligned}$$

(85)

Denote

$$\begin{aligned} F^k&= \frac{\tau ^{1-\alpha }}{2\varGamma (2-\alpha )} \sum ^k_{\ell =1} a^{(\alpha )}_{k-\ell } \bigg [ \frac{\eta ^{\alpha }_1}{\varGamma (1+\alpha )}\Vert \delta _t e^{\ell -\frac{1}{2}}\Vert ^2_l +\frac{\eta ^{\alpha }_2}{\varGamma (1+\alpha )} \Vert \delta _t e^{\ell -\frac{1}{2}}\Vert ^2_r\bigg ] \nonumber \\&\quad +\frac{ K^2_{\mathrm{n,1}}}{6\eta _1}\Big [\Vert \delta _x e^k\Vert ^2_l + \frac{1}{\gamma _1K_{\mathrm{n,1}}}(e^k_0)^2\Big ] + \frac{ K^2_{\mathrm{n,2}}}{6\eta _2}\Big [\Vert \delta _x e^k\Vert ^2_r + \frac{1}{\gamma _2K_{\mathrm{n,2}}}(e^k_M)^2\Big ],\quad 1\le k\le N. \end{aligned}$$

(86)

Fig. 9

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.5,0.9)$

Fig. 10

Temperature profiles at $t=0.2\,(\mathrm{ps}),0.25\,(\mathrm{ps}),0.5\,(\mathrm{ps}),2\,(\mathrm{ps})$ for $(\alpha ,\beta )=(0.9,0.9)$

Fig. 11

Temperature profiles at $x=0\,(\mathrm{nm})$ for different parameters $(\alpha ,\beta )$ and G

Inserting (84) and (85) into (83), replacing the superscript k with s and summing up s from 1 to k on both sides of the result, then using Lemma  6 with $\epsilon = \frac{1}{2}$ and Lemma 7, we get

$$\begin{aligned}&F^k+\tau \sum ^{k}_{s=1} \big ( \Vert \delta _t e^{s-\frac{1}{2}}\Vert ^2_l + \Vert \delta _t e^{s-\frac{1}{2}}\Vert ^2_r \big )\nonumber \\&\quad \le \tau \sum ^k_{s=1} \Big [\big ((R^t)^{s-\frac{1}{2}}+(R^x)^{s-\frac{1}{2}},\delta _t e^{s-\frac{1}{2}}\big )_l+\big ((S^t)^{s-\frac{1}{2}}+(S^x)^{s-\frac{1}{2}},\delta _t e^{s-\frac{1}{2}}\big )_r\Big ] \nonumber \\&\quad = \tau \sum ^{k}_{s=1} \big ( (R^t)^{s-\frac{1}{2}}, \delta _t e^{s-\frac{1}{2}} \big )_l + \tau \sum ^{k}_{s=1} h_1 \sum ^{m-1}_{i=1} (R^x)^{s-\frac{1}{2}}_i \cdot \delta _t e^{s-\frac{1}{2}}_i \nonumber \\&\qquad + \tau \sum ^{k}_{s=1} \frac{h_1}{2} \sum _{i=0,m} (R^x)^{s-\frac{1}{2}}_i\cdot \delta _t e^{s-\frac{1}{2}}_i \nonumber \\&\qquad + \tau \sum ^{k}_{s=1} \big ( (S^t)^{s-\frac{1}{2}}, \delta _t e^{s-\frac{1}{2}} \big )_r + \tau \sum ^k_{s=1} h_2 \sum ^{M-1}_{i=m+1} (S^x)^{s-\frac{1}{2}}_i\cdot \delta _t e^{s-\frac{1}{2}}_i \nonumber \\&\qquad + \tau \sum ^k_{s=1} \frac{h_2}{2} \sum _{i=m,M} (S^x)^{s-\frac{1}{2}}_i \cdot \delta _t e^{s-\frac{1}{2}}_i. \end{aligned}$$

(87)

For the first two terms on the right hand side of (87), by using the Cauchy–Schwarz inequality, we have

$$\begin{aligned}&\tau \sum ^{k}_{s=1} \big ( (R^t)^{s-\frac{1}{2}}, \delta _t e^{s-\frac{1}{2}} \big )_l + \tau \sum ^{k}_{s=1} h_1 \sum ^{m-1}_{i=1} (R^x)^{s-\frac{1}{2}}_i \cdot \delta _t e^{s-\frac{1}{2}}_i \nonumber \\&\quad \le \tau \sum ^k_{s=1} \Vert \delta _t e^{s-\frac{1}{2}}\Vert ^2_l +\frac{\tau }{2} \sum ^{k}_{s=1} \Big [ \Vert (R^t)^{s-\frac{1}{2}}\Vert ^2_l + h_1 \sum ^{m-1}_{i=1} \big ( (R^x)^{s-\frac{1}{2}}_i \big )^2 \Big ]. \end{aligned}$$

(88)

Similarly, applying the Cauchy–Schwarz inequality for the fourth term and fifth term on the right hand side of (87) leads to

$$\begin{aligned}&\tau \sum ^{k}_{s=1} \big ( (S^t)^{s-\frac{1}{2}}, \delta _t e^{s-\frac{1}{2}} \big )_r + \tau \sum ^{k}_{s=1} h_1 \sum ^{m-1}_{i=1} (S^x)^{s-\frac{1}{2}}_i \cdot \delta _t e^{s-\frac{1}{2}}_i \nonumber \\&\quad \le \tau \sum ^k_{s=1} \Vert \delta _t e^{s-\frac{1}{2}}\Vert ^2_r +\frac{\tau }{2} \sum ^{k}_{s=1} \Big [ \Vert (S^t)^{s-\frac{1}{2}}\Vert ^2_r + h_2 \sum ^{M-1}_{i=m+1} \big ( (S^x)^{s-\frac{1}{2}}_i \big )^2 \Big ]. \end{aligned}$$

(89)

To provide convergence order in space, we adopt a strategy different from (88), specifically,

$$\begin{aligned}&\frac{\tau }{2} \sum ^k_{s=1} h_1\sum _{i=0,m} (R^x)^{s-\frac{1}{2}}_i\cdot \delta _t e^{s-\frac{1}{2}}_i \nonumber \\&\quad = \frac{h_1}{2} \sum _{i=0,m} \bigg [(R^x)^{k-\frac{1}{2}}_i\cdot e^k_i -\tau \sum ^{k-1}_{s=1} \frac{(R^x)^{s+\frac{1}{2}}_i-(R^x)^{s-\frac{1}{2}}_i}{\tau } \cdot e^s_i \bigg ] \nonumber \\&\quad \le \frac{c_1}{2} \Big [ \Vert e^k\Vert ^2_{l,\infty } + \tau \sum ^{k-1}_{s=1}\Vert e^s\Vert ^2_{l,\infty } \Big ]+ \frac{h^2_1}{4c_1} \sum _{i=0,m} \Big [ |(R^x)^{k-\frac{1}{2}}_i|^2 \nonumber \\&\qquad +\, \tau \sum ^{k-1}_{s=1} \Big | \frac{(R^x)^{s+\frac{1}{2}}_i-(R^x)^{s-\frac{1}{2}}_i}{\tau }\Big |^2\Big ], \end{aligned}$$

(90)

and

$$\begin{aligned}&\frac{\tau }{2} \sum ^k_{s=1} h_2\sum _{i=m,M} (S^x)^{s-\frac{1}{2}}_i\cdot \delta _t e^{s-\frac{1}{2}}_i \nonumber \\&\quad = \frac{h_2}{2} \sum _{i=m,M} \bigg [(S^x)^{k-\frac{1}{2}}_i\cdot e^k_i -\tau \sum ^{k-1}_{s=1} \frac{(S^x)^{s+\frac{1}{2}}_i-(S^x)^{s-\frac{1}{2}}_i}{\tau } \cdot e^s_i \bigg ] \nonumber \\&\quad \le \frac{c_2}{2} \Big [ \Vert e^k\Vert ^2_{r,\infty } + \tau \sum ^{k-1}_{s=1}\Vert e^s\Vert ^2_{r,\infty } \Big ]+ \frac{h^2_2}{4c_2} \sum _{i=m,M} \Big [ |(S^x)^{k-\frac{1}{2}}_i|^2 \nonumber \\&\qquad + \,\tau \sum ^{k-1}_{s=1} \Big | \frac{(S^x)^{s+\frac{1}{2}}_i-(S^x)^{s-\frac{1}{2}}_i}{\tau }\Big |^2\Big ], \end{aligned}$$

(91)

where $c_1$ and $c_2$ are defined in (76). Here, we have used the fact $e^0_i=0,~ i \in \omega _l \cup \omega _r.$

Inserting the above estimations (88)–(91) into (87), and noticing the truncation errors (53)–(59) and $F^k\ge c_1\Vert e^k\Vert ^2_{l,\infty }+ c_2\Vert e^k\Vert ^2_{r,\infty }$, we have

$$\begin{aligned}&c_1\Vert e^k\Vert ^2_{l,\infty }+ c_2\Vert e^k\Vert ^2_{r,\infty } \le \tau \sum ^{k-1}_{s=1} \Big [c_1\Vert e^s\Vert ^2_{l,\infty }+ c_2\Vert e^s\Vert ^2_{r,\infty }\Big ] \\&\quad +\, c_4(\tau ^{\min \{2-\alpha ,2-\beta \}}+h_1^2+h^2_2)^2,\quad 1\le k\le N, \end{aligned}$$

where $c_4$ is a positive constant independent of $\tau$, $h_1$ and $h_2$.

From the Gronwall inequality, the conclusion holds. $\square$

6 Numerical Experiments

In this section, we carry out some numerical experiments to study the performance and the convergence accuracy of the present numerical scheme. Then, we illustrate the applicability of the model (10)–(16) by predicting the temperature in a double-layered nanoscale thin film, where a gold layer is on a chromium padding layer exposed to an ultrashort-pulsed laser heating. All numerical computations were carried out by using MATLAB. And we exploit the Thomas algorithm to obtain the numerical results.

For simplicity, we set the space step sizes $h_1=h_2=h.$ Since the analytical solution is not available for the general case, we use

$$\begin{aligned} \mathrm{Error}_h(\tau )= & {} \max _{0\le i \le M} \left| u^{N}_i(h,\tau ) -u^{2N}_i\left( h,\frac{\tau }{2}\right) \right| , \\ \mathrm{Error}_{\tau }(h)= & {} \max _{0\le i \le M} \left| u^{N}_{i}(h,\tau ) -u_{2i}^N\left( \frac{h}{2},\tau \right) \right| \end{aligned}$$

to measure the numerical errors in time and in space, respectively, where $u^N_i(h,\tau )$ denotes the numerical solution at grids $(x_i,t_N)$. The corresponding temporal and spatial convergence orders are defined by

$$\begin{aligned} \mathrm{Order}_{h}(\tau )= \log _2\frac{\mathrm{Error}_h(2\tau )}{\mathrm{Error}_h(\tau )},\quad \mathrm{Order}_{\tau }(h) = \log _2 \frac{\mathrm{Error}_{\tau }(2h)}{\mathrm{Error}_{\tau }(h)}. \end{aligned}$$

Example 1

Consider (10)–(16) with $\psi _1(x)=0$, $\psi _2(x)=0$, $\phi _1(t)=0,$$\phi _2(t)=0$, and the source term

$$\begin{aligned} {\left\{ \begin{array}{ll} f_1(x,t)= \sin x,\quad 0 \le x \le 0.5, \\ f_2(x,t)= \cos x, \quad 0.5\le x\le 1. \end{array}\right. } \end{aligned}$$

In this numerical experiment, the parameters are chosen as follows: $L_t=1,$$l_x=0.5,$$L_x=1,$$B_1=1$, $B_2=1$, $\eta _1=1$, $\eta _2=1$, $\gamma _1=1,$$\gamma _2=1$.

We use the scheme (61)–(64) to obtain the numerical solution. Fixing $\tau =\frac{1}{500}$, Table 1 shows the numerical errors and the spatial convergence accuracy of the scheme for the Knudsen numbers $K_{\mathrm{n,1}}=K_{\mathrm{n,2}}=0.1,1,10$, respectively. It should be pointed out that when the mean free path is constant, the larger the Knudsen number is, the smaller the characteristic length is. From Table 1, we see that the scheme achieves second-order convergence accuracy in space. Let $h=\frac{1}{100}$. Table 2 presents the corresponding numerical results in the temporal direction. As expected, our scheme gives the convergence accuracy of order $\min \{2-\alpha ,2-\beta \}$ in time.

Example 2

We considered the 1D time fractional DPL heat conduction equation in a double-layered nanoscale thin film where a gold layer is on a chromium padding layer exposed to an ultrashort-pulsed laser heating. Each thickness of the gold layer and the chromium layer is $1 (\mathrm{nm})$, implying that $L_x = 2(\mathrm{nm}), l_x= 1(\mathrm{nm})$. The thermal properties of gold and chromium used in the analysis are listed in Table 3.

The heat source for both layers was given as

$$\begin{aligned} S(x,t) = 0.94J\frac{1-R}{t_p \delta } \exp \left[ -\frac{x}{\delta } -2.77 \left( \frac{t-2t_p}{t_p^2}\right) ^2 \right] \end{aligned}$$

where $J = 13.7 (\mathrm{J/m^2})$, $\delta = 15.3(\mathrm{nm})$, $t_p = 0.1(\mathrm{ps})$ and $R=0.93$. The initial temperature was chosen to be $T_0=300(\mathrm{K})$. In this case, we chose $f_1(x,t) = S(x,t) + \frac{(\tau _{q,1})^{\alpha }}{\varGamma (1+\alpha )} {}^C_0 D^{\alpha }_t S(x,t)$ and $f_2(x,t) = S(x,t) + \frac{(\tau _{q,2})^{\alpha }}{\varGamma (1+\alpha )} {}^C_0 D^{\alpha }_t S(x,t)$ as given in (4). For simplicity, we fixed the following boundary conditions and initial conditions

$$\begin{aligned} -G T_x(0,t) + T(0,t)= & {} T_0,\quad GT_x(L_x,t) + T(L_x,t) = T_0,\quad 0< t < L_t,\\ T(x,0)= & {} T_0,\quad T_t (x,0) =0, \quad 0\le x \le L_x. \end{aligned}$$

We used the scheme (61)–(64) to compute the numerical solution of this example. And here, we only considered the parameters $(\alpha ,\beta )=(0.7,0.3)$ and the computational accuracy at time $L_t = 1$ to check the convergence orders. We fixed the number of subintervals $N=500$ and varied the space step size h to test the convergence order in space. On the other hand, in order to verify the convergence order in time, we used an adequately small space step size $h= \frac{1}{200}$ with different time step sizes. The numerical results in Table 4 and Table  5 indicate that the scheme (61)–(64) guarantees second-order accuracy in space and $\min \{2-\alpha ,2-\beta \}$ order accuracy in time which coincide with the theoretical results in Theorem  3.

Figures 2, 3, 4, 5, 6, 7, 8, 9 and 10 show the temperature profiles along the spatial direction for $G=0.1,1,10$, at t = 0.2 (ps), 0.25 (ps), 0.5 (ps), 2(ps), respectively, with various pair $(\alpha ,\beta )$. Numerical results were computed by using the time increment 0.025 (ps) and 40 grid points in space. From those figures, one can distinguish the temperature difference at the interface between these two layers at the beginning since thermal properties of gold and chromium are different. At $t=2(ps)$, the temperature reaches the steady-state due to the nanoscale thickness. Furthermore, we see that the temperature level for $G=10$ is highest, the temperature level for $G=1.0$ is next, and the lowest one is for $G=0.1$. This is because when G is larger, the boundary condition tends to be insulated, on the other hand, when G is smaller, the boundary condition tends to be the Dirichlet boundary condition. Fixing the fractional order $\beta =0.1$ and varying $\alpha =0.1,0.5,0.9$, we see that the temperature level heightens slightly as $\alpha$ increases. Similar results can be seen for $\beta =0.5,0.9$ cases.

Based on the same mesh girds of the above figures, Fig. 11 shows the temperature profiles along the temporal direction for all $(\alpha ,\beta )= (0,0)$, (0.1, 0.1), $(0.5,0.5),(0.9,0.9),(0.1,0.9),(0.9,0.1)$, (0.99, 0.99), (1, 1) and $G=0.1,1,10$ at $x=0(\mathrm nm)$. From this figure, we see that the temperature changes from the traditional heat conduction equation to the DPL heat conduction equation with integer order derivatives. This vividly explains that the new model (10)–(16) may be used for modeling some type of sub-DPL heat conduction behavior. In addition, from Fig. 11, one may see that the temperature rises and reaches quickly the steady-state within $0\le t\le 0.3 (\mathrm{ps})$, because of the very thin film. The maximum temperature is round about $324(\mathrm{K})$ when $\alpha =\beta =0.99$ and $G=10$, which are close to the integer order derivatives and the insulated boundary condition. The maximum temperature is almost identical to that obtained in [2], where the DPL equation has not fractional order derivatives.

7 Conclusion

We have developed a well-posed time fractional dual-phase-lagging (DPL) heat conduction equation in a double-layered nanoscale thin film with the temperature-jump boundary condition. An accurate finite difference scheme has been presented for solving this model. Unconditional stability and convergence of the scheme are proved in the maximum norm. Numerical results support the theoretical analysis and show the applicability to the nanoscale heat transfer in a double-layered thin film. Further research will focus on the development of high order accurate numerical schemes for the fractional DPL model in a double layered thin film, where the challenge lies in the interface. As such, one may obtain a reasonably accurate solution using a relatively coarse mesh. This is particularly interesting for thermal analysis in nanoscale heat conduction.