1 Introduction

The partial differential equations of fractional order begun to play an important role in particular in modeling of the so called anomalous phenomena and in the theory of the complex systems during the last few decades. The subject of a huge number of papers and a number of monographs (see [6, 9, 10]) are different applications of the fractional differential equations in physics, mechanics, chemistry, technique, engineering, biology and astrophysics.

The time fractional diffusion equation (TFDE) is obtained from the classical diffusion equation by replacing the first order time derivative by fractional derivative of order \(\alpha \) with \(\alpha \in (0,1)\). The TFDE have been investigated in analytical and numerical frames. But, analytic solutions of the most fractional differential equations cannot be obtained explicitly. So many authors proposed discrete numerical approximations and discussed about its stability and convergence (see [1, 3, 7]).

2 Fractional Derivatives and Subdiffusion Equation

The most popular two definitions of fractional derivative are the Rimann–Liouville and the Caputo definitions.

The left Riemann-Liouville fractional derivative of order \(\alpha \) is defined as

$$\begin{aligned} D_{a+}^{\alpha }u(t)=\frac{1}{\varGamma (n-\alpha )}\frac{d^{n}}{dt^{n}} \int _{a}^{t}\frac{u(s)}{(t-s)^{\alpha +1-n}}\,ds , \quad t \ge a, \end{aligned}$$

where \(n-1\le \alpha <n, n\in {N}\) and \(\varGamma (\cdot )\) is the Gamma function.

By interchanging the derivative and integral operators in the previous definition one obtains so called the left Caputo fractional derivative

$$ {}^CD_{a+}^{\alpha }u(t)=\frac{1}{\varGamma (n-\alpha )} \int _{a}^{t}\frac{u^{(n)}(s)}{(t-s)^{\alpha +1-n}}\,ds\quad t \ge a. $$

In particular, \(D_{a+}^{\alpha }u(t)= {}^CD_{a+}^{\alpha }u(t)\,\) if \(u(a) = u'(a) = \dots = u^{(n-1)}(a) = 0\). Analogously one defines the right derivatives. We use some special spaces and norms and inner products in them. For \(\alpha >0\) we set

$$ |u|_{C_{+}^{\alpha }[a,b]}=\Vert D_{a+}^{\alpha }u\Vert _{C[a,b]},\qquad |u|_{C_{-}^{\alpha }[a,b]}= \Vert D_{b-}^{\alpha }u\Vert _{C[a,b]}, $$
$$ \Vert u\Vert _{C_{\pm }^{\alpha }[a,b]}=\left( \Vert u\Vert ^{2}_{C^{[\alpha ]^-}[a,b]} +|u|_{C_{\pm }^{\alpha }[a,b]}^2\right) ^{1/2}, $$
$$ |u|_{H_{+}^{\alpha }(a,b)}=\Vert D_{a+}^{\alpha }u\Vert _{L^{2}(a,b)},\qquad |u|_{H_{-}^{\alpha }(a,b)}= \Vert D_{b-}^{\alpha }u\Vert _{L^{2}(a,b)} $$

and

$$ \Vert u\Vert _{H_{\pm }^{\alpha }(a,b)}=\left( \Vert u\Vert ^{2}_{H^{[\alpha ]^-}(a,b)} +|u|_{H_{\pm }^{\alpha }(a,b)}^2\right) ^{1/2}, $$

where \([\alpha ]^-\) denotes the largest integer \(< \alpha \). Then we define \(C^{\alpha }_{\pm }[a,b]\) as the space of functions \(u \in C^{[\alpha ]^-}[a,b]\) with finite norm \(\Vert u\Vert _{C_{\pm }^{\alpha }[a,b]}\). The space \(H^{\alpha }_{\pm }(a,b)\) is defined analogously, while the space \(\dot{H}^{\alpha }_{\pm }(a,b)\) is defined as the closure of \(\dot{C}^{\infty }(a,b) = {C}_0^{\infty }(a,b)\) with the respect to the norm \(\Vert \cdot \Vert _{H_{\pm }^{\alpha }(a,b)}\).

Let \(0<\alpha <1,\) \(\varOmega =(0,1)\times (0,1)\) and \(Q=\varOmega \times (0,T).\) We shall consider the next equation

$$\begin{aligned} D_{t,0+}^{\alpha }u + \mathcal L u = f(x,t),\quad x =(x_1,x_2)\in \varOmega ,\quad t\in (0,T), \end{aligned}$$
(1)

subject to boundary and initial conditions

$$\begin{aligned} u(x,t)=0,\quad x\in \partial \varOmega ,\quad t\in (0,T), \end{aligned}$$
(2)
$$\begin{aligned} u(x,0)=0,\quad x\in \bar{\varOmega }, \end{aligned}$$
(3)

where \( \mathcal L u = - \sum _{i=1}^2 \frac{\partial }{\partial x_i}\left( a_{i} \frac{\partial u}{\partial x_i}\right) . \) We also assume, unless otherwise stated, that \(a_i=a(x_1, x_2, t)\) are differentiable functions which satisfy \(0<c_0<a_i<c_1,\) \(i=1,2.\)

By multiplying (1) with a test function v and after partial integration, we obtain the following weak formulation of the problem (1)–(3):

Find \(u\in \dot{H}^{1,\alpha /2}(Q)=L^{2}((0,T),\dot{H}^{1}(\varOmega )) \cap \dot{H}^{\alpha /2}((0,T),L^{2}(\varOmega ))\) such that

$$ a(u,v)=f(v),\quad \forall v\in \dot{H}^{1,\alpha /2}(Q), $$

where \( a(u,v)= \left( D^{\alpha /2}_{t,0+}u,D^{\alpha /2}_{t,T-}v\right) _{L^2(Q)}+\left( a_1\frac{\partial u}{\partial x_1},\frac{\partial v}{\partial x_1}\right) _{L^2(Q)} +\left( a_2\frac{\partial u}{\partial x_2},\frac{\partial v}{\partial x_2}\right) _{L^2(Q)} \) and \( f(v)=(f,v)_{L^2(Q)}. \)

Theorem 1

(see [2, 8]). Let \(f\in L^2(Q),\) \(a_i\in L^{\infty }(Q),\) and let \(0<c_0<a_i<c_1.\) Then the problem (1)–(3) is well posed in \(\dot{H}^{1,\alpha /2}(Q) \) and its solution satisfy the a priori estimate

$$ \Vert u\Vert _{H^{1,\alpha /2}(Q)}\le C\Vert f\Vert _{L^2(Q)}. $$

From Theorem 1 it immediately follows the weaker a priori estimate

$$ \Vert u\Vert _{B^{1,\alpha /2}(Q)}\le C\Vert f\Vert _{L^2(Q)}, $$

where tne norm \(\Vert \cdot \Vert _{B^{1,\alpha /2}(Q)}\) is defined by

$$ \Vert u\Vert _{B^{1,\alpha /2}(Q)}^2 = \int _0^T \left[ (T-t)^{-\alpha } \Vert u(\cdot ,t)\Vert _{L^2(\varOmega )}^2 + \Vert u(\cdot ,t)\Vert _{H^1(\varOmega )}^2\right] dt. $$

3 Approximation

We define the uniform mesh \(\bar{Q}_{h\tau }=\bar{\omega }_h\times \bar{\omega }_{\tau }\) where \(\bar{\omega }_h=\{x=(n_1h,n_2h): n_1,n_2=0,1,...,N; h=1/N\}\) and \(\bar{\omega }_{\tau }=\{t_k=k\tau : k=0,1,...,M; \tau =T/M\}.\) We also define \(\omega _h=\bar{\omega }_h\cap \varOmega ,\) \(\omega _{1h}=\bar{\omega }_h\cap ((0,1]\times (0,1)),\) \(\omega _{2h}=\bar{\omega }_h\cap ((0,1)\times (0,1]),\) \(\omega _{\tau }=\bar{\omega }_{\tau }\cap (0,T),\) \(\omega _{\tau }^-=\bar{\omega }_{\tau }\cap [0,T),\) \(\omega _{\tau }^+=\bar{\omega }_{\tau }\cap (0,T]\) and \(\gamma _h=\bar{\omega }_h\setminus \omega _h.\) We shall use standard notation from the theory of the finite difference schemes (see [11]):

$$ v=v(x,t),\quad \hat{v}=v(x,t+\tau ), \quad \check{v}=v(x,t-\tau ), \quad v^{k}=v(x,t_{k}), \quad x\in \bar{\omega }_{h}, $$
$$ v_{x_i}={v(x+he_i,t)-v(x,t)}/{h} = v_{\bar{x}_i}(x+he_i,t),\ i=1,2, $$
$$ v_{t}={v(x,t+\tau )-v(x,t)}/{\tau } = v_{\bar{t}}(x,t+\tau ) = \hat{v}_{\bar{t}}, $$

where \(e_i\) denotes the unit vector of the axis \(0x_i\).

For a function u defined on \(\bar{Q}\) which satisfies a homogeneous initial condtion, we use two discrete approximations of fractional derivative. The first is

$$\begin{aligned} {}^C D^{\alpha }_{t,0+,\tau }u^k =\frac{\tau ^{1-\alpha }}{\varGamma (2-\alpha )}\sum _{l=0}^{k-1}d_{k-l}u_t^l, \end{aligned}$$
(4)

where \( d_{k-l}=(k-l)^{1-\alpha }-(k-l-1)^{1-\alpha },\quad 0\le l<k\le M, \) and the second is

$$\begin{aligned} D^{\alpha }_{t,0+,\tau }u^k =\frac{1}{\tau \varGamma (1-\alpha )}\left( b_0u^k+\sum _{l=1}^{k-1}[b_{k-l}-b_{k-l-1}]u^l\right) \end{aligned}$$
(5)

where \( b_{k-l}=\frac{\tau ^{1-\alpha }}{1-\alpha }d_{k-l+1}\). Note that in our case \({}^C D^{\alpha }_{t,0+,\tau }u^k=D^{\alpha }_{t,0+,\tau }u^k\).

Lemma 1

[12]. Suppose that \(\alpha \in (0,1)\), \(u\in C^{2}([0,t], C(\bar{\varOmega }))\) and \(t\in \omega _{\tau }^+.\) Then

$$ |{}^C D_{t,0+}^{\alpha }u-{}^C D_{t,0+,\tau }^{\alpha }u| \le \frac{\tau ^{2-\alpha }}{1-\alpha }\left[ \frac{1-\alpha }{12}+\frac{2^{2-\alpha }}{2-\alpha }-(1+2^{-\alpha })\right] \max _{\bar{Q}_t}\left| \frac{\partial ^2 u}{\partial t^2}\right| , $$

where denoted \(Q_t = (0,t)\times \varOmega \).

Lemma 2

[5]. For \(0<\alpha <1\) and any function v(t) defined on \(\bar{\omega }_{\tau }\) which satisfies \(v(0)=0\) the following inequality is valid

$$\begin{aligned} v^{k}(D^{\alpha }_{t,0+,\tau }v)^{k}\ge \frac{1}{2}\,\left( D^{\alpha }_{t,0+,\tau }(v^{2})\right) ^{k} + \frac{\tau ^{2-\alpha }(1-2^{-\alpha })}{\varGamma (2-\alpha )}\big (v^{k-1}_{t}\big )^2. \end{aligned}$$
(6)

We shall use the following inner products and norms.

$$ (v,w)_{h}=(v,w)_{L^{2}(\omega _{h})}=h^{2}\!\sum _{x\in \omega _h}\!vw, \quad \Vert v\Vert _{h}=\Vert v\Vert _{L^{2}(\omega _{h})}=(v,v)_{h}^{1/2}, $$
$$ (v,w)_{ih}\!=(v,w)_{L^{2}(\omega _{ih})}\!=h^{2}\!\sum _{x\in \omega _{ih}}\!vw, \ \ \Vert v\Vert _{ih}\!=\Vert v\Vert _{L^{2}(\omega _{ih})}\!=(v,v)_{ih}^{1/2}, \ \ i=0,1,2, $$
$$ |v|^{2}_{H^{1}(\omega _{h})}= \sum _{i=1}^2 \Vert v_{\bar{x}_i}\Vert ^{2}_{ih} , \quad \Vert v\Vert ^{2}_{H^{1}(\omega _{h})}=|v|^{2}_{H^{1}(\omega _{h})} + \Vert v\Vert ^{2}_{h}, $$
$$ \Vert v\Vert _{L^{2}(Q_{h\tau })}^2 = \tau \sum _{k=1}^{M}\Vert v^{k}\Vert _{h}^{2} , \quad \Vert v\Vert _{L^{2}(Q_{ih\tau })}^2 = \tau \sum _{k=1}^{M}\Vert v^{k}\Vert _{ih}^{2} ,\quad i=0,1,2, $$

3.1 Additive Scheme

Let \(M=2m\). We approximate (1)–(3) with the following additive difference scheme:

$$\begin{aligned} D^{\alpha }_{t,0+,\tau }v^{2k-1}-2(\tilde{a}_1v_{\bar{x}_1})^{2k-1}_{x_1}=\bar{f}^{2k-1},\quad x\in \omega _h,\quad k=1,2,\ldots ,m \end{aligned}$$
(7)
$$\begin{aligned} D^{\alpha }_{t,0+,\tau }v^{2k}-2(\tilde{a}_2v_{\bar{x}_2})^{2k}_{x_2}=\bar{f}^{2k},\quad x\in \omega _h,\quad k=1,2,\ldots ,m \end{aligned}$$
(8)
$$\begin{aligned} v(x,t)=0,\quad (x,t)\in \gamma _h\times \omega ^{+}_{h}, \end{aligned}$$
(9)
$$\begin{aligned} v(x,0)=0,\quad x\in \bar{\omega }_{h}, \end{aligned}$$
(10)

where \(\tilde{a}_i(x,t) = a_i(x-0.5h e_i,t)\). The idea of this difference scheme is to reduce two dimensional problem to one dimensional. On each time level we have to solve a sequence of linear systems with three-diagonal matrices. Because of that, this scheme is economical.

When the right-hand side f is a continuous function, we set \(\bar{f} = f\), otherwise we must use some averaged value, for example \(\bar{f} = T_1T_2 f\), where \(T_i\) are Steklov averaging operators: \( T_if(x,t) = \int \limits _{-1/2}^{1/2} f(x + hse_i,t)\,ds, \quad i =1,2 . \)

Here, we define the following norm

$$\begin{aligned} \Vert v\Vert _{B^{1,\alpha /2}(Q_{h\tau })}\!=\!\Bigg [\tau \sum _{k=1}^{2m}\Big (D^{\alpha }_{t,0+,\tau } \big (\Vert v\Vert ^{2}_{h}\big )\Big )^{k}\!+\tau \sum _{k=1}^{m} \Big (\Vert v^{2k-1}_{\bar{x}_1}\Vert ^2_{L^{2}(\omega _{1h})} +\Vert v^{2k}_{\bar{x}_2}\Vert ^2_{L^{2}(\omega _{2h})}\Big ) \Bigg ]^{1/2}\!\! . \end{aligned}$$

Theorem 2

Let \(0<\alpha <1\) and \(f\in L^2(Q).\) Let also \(a_i\) be continuous functions. Then the difference scheme (7)–(10) is absolutely stable and its solution satisfies the following a priori estimate

$$\begin{aligned} \Vert v\Vert _{B^{1,\alpha /2}(Q_{h\tau })}\le C\Vert \bar{f}\Vert _{L^2(Q_{h\tau })}. \end{aligned}$$
(11)

Proof

Taking the inner products of Eqs. (7) and (8) with \(v^{2k-1}\) and \(v^{2k}\), respectively, we obtain

$$\left( v^{2k-1},D^{\alpha }_{t,0+,\tau }v^{2k-1}\right) _h + 2\left( \tilde{a}_1v_{\bar{x}_1}^{2k-1},v^{2k-1}_{\bar{x}_1}\right) _h = \left( \bar{f}^{2k-1},v^{2k-1}\right) _h, $$
$$\left( v^{2k},D^{\alpha }_{t,0+,\tau }v^{2k}\right) _h +2\left( \tilde{a}_2v_{\bar{x}_2}^{2k},v^{2k}_{\bar{x}_2}\right) _h= \left( \bar{f}^{2k},v^{2k}\right) _h.$$

From here, applying Lemma 2, it follows that

$$ \frac{1}{2}\,\Big (D^{\alpha }_{t,0+,\tau }\big (\Vert v\Vert _{h}^{2}\big )\Big )^{2k-1} +2c_0\,\big \Vert v_{\bar{x}_1}^{2k-1}\big \Vert ^{2}_{1h} \le \varepsilon \,\big \Vert v^{2k-1}\big \Vert _{h}^{2} +\frac{1}{4\varepsilon }\,\big \Vert \bar{f}^{2k-1}\big \Vert _{h}^{2}, $$
$$ \frac{1}{2}\,\Big (D^{\alpha }_{t,0+,\tau }\big (\Vert v\Vert _{h}^{2}\big )\Big )^{2k} +2c_0\,\big \Vert v_{\bar{x}_2}^{2k}\big \Vert ^{2}_{2h}\le \varepsilon \,\big \Vert v^{2k}\big \Vert _{h}^{2} +\frac{1}{4\varepsilon }\,\big \Vert \bar{f}^{2k}\big \Vert _{h}^{2}. $$

By using the Poincaré inequality on the right-hand side and for \(\varepsilon =8 c_0\), we obtain

$$ \frac{1}{2}\,\Big (D^{\alpha }_{t,0+,\tau }\big (\Vert v\Vert _{h}^{2}\big )\Big )^{2k-1} +c_0\,\big \Vert v_{\bar{x}_1}^{2k-1}\big \Vert ^{2}_{1h}\le \frac{1}{32c_0}\big \Vert \bar{f}^{2k-1}\big \Vert _{h}^{2}, $$
$$ \frac{1}{2}\,\Big (D^{\alpha }_{t,0+,\tau }\big (\Vert v\Vert _{h}^{2}\big )\Big )^{2k} +c_0\,\big \Vert v_{\bar{x}_2}^{2k}\big \Vert ^{2}_{2h}\le \frac{1}{32c_0}\big \Vert \bar{f}^{2k}\big \Vert _{h}^{2}, $$

Multiplying with \(32c_0 \tau \) and summing these two inequalities through \(k=1,\ldots ,m\), the proof is completed where \( C=\left( 16c_0\min \left\{ 1,2c_0\right\} \right) ^{-1/2}. \)    \(\square \)

Let us analyze the error and the accuracy of the proposed scheme. Let u be the solution of the initial-boundary-value problem (1)–(3) and v the solution of the difference problem (7)–(10) with \(\bar{f} = T_1T_2 f\), then the error \(z=u-v\) is defined on the mesh \(\bar{\omega }_h\times \bar{\omega }_{\tau }\). Putting \(v=u-z\) into (7)–(10) it follows that error satisfies

$$\begin{aligned} D_{t,0+,\tau }^{\alpha }z^{2k-1}-2(\tilde{a}_1z_{\bar{x}_1})_{x_1}^{2k-1}=\psi _{1}^{2k-1}, \quad x\in \omega _{h}, \quad k=1,2,\dots ,m, \end{aligned}$$
(12)
$$\begin{aligned} D_{t,0+,\tau }^{\alpha }z^{2k}-2(\tilde{a}_2z_{\bar{x}_2})_{x_2}^{2k}=\psi _{2}^{2k}, \quad x\in \omega _{h}, \quad k=1,2,\dots ,m, \end{aligned}$$
(13)
$$\begin{aligned} z=0,\quad x\in \gamma _{h}, \quad t\in \bar{\omega }_{\tau }, \end{aligned}$$
(14)
$$\begin{aligned} z^{0}=z(x,0) = 0, \quad x\in \omega _{h}. \end{aligned}$$
(15)

where

$$ \psi _{1}^{2k-1}=\xi ^{2k-1}+ 2\eta _1^{2k-1}+\chi ^{2k-1},\qquad \psi _{2}^{2k}= {\xi }^{2k}+ 2\eta _2^{2k}-\chi ^{2k}. $$

Here we denoted

$$ \xi =D_{t,0+,\tau }^{\alpha }u -T_1T_2D^{\alpha }_{t,0+}u, $$
$$ \eta _i= T_1T_2\left( \frac{\partial }{\partial x_i}\left( a_i\frac{\partial u}{\partial x_i}\right) \right) - (\tilde{a}_iu_{\bar{x}_i})_{x_i}, \quad i=1,2, $$
$$ \chi =T_1T_2\left( \frac{\partial }{\partial x_2}\left( a_2\frac{\partial u}{\partial x_2}\right) -\frac{\partial }{\partial x_1}\left( a_1\frac{\partial u}{\partial x_1}\right) \right) . $$

Further, using the properties of Steklov averaging operators, we obtain \(\eta _1=\zeta _{1,x_1}\) and \(\eta _2=\zeta _{2,x_2}\) where

$$ \zeta _i=T_{3-i}\left( a_i\frac{\partial u}{\partial x_i}\right) \bigg |_{(x-0.5 he_i,t)}-a_i(x-0.5 he_i,t)\,u_{\bar{x}_i}(x,t),\quad i =1,2. $$

Lemma 3

[5]. Additive difference scheme (12)–(15) is absolutely stable and the following a priori estimate holds

(16)

Proof

The proof is similar to the proof of Lemma 6.1 in [5].    \(\square \)

Theorem 3

Let the solution u of initial-boundary value problem (1)–(3) belong to the space \(C^2([0,T], C(\bar{\varOmega }))\cap C^1([0,T], H^2(\varOmega ))\cap C([0,T], H^3(\varOmega ))\) and \(a_i \in C([0,T],\) \( H^2(\varOmega ))\), \(i=1,2\). Then the solution v of finite difference scheme (7)–(10) with \(\bar{f} =T_1T_2 f\) converges to u and the following convergence rate estimate holds:

$$ \Vert u-v\Vert _{B^{1,\alpha /2}(Q_{h\tau })} = O(h^2 + {\tau }^{\alpha /2}). $$

Proof

We must estimate the terms on the right-hand side of the last inequality. Let us set \(\xi = \xi _1 + \xi _2\), where

$$ \xi _1 = D_{t,0+,\tau }^{\alpha }u - D^{\alpha }_{t,0+}u, \qquad \xi _2 = D^{\alpha }_{t,0+}u - T_1T_2D^{\alpha }_{t,0+}u = D^{\alpha }_{t,0+}(u - T_1T_2u). $$

From Lemma 1 and from integral representation of \(u-T_1T_2u,\) one can obtains

$$\begin{aligned} \Bigg (\tau \sum _{k=1}^{2m}\Vert \xi _1^{k}\Vert _{h}^{2} \Bigg )^{1/2} \le C\tau ^{2-\alpha }\,\Vert u\Vert _{C^2([0,T], C(\bar{\varOmega }))}. \end{aligned}$$
(17)
$$\begin{aligned} \Bigg (\tau \sum _{k=1}^{2m}\Vert \xi _2^{k}\Vert _{h}^{2}\Bigg )^{1/2} \le Ch^{2}\,\Vert u\Vert _{C^{\alpha }_+([0,T], H^2(\varOmega ))}. \end{aligned}$$
(18)

The terms \(\zeta _i,\) \(i=1,2,\) we decompose in the next way

$$ \begin{array}{l} \displaystyle \zeta _{i} = \zeta _{i1} + \zeta _{i2} + \zeta _{i3} , \qquad \text{ where } \\[3mm]\displaystyle \zeta _{i1} = T_{3-i}\Big (a_{i}\frac{\partial u}{\partial x_i}\Big )\Big |_{(x-0.5he_i,t)} - T_{3-i}\left( a_{i}\right) \Big |_{(x-0.5he_i,t)} T_{3-i}\Big (\frac{\partial u}{\partial x_i}\Big )\Big |_{(x-0.5he_i,t)}, \\[4mm]\displaystyle \zeta _{i2} = T_{3-i}\left( a_{i}\right) \Big |_{(x-0.5he_i,t)}\Big [ T_{3-i}\Big (\frac{\partial u}{\partial x_i}\Big )\Big |_{(x-0.5he_i,t)} - u_{\bar{x}_i}(x,t)\Big ] , \displaystyle \\[4mm]\displaystyle \zeta _{i3} = \Big [ T_{3-i}\left( a_{i}\right) \Big |_{(x-0.5he_i,t)} - a_{i}(x-0.5he_i,t) \Big ]u_{\bar{x}_i}(x,t). \end{array} $$

These terms we can estimate like in [2]

$$\begin{aligned} \left\| \zeta _{ij}\right\| _{L^2(Q_{ih\tau })}\!\le \! Ch^2\left\| a_{i}\right\| _{C([0,T],H^2(\varOmega ))}\left\| u\right\| _{C([0,T],H^3(\varOmega ))},\,\,\,\, i=1,2,\,\,\, j=1,2,3, \end{aligned}$$
(19)

while the terms \(\chi \) and \(\chi _t\) are bounded and them we can estimate directly.   \(\square \)

3.2 The Modified Factorized Scheme

The factorized difference scheme for a 2D fractional in time subdiffusion equation is proposed in [2, 4]. By multiplying discretized fractional derivative with operator \((I+\theta \tau ^\alpha A_1)(I+\theta \tau ^\alpha A_2),\) where \(A_i v=-v_{x_i\bar{x}_i},\) and \(\theta \) positive parameter, we add a small error term, but achieved an economical scheme. In [13], when the operator \(\mathcal {L}\) with constant coefficients does not contain mixed partial derivatives, the authors used the part of this factorized operator and obtained better convergence rate in time direction. Here, we consider (1)–(3), where \(a_i=a_i(x_i).\) The operator \(\mathcal {L}\) we approximate with \( L_h =L_{h,1}+L_{h,2}\), where \(L_{h,i}v=-(\tilde{a}_iv_{\bar{x}_i})_{x_i}\) and \(\tilde{a}_i= \tilde{a}_i(x_i) = a_i(x_i-0.5 h)\). The problem (1)–(3) we approximate with the following scheme

$$\begin{aligned} \left( I+\mu ^2L_{h,1}L_{h,2}\right) D^{\alpha }_{t,0+,\tau }v^{k}+L_hv^{k}=\bar{f}^{k}, \quad x\in \omega _{h}, \end{aligned}$$
(20)

where \(\mu =\varGamma (2-\alpha )\tau ^{\alpha },\) subject to homogeneous boundary and initial conditions (9), (10). By using formula (5), it follows from (20)

$$\begin{aligned} \left( I+\mu L_{h,1}\right) \left( I+\mu L_{h,2}\right) v^k=-\sum _{l=1}^{k-1}\left[ d_{k-l+1}-d_{k-l}\right] \left( v^l+\mu ^2L_{h,1}L_{h,2}v^l\right) +\mu \bar{f}^k. \end{aligned}$$

This scheme is also economical. To compute the solution on time level k, we have to solve two systems of linear equations with three-diagonal matrices. Now, we define the norm \(\Vert \cdot \Vert _{B^{1,\alpha /2}(Q_{h\tau })}\) on the following way

$$\begin{aligned} \Vert v\Vert _{B^{1,\alpha /2}(Q_{h\tau })}^2 = \tau \sum _{k=1}^{M} \left[ \Big (D^{\alpha }_{t,0+,\tau } \big (\Vert v\Vert _h^{2}\big )\!\Big )^{\! k} + |v^{k}|^2_{H^{1}(\omega _{h})} \right] . \end{aligned}$$

Theorem 4

Let \(0<\alpha <1\) and \(a_i\in C[0,1].\) Then the difference scheme (20), (9), (10) is absolutely stable and its solution satisfies the a apriori estimate

$$\begin{aligned} \Vert v\Vert _{B^{1,\alpha /2}(Q_{h\tau })}\le C \Vert \bar{f}\Vert _{L^2(Q_{h\tau })}. \end{aligned}$$
(21)

Proof

If we denote \(B=I+\mu ^2L_{h,1}L_{h,2},\) it is obviously that \(B>I\) and because of that \(\Vert v\Vert ^2_{B} := (Bv,v)_h \ge \Vert v\Vert ^2_h.\) Multiplying (20) with \(v^k,\) we obtain

$$ \frac{1}{2}\left( D^{\alpha }_{t,0+,\tau }(\Vert v\Vert _h^2)\right) ^k+c_0|v^k|^2_{H^1(\omega _h)}\le \varepsilon \Vert v^k\Vert ^2+\frac{1}{4\varepsilon }\Vert \bar{f}^k\Vert ^2_h. $$

Like in the proof of the Theorem 2, for \(\varepsilon =8c_0,\) we show that (21) is valid, where \(C=\left( 16c_0\min \left\{ 1,c_0\right\} \right) ^{-1/2}.\)    \(\square \)

If u is the solution of the initial-boundary-value problem (1)–(3) and v the solution of the difference problem (20), (9), (10) with \(\bar{f} = T_1T_2 f\), then the error \(z=u-v\) satisfies

$$\begin{aligned} \left( I+\mu ^2L_{h,1}L_{h,2}\right) D^{\alpha }_{t,0+,\tau }z^{k}+L_hz^{k}=\psi ^{k}, \quad x\in \omega _{h}, \end{aligned}$$
(22)

where \( \psi ^{k}=\xi ^{k}+\frac{1}{2}\left( \zeta _{1,x_1}^{k}+\zeta _{2,x_2}^{k}\right) + \nu ^{k}_{1,x_1}+ \nu ^{k}_{2,x_2}. \) The terms \(\xi \) and \(\zeta _i\) are already defined and \( {\nu }_i=\frac{1}{2}\mu ^2L_{h,3-i}D^{\alpha }_{t,0+,\tau }(\tilde{a}_iu_{\bar{x}_i}^k)\). The next assertion holds.

Lemma 4

Difference scheme (22),(14),(15) is apsolutely stable and the following a priori estimate is valid

$$\begin{aligned} \Vert z\Vert _{B^{1,\alpha /2}(Q_{h\tau })}\le C\,\left( \tau \sum _{k=1}^{M}\left( \Vert \xi ^{k}\Vert _{h}^{2}+\sum _{i=1}^{2}\left( \Vert \zeta _i^{k}\Vert _{ih}^{2}+ \Vert \nu _i^{k}\Vert _{ih}^{2}\right) \right) \right) ^{1/2}. \end{aligned}$$
(23)

Now, for the convergence rate of the proposed scheme, we have to estimate the terms on the right-hand side in (23). It can be seen that the temporal convergence rate is better here than in additive scheme.

Theorem 5

Let the solution u of the initial-boundary value problem (1)–(3) belong to the space \(C^2([0,T], C(\bar{\varOmega }))\cap C^1([0,T], H^3(\varOmega ))\) and \(a_i\in C^1[0,1].\) Then the solution v of the difference scheme (20), (14), (15) with \(\bar{f}=T_1T_2f\) convergens to the u and the following convergence rate estimate holds

$$ \Vert u-v\Vert _{B^{1,\alpha /2}(Q_{h\tau })} = O(h^2 + \tau ^{\min \{{2\alpha },{2-\alpha }\}}). $$

Proof

The result follows from (17)–(19) and from

$$ \qquad \qquad \,\,\, \left( \tau \sum _{k=1}^{M}\left\| {\nu }_i\right\| ^2_{ih}\right) ^{1/2}\le C\tau ^{2\alpha }\max _{j=1,2}\Vert a_j\Vert _{C^1[0,1]}\left\| u\right\| _{C^1([0,T],H^3(\varOmega ))}. $$

               \(\square \)