Abstract
An additive and modified factorized finite-difference scheme for an initial-boundary value problem for a two-dimensional subdiffusion equation are proposed. Its stability and convergence are investigated.
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
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
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
and
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
subject to boundary and initial conditions
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
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
From Theorem 1 it immediately follows the weaker a priori estimate
where tne norm \(\Vert \cdot \Vert _{B^{1,\alpha /2}(Q)}\) is defined by
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]):
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
where \( d_{k-l}=(k-l)^{1-\alpha }-(k-l-1)^{1-\alpha },\quad 0\le l<k\le M, \) and the second is
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
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
We shall use the following inner products and norms.
3.1 Additive Scheme
Let \(M=2m\). We approximate (1)–(3) with the following additive difference scheme:
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
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
Proof
Taking the inner products of Eqs. (7) and (8) with \(v^{2k-1}\) and \(v^{2k}\), respectively, we obtain
From here, applying Lemma 2, it follows that
By using the Poincaré inequality on the right-hand side and for \(\varepsilon =8 c_0\), we obtain
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
where
Here we denoted
Further, using the properties of Steklov averaging operators, we obtain \(\eta _1=\zeta _{1,x_1}\) and \(\eta _2=\zeta _{2,x_2}\) where
Lemma 3
[5]. Additive difference scheme (12)–(15) is absolutely stable and the following a priori estimate holds
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:
Proof
We must estimate the terms on the right-hand side of the last inequality. Let us set \(\xi = \xi _1 + \xi _2\), where
From Lemma 1 and from integral representation of \(u-T_1T_2u,\) one can obtains
The terms \(\zeta _i,\) \(i=1,2,\) we decompose in the next way
These terms we can estimate like in [2]
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
where \(\mu =\varGamma (2-\alpha )\tau ^{\alpha },\) subject to homogeneous boundary and initial conditions (9), (10). By using formula (5), it follows from (20)
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
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
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
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
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
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
Proof
The result follows from (17)–(19) and from
\(\square \)
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Acknowledgement
This research was supported by Ministry of Education, Science and Technological Development of Republic of Serbia thorough the project No. 174015.
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Živanovic, S., Jovanović, B.S. (2017). ADI Schemes for 2D Subdiffusion Equation. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2016. Lecture Notes in Computer Science(), vol 10187. Springer, Cham. https://doi.org/10.1007/978-3-319-57099-0_38
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