Advanced Theoretical and Applied Studies of Fractional Differential Equations
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Ibrahim Karatay, Serife R. Bayramoglu, "A Characteristic Difference Scheme for TimeFractional Heat Equations Based on the CrankNicholson Difference Schemes", Abstract and Applied Analysis, vol. 2012, Article ID 548292, 11 pages, 2012. https://doi.org/10.1155/2012/548292
A Characteristic Difference Scheme for TimeFractional Heat Equations Based on the CrankNicholson Difference Schemes
Abstract
We consider the numerical solution of a timefractional heat equation, which is obtained from the standard diffusion equation by replacing the firstorder time derivative with RiemannLiouville fractional derivative of order α, where . The main purpose of this work is to extend the idea on CrankNicholson method to the timefractional heat equations. We prove that the proposed method is unconditionally stable, and the numerical solution converges to the exact one with the order . Numerical experiments are carried out to support the theoretical claims.
1. Introduction
Fractional calculus is one of the most popular subjects in many scientific areas for decades. Many problems in applied science, physics and engineering are modeled mathematically by the fractional partial differential equations (FPDEs). We can see these models adoption in viscoelasticity [1, 2], finance [3, 4], hydrology [5, 6], engineering [7, 8], and control systems [9–11]. FPDEs may be investigated into two fundamental types: timefractional differential equations and spacefractional differential equations.
Several different methods have been used for solving FPDEs. For the analytical solutions to problems, some methods have been proposed: the variational iteration method [12, 13], the Adomian decomposition method [13–16], as well as the Laplace transform and Fourier transform methods [17, 18].
On the other hand, numerical methods which based on a finitedifference approximation to the fractional derivative, for solving FDPEs [19–24], have been proposed. A practical numerical method for solving multidimensional fractional partial differential equations, using a variation on the classical alternatingdirections implicit (ADI) Euler method, is presented in [25]. Many finitedifference approximations for the FPDEs are only firstorder accurate. Some secondorder accurate numerical approximations for the spacefractional differential equations were presented in [26–28]. Here, we propose a CrankNicholsontype method for timefractional differential heat equations with the accuracy of order .
In this work, we consider the following timefractional heat equation: Here, the term denotes ordermodified RiemannLiouville fractional derivative [29] given with the formula: where is the Gamma function.
Remark 1.1. If , then the RiemannLiouville and the modified RiemannLiouville fractional derivatives are identical, since the RiemannLiouville derivative is given by the following formula: If is nonzero, then there are some problems about the existence of the solutions for the heat equation (1.1). To rectify the situation, two main approaches can be used: the modified RiemannLiouville fractional derivative can be used [29] or the initial condition should be modified [30]. We chose the first approach in our work.
2. Discretization of the Problem
In this section, we introduce the basic ideas for the numerical solution of the timefractional heat equation (1.1) by CrankNicholson difference scheme.
For some positive integers and , the grid sizes in space and time for the finitedifference algorithm are defined by and , respectively. The grid points in the space interval are the numbers , , and the grid points in the time interval are labeled , . The values of the functions and at the grid points are denoted and , respectively.
As in the classical CrankNicholson difference scheme, we will obtain a discrete approximation to the fractional derivative at . Let Then, we have Now, we will find the approximations for and : where
Similarly, we can obtain where and
Then, we can write the following approximation: where
On the other hand, using the meanvalue theorem, we get where and . So, we obtain the following secondorder approximation for the modified RiemannLiouville derivative:
3. CrankNicholson Difference Scheme
Using the approximation above, we obtain the following difference scheme which is accurate of order :
We can arrange the system above to obtain
The difference scheme above can be written in matrix form: where , , , , , and .
Here, and are the matrices of the form
We note that the unspecified entries are zero at the matrices above.
Using the idea on the modified GaussElimination method, we can convert (3.3) into the following form:
This way, the twostep form of difference schemes in (3.3) is transformed to onestep method as in (3.5).
Now, we need to determine the matrices and satisfying the last equality. Since , we can select and . Combining the equalities and and the matrix equation (3.3), we have Then, we write where .
So, we obtain the following pair of formulas: where .
4. Stability of the Method
The stability analysis is done by using the analysis of the eigenvalues of the iteration matrix () of the scheme (3.5).
Let denote the spectral radius of a matrix , that is, the maximum of the absolute value of the eigenvalues of the matrix .
We will prove that , (), by induction.
Since is a zero matrix .
Moreover, , since is of the form therefore, .
Now, assume . After some calculations, we find that and we already know that and for :
Since , it follows that . So, for any , where .
Remark 4.1. The convergence of the method follows from the Lax equivalence theorem [31] because of the stability and consistency of the proposed scheme.
5. Numerical Analysis
Example 5.1. Consider
Exact solution of this problem is . The solution by the CrankNicholson scheme is given in Figure 1. The errors when solving this problem are listed in the Table 1 for various values of time and space nodes.
The errors in the table are calculated by the formula and the error rate formula is .

(a)
(b)
Example 5.2. Consider
Exact solution of this problem is . The solution by the CrankNicholson scheme is given in Figure 2. The errors when solving this problem are listed in Table 2 for various values of time and space nodes and several values of .
It can be concluded from the tables and the figures that when the step size is reduced by a factor of 1/2, the error decreases by about 1/4. The numerical results support the claim about the order of the convergence.

(a)
(b)
6. Conclusion
In this work, the CrankNicholson difference scheme was successfully extended to solve the timefractional heat equations. A secondorder approximation for the RiemannLiouville fractional derivative is obtained. It is proven that the timefractional CrankNicholson difference scheme is unconditionally stable and convergent. Numerical results are in good agreement with the theoretical results.
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Copyright © 2012 Ibrahim Karatay and Serife R. Bayramoglu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.