Abstract
The second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation are presented using by rmodified CrankNicholson difference scheme. Stability estimate for the solution of this difference scheme is obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of onedimensional fractional parabolic partial differential equations. Numerical results for this scheme and the CrankNicholson scheme are compared in test examples.
1 Introduction
At present, there is a huge number of theoretical and applied works devoted to the study of fractional differential equations. Solutions of various problems for fractional differential equations can be found, for example, in the monographs of Podlubny [1], Kilbas, Srivastava, and Trujillo [2], Diethelm [3], and in [411]. These problems were studied in various directions: qualitative properties of solutions, spectral problems, various statements of boundary value problems, and numerical investigations.
Many problems in fluid flow, dynamical and diffusion processes, control theory, mechanics, and other areas of physics can be reduced fractional partial differential equations.
In [12] the simple connection of fractional derivatives with fractional powers of first order differential operator was presented. This approach is important to obtain the formula for the fractional difference derivative. Presently, many mathematicians apply this approach and operator tools to investigate various problems for fractional partial differential equations which appear in applied problems (see, e.g., [1320] and the references therein).
In previous paper [17] authors investigated stability estimates for CrankNicholson schemes for the Dirichlet problem for the fractional parabolic equation
Here is the standard RiemannLiouville’s derivative of order and Ω is the open cube in the mdimensional Euclidean space
with boundary S, , () and (, ) are given smooth functions and , .
In [18] the authors investigated stability estimates for CrankNicholson schemes for the Neumann problem for the fractional parabolic equation
The role played by stability inequalities (well posedness) in the study of boundaryvalue problems for parabolic partial differential equations is well known (see, e.g., [2126]).
In the present paper, we consider an rmodified CrankNicholson difference scheme of the above mentioned two problems (1.1), (1.2). This rmodified scheme is of the second order of accuracy in t and in space variables difference schemes for the approximate solution of problems. The stability estimate for the solution of this difference scheme is established. We use a procedure of a modified Gauss elimination method for solving this difference scheme in the case of onedimensional fractional parabolic partial differential equations.
2 Stability of difference scheme
Let us define the grid space
We introduce the Hilbert space of the grid function defined on , equipped with the norm
To the differential operator generated by problem (1.1) or (1.2), respectively, we assign the difference operator by the formula
acting in the space of grid functions , satisfying the conditions or (). It is known that is a selfadjoint positive definite operator in . Here,
With the help of , we arrive at the initial boundary value problem
for a finite system of ordinary fractional differential equations.
We denote
Applying the second order of the approximation formula
for
(see [16]) and the CrankNicholson difference scheme for parabolic equations, one can present the second order of accuracy difference scheme with respect to t and to x. Here,
Now, we introduce the second order accuracy rmodified CrankNicholson difference scheme in the following form:
for the approximate solution of problem (2.2).
Theorem 2.1Letτandbe sufficiently small positive numbers. Then the solutions of the difference scheme (2.4) satisfy the following stability estimate:
whereCdoes not depend onτ, h, and, .
Proof Consider the difference scheme (2.4). We have
where
We obtain
Let us write . Using (2.6), we have
Now, let us estimate , . From the triangle inequality, it follows that
Applying the triangle inequality and the estimates [24]
we have
Hence, applying the difference analog of the integral inequality and inequalities (2.8), (2.10), and (2.11), we get
The proof of estimate (2.5) for the solution of (2.4) follows from (2.6), (2.9), and (2.12). Note that , are independent from τ, h, and , . Theorem 2.1 is proved. □
3 Numerical analysis
We consider two examples for numerical results.
Example 3.1 We consider the following initial boundary value problem with Dirichlet condition for the onedimensional fractional parabolic partial differential equation:
It is clear that the exact solution of problem (3.1) is
Applying the rmodified CrankNicholson difference scheme (2.4), we get
where is defined by (2.3) for any n, . We can rewrite it in the system of equations with matrix coefficients
Here and in the sequel is the zero matrix,
For solving (3.2) we use a modified Gauss elimination method [27]. Hence, we seek a solution of the matrix equation in the following form:
where () are square matrices and () are column matrices defined by
where , is the zero matrix and is the zero matrix and .
Example 3.2 We consider the following initial boundary value problem with Neumann condition for the onedimensional fractional parabolic partial differential equation:
It is clear that the exact solution of problem (3.10) is
Applying the rmodified CrankNicholson difference scheme (2.4), we get
where is defined by (2.3). We can rewrite it in the form of a system of equations with matrix coefficients
Here, A, B, C are defined by (3.3), (3.4), and (3.5):
For solving this matrix equation we will use the same method as for Example 3.1. Namely, we use (3.7), (3.8), (3.9), and
Finally, we give the results of the numerical analysis. The numerical solutions are recorded for different values of the modification parameter r, and discretization parameters N and M. Besides represents the numerical solutions of these difference schemes at . The error is computed by the following formula:
Table 1 and Table 2 are constructed for , respectively. As can be seen from Table 1, the rmodified CrankNicholson difference scheme is more accurate than the CrankNicholson and Rothe difference schemes. Table 2 shows that the rmodified CrankNicholson difference scheme has the same order error as the CrankNicholson difference scheme.
4 Conclusion
In this study, the second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation is investigated. We have obtained a stability estimate for the solution of this difference scheme. The theoretical statements for the solution of this difference scheme for onedimensional parabolic equations are supported by numerical examples obtained by computer.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZC proposed the main idea of this paper, obtained the theoretical and numerical results. AA designed the study and interpreted the results. All authors read and approved the final manuscript.
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