On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations
The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this difference scheme and for the first and second orders difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.
It is known that various problems in fluid mechanics (dynamics, elasticity) and other areas of physics lead to fractional partial differential equations. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [1–11] and the references given therein).
The role played by stability inequalities (well posedness) in the study of boundary-value problems for hyperbolic partial differential equations is well known (see, e.g., [12–25]). In the present paper, the mixed boundary value problem for the multidimensional fractional hyperbolic equation
is considered. Here is the standard Riemann-Lioville's derivative of order and is the unit open cube in the -dimensional Euclidean space with boundary , and are given smooth functions and .
The first order of accuracy in and the second order of accuracy in space variables for the approximate solution of problem (1.1) are presented. The stability estimates for the solution of this difference scheme and its first and second ordes difference derivatives are established. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.
2. The Difference Scheme and Stability Estimates
The discretization of problem (1.1) is carried out in two steps. In the first step, let us define the grid space
We introduce the Banach space of the grid functions defined on equipped with the norm
To the differential operator generated by problem (1.1), we assign the difference operator by the formula
acting in the space of grid functions satisfying the conditions for all It is known that is a self-adjoint positive definite operator in . With the help of we arrive at the initial boundary value problem
for an finite system of ordinary fractional differential equations.
In the second step, applying the first order of approximation formula for (see ) and using the first order of accuracy stable difference scheme for hyperbolic equations (see ), one can present the first order of acuraccy difference scheme
for the approximate solution of problem (2.4). Here
Theorem 2.1. Let and be sufficiently small numbers. Then, the solutions of difference scheme (2.5) satisfy the following stability estimates: Here and do not depend on and
The proof of Theorem 2.1 is based on the self-adjointness and positive definitness of operator in and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in
Theorem 2.2. For the solutions of the elliptic difference problem the following coercivity inequality holds :
Remark 2.3. The stability estimates of Theorem 2.1 are satisfied in the case of operator with Dirichlet condition in and is the standard Riemann-Lioville's derivative of order In this case, is not self-adjoint operator in Nevertheless, and is a self-adjoint positive definite operator in and is bounded in The proof of this statement is based on the abstract results of  and difference analogy of integral inequality.
Remark 2.4. The stability estimates of Theorem 2.1 permit us to obtain the estimate of convergence of difference scheme of the first order of accuracy for approximate solutions of the initial-boundary value problem for semilinear fractional hyperbolic partial differential equations.
Note that, one has not been able to obtain a sharp estimate for the constants figuring in the stability estimates of Theorem 2.1. Therefore, our interest in the present paper is studying the difference scheme (2.5) by numerical experiments. Applying this difference scheme, the numerical methods are proposed in the following section for solving the one-dimensional fractional hyperbolic partial differential equation. The method is illustrated by numerical experiments.
3. Numerical Results
For the numerical result, the mixed problem
for the one-dimensional fractional hyperbolic partial differential equation is considered. Applying difference scheme (2.5), we obtain
We get the system of equations in the matrix form
So, we have the second-order difference equation with respect to matrix coefficients. This type system was developed by Samarskii and Nikolaev . To solve this difference equation we have applied a procedure of modified Gauss elimination method for difference equation with respect to matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:
where are square matrices and are column matrices defined by
Now, we will give the results of the numerical analysis. First, as we noted above one can not obtain a sharp estimate for the constants and figuring in the stability estimates of Theorem 2.1. We have
The constants and in the case of numerical solution of initial-boundary value problem (3.1) are computed.
The numerical solutions are recorded for different values of and represents the numerical solutions of this difference scheme at The constants and are given in Table 1 for and , respectively.
Recall that we have not been able to obtain a sharp estimate for the constants and figuring in the stability estimates. The numerical results in the Tables 1 and 2 give and respectively. That means the constants and figuring in the stability estimates in the case of numerical solution of initial-boundary value problem (3.1) of this difference scheme is stable with no large constants.
Second, for the accurate comparison of the difference scheme considered, the errors computed by
of the numerical solution are recorded for higher values of and where represents the exact solution and represents the numerical solution at The errors and results are shown in Table 2 for and , respectively.
R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 223–276, Springer, Vienna, Austria, 1997.View at: Google Scholar | MathSciNet
D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in System Application, vol. 2, Lille, France, 1996.View at: Google Scholar
I. Podlubny and A. M. A. El-Sayed, On Two Definitions of Fractional Calculus, Slovak Academy of Science-Institute of Experimental Physics, 1996.
S. G. Krein, Linear Differential Equations in a Banach Space, Nauka, Moscow, Russia, 1967.View at: MathSciNet
P. E. Sobolevskii and L. M. Chebotaryeva, “Approximate solution by method of lines of the Cauchy problem for an abstract hyperbolic equations,” Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 5, pp. 103–116, 1977 (Russian).View at: Google Scholar
A. Ashyralyev, M. Martinez, J. Paster, and S. Piskarev, “Weak maximal regularity for abstract hyperbolic problems in function spaces, further progress in analysis,” in Proceedings of the 6th International ISAAC Congress, pp. 679–689, World Scientific, Ankara, Turkey, August 2007.View at: Google Scholar
A. Ashyralyev and I. Muradov, “On difference schemes a second order of accuracy for hyperbolic equations,” in Modelling Processes of Explotation of Gas Places and Applied Problems of Theoretical Gasohydrodynamics, pp. 127–138, Ilim, Ashgabat, Turkmenistan, 1998.View at: Google Scholar
A. Ashyralyev and O. Yildirim, “On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations,” Taiwanese Journal of Mathematics, vol. 13, 22 pages, 2009.View at: Google Scholar
A. A. Samarskii, I. P. Gavrilyuk, and V. L. Makarov, “Stability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces,” SIAM Journal on Numerical Analysis, vol. 39, no. 2, pp. 708–723, 2001.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Ashyralyev and P. E. Sobolevskii, “Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations,” Discrete Dynamics in Nature and Society, vol. 2005, no. 2, pp. 183–213, 2005.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Ashyralyev and M. E. Koksal, “On the stability of the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 57491, 26 pages, 2007.View at: Google Scholar
P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdatelstvo Voronezhskogo Gosud Universiteta, Voronezh, Russia, 1975.
A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2 of Iterative Methods, Birkhäuser, Basel, Switzerland, 1989.View at: MathSciNet