Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition
Allaberen Ashyralyev1,2and Fadime Dal3
Academic Editor: Chuanxi Qian
Received19 Dec 2011
Accepted18 Apr 2012
Published28 Jun 2012
Abstract
The numerical and analytic solutions of the mixed
problem for multidimensional fractional hyperbolic partial
differential equations with the Neumann condition are presented. The
stable difference scheme for the numerical solution of the mixed
problem for the multidimensional fractional hyperbolic equation with
the Neumann condition is presented. Stability estimates for the
solution of this difference scheme and for the first- and second-order 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. He's variational iteration method is
applied. The comparison of these methods is presented.
1. Introduction
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., [1โ15] 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., [16โ29]).
In the present paper, finite difference and He's iteration methods for the approximate solutions of the mixed boundary-value problem for the multidimensional fractional hyperbolic equation
are studied. Here is the unit open cube in the -dimensional Euclidean space: with boundary ; and are given smooth functions and .
1.1. Definition
The Caputo fractional derivative of order of a continuous function is defined by
where is the gamma function.
2. The Finite Difference Method
In this section, we consider the first order of accuracy in and the second-orders of accuracy in space variablesโ stable difference scheme for the approximate solution of problem (1.1). The stability estimates for the solution of this difference scheme and its first- and second-order 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.
2.1. The Difference Scheme: 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 infinite system of ordinary fractional differential equations.
In the second step, we replace problem (2.4) by the first order of accuracy difference scheme
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 [30]:
Finally, 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 examples.
2.2. Numerical Results
For the numerical result, the mixed problem
for solving the one-dimensional fractional hyperbolic partial differential equation is considered. Applying difference scheme (2.5), we obtained
We get the system of equations in the matrix form:
where
Here
So, we have the second-order difference equation with respect to matrix coefficients. 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:
,โโ are square matrices, and are column matrices defined by
where
Now, we will give the results of the numerical analysis. First, we give an 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 (2.9) are computed. The constants and are given in Table 1 for , and , respectively.
Second, for the accurate comparison of the difference scheme considered, the errors computed by
of the numerical solutions are recorded for higher values of , where represents the exact solution and represents the numerical solution at . The errors and results are shown in Table 2 for and , respectively.
The figure of the difference scheme solution of (2.9) is given by the Figure 2. The exact solution of (2.9) is given by as follows:
The figure of the exact solution of (2.9) is shown by the Figure 1.
3. He's Variational Iteration Method
In the present paper, the mixed boundary value problem for the multidimensional fractional hyperbolic equation (1.1) is considered. The correction functional for (1.1) can be approximately expressed as follows:
where is a general Lagrangian multiplier (see, e.g., [31]) and is considered as a restricted variation as a restricted variation (see, e.g., [32]); that is, , is its initial approximation. Using the above correction functional stationary and noticing that , we obtain
From the above relation for any , we get the Euler-Lagrange equation:
with the following natural boundary conditions:
Therefore, the Lagrange multiplier can be identified as follows:
Substituting the identified Lagrange multiplier into (3.1), the following variational iteration formula can be obtained:
In this case, let an initial approximation . Then approximate solution takes the form .
3.1. Variational Iteration Solution 1
For the numerical result, the mixed problem
for solving the one-dimensional fractional hyperbolic partial differential equation is considered.
According to formula (3.6), the iteration formula for (3.7) is given by
Now we start with an initial approximation
Using the above iteration formula (3.8), we can obtain the other components as
The figure of (3.10) is given by the Figure 3.
3.2. Variational Iteration Solution 2
For the numerical result, the mixed problem
for solving the two-dimensional fractional hyperbolic partial differential equation is considered.
According to formula (3.6), the iteration formula for (3.11) is given by
we start with an initial approximation
Using the above iteration formula (3.12), we can obtain the other components as
The exact solution of (3.11) is given by as follows:
The figure of the exact solution of (3.11) is shown by the Figure 5.
The figure of (3.14) is given by the Figure 4, and so on; in the same manner the rest of the components of the iteration formula (3.12) can be obtained using the Maple package.
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