Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition

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 firstand 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.


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., 1-15 and the references given therein .
The role played by stability inequalities well posedness in the study of boundaryvalue problems for hyperbolic partial differential equations is well known see, e.g., 16-29 .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 a r x u x r x r D 1/2 t u t, x σu t, x f t, x , x x 1 , . . ., x m ∈ Ω, 0 < t < 1, u 0, x 0, u t 0, x 0, x ∈ Ω; ∂u t, x ∂n 0, x ∈ S, Here Ω is the unit open cube in the m-dimensional Euclidean space: R m : {Ω x x 1 , . . ., x m : 0 < x j < 1, 1 ≤ j ≤ m} with boundary S, Ω Ω ∪ S; a r x x ∈ Ω and f t, x t ∈ 0, 1 , x ∈ Ω are given smooth functions and a r x ≥ a > 0.

Definition
The Caputo fractional derivative of order α > 0 of a continuous function u t, x is defined by where Γ • is the gamma function.

The Finite Difference Method
In this section, we consider the first order of accuracy in t and the second-orders of accuracy in space variables' 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.

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

2.1
We introduce the Banach space L 2h L 2 Ω h of the grid functions ϕ h x {ϕ h 1 r 1 , . . ., h m r m } defined on Ω h , equipped with the norm

2.2
To the differential operator A x generated by problem 1.1 , we assign the difference operator A x h by the formula acting in the space of grid functions u h x , satisfying the conditions D h u h x 0 for all x ∈ S h .It is known that A x h is a self-adjoint positive definite operator in L 2 Ω h .With the help of A x h 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

2.5
Here • • • h 2 m be sufficiently small numbers.Then, the solutions of difference scheme 2.5 satisfy the following stability estimates: .

2.6
Here C 1 and C 2 do not depend on τ, h, and The proof of Theorem 2.1 is based on the self-adjointness and positive definitness of operator A x h in L 2h and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L 2h .

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.

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

2.13
So, we have the second-order difference equation with respect to n matrix coefficients.To solve this difference equation, we have applied a procedure of modified Gauss elimination method for difference equation with respect to k matrix coefficients.Hence, we seek a solution of the matrix equation in the following form: 14 n M − 1, . . ., 2, 1, α j j 1, . . ., M are N 1 × N 1 square matrices, and β j j 1, . . ., M are N 1 × 1 column matrices defined by where .

2.16
Now, we will give the results of the numerical analysis.First, we give an estimate for the constants C 1 and C 2 figuring in the stability estimates of Theorem 2.1.We have

2.17
The constants C t1 and C t2 in the case of numerical solution of initial-boundary value problem 2.9 are computed.The constants C t1 and C t2 are given in Table 1 for N 20, 40, 80, and M 80, 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 N M, where u t k , x n represents the exact solution and u k n represents the numerical solution at t k , x n .The errors E 0 , E 1 and E 2 results are shown in Table 2 for N 20, 40, 60 and M 60, 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: u t, x t 2 cos πx .

2.19
The figure of the exact solution of 2.9 is shown by the Figure 1.

Figure 1:
The surface shows the exact solution u t, x for 2.9 .

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:

3.4
Therefore, the Lagrange multiplier can be identified as follows: λ t, s s − t.

3.5
Substituting the identified Lagrange multiplier into 3.1 , the following variational iteration formula can be obtained: a r x u n xr x r D 1/2 s u n s, x u n s, x −f s, x ds.

3.6
In this case, let an initial approximation u 0 t, x u 0, x tu t 0, x .Then approximate solution takes the form u t, x lim n → ∞ u n t, x .

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

3.8
Now we start with an initial approximation u 0 t, x u 0, x tu t 0, x .

3.10
The figure of 3.10 is given by the Figure 3.

Figure 5 :
Figure 5:The surface shows the exact solution u t, x, y for 3.11 .
t dt.

Table 2 :
Comparison of the errors for the difference scheme.