Numerical Solution of Stochastic Hyperbolic Equations

and Applied Analysis 3 Theorem 2.1. Let v tk be the solution of the initial value problem 1.2 at the grid points t tk. Then, {v tk }0 is the solution of the initial value problem for the following difference equation: 1 τ2 v tk 1 − 2v tk v tk−1 2 τ2 I − c τ v tk 1 τ ( f1,k 1 s τ f2,k − c τ f1,k ) , f1,k 1 τ ∫ tk tk−1 s tk − z f z dwz, f2,k 1 τ ∫ tk tk−1 c tk − z f z dwz, 1 ≤ k ≤N − 1, v 0 φ, v τ c τ φ s τ ψ τf1,1. 2.2 Proof. Putting t tk into the formula 1.3 , we can write v tk c tk φ s tk ψ ∫ tk 0 s tk − z f z dwz. 2.3 Using 2.3 , the definition of the sine and cosine operator function, we obtain v tk c tk φ s tk ψ k ∑


Introduction
Stochastic partial differential equations have been studied extensively by many researchers.For example, the method of operators as a tool for investigation of the solution to stochastic equations in Hilbert and Banach spaces have been used systematically by several authors see, 1-7 and the references therein .Numerical methods and theory of solutions of initial boundary value problem for stochastic partial differential equations have been studied in 8-16 .Moreover, the authors of 17 presented a two-step difference scheme for the numerical solution of the following initial value problem: for stochastic hyperbolic differential equations.We have the following.i w t is a standard Wiener process given on the probability space Ω, F, P .
ii For any z ∈ 0, T , f z is an element of the space M 2 w 0, T , H 1 , where H 1 is a subspace of H.
The convergence estimates for the solution of the difference scheme are established.In the present work, we consider the following initial value problem: for stochastic hyperbolic equation in a Hilbert space H with a self-adjoint positive definite operator A with A ≥ δI, where δ > δ 0 > 0. In addition to i and ii , we put the following.
iii ϕ and ψ are elements of the space M 2 w 0, T , H 2 of H 2 -valued measurable processes, where H 2 is a subspace of H.
By the solutions provided in 19 page 423, 0.4 and in 20 page 1005, 2.9 , under the assumptions i , ii , and iii , the initial value problem 1.2 has a unique mild solution given by the following formula: v t c t ϕ s t ψ t 0 s t − z f z dw z .

1.3
For the theory of cosine and sine operator-function we refer to 21, 22 .Our interest in this study is to construct and investigate the difference scheme for the initial value problem 1.2 .The convergence estimate for the solution of the difference scheme is proved.In applications, the theorems on convergence estimates for the solution of difference schemes for the numerical solution of initial-boundary value problems for hyperbolic equations are established.The theoretical statements for the solution of this difference scheme are supported by the result of the numerical experiments.
Theorem 2.1.Let v t k be the solution of the initial value problem 1.2 at the grid points t t k .Then, {v t k } N 0 is the solution of the initial value problem for the following difference equation:

2.2
Proof.Putting t t k into the formula 1.3 , we can write Using 2.3 , the definition of the sine and cosine operator function, we obtain

2.4
It follows that

2.5
Hence, we get the relation between v t k and v t k±1 as This relation and equality 2.2 are equivalent.Theorem 2.1 is proved.

Convergence of the Difference Scheme
For the approximate solution of problem 1.2 , we need to approximate the following expressions:

3.1
Using Taylor's formula and Pade approximation of the function exp −z at z 0, we get

3.2
Applying the difference scheme 2.2 and formula 3.2 , we can construct the following difference scheme: for the approximate solution of the initial value problem 1.2 .Using the definition of c τ τ and s τ τ , we can write 3.3 in the following equivalent form:

3.5
Now, let us give the lemma we need in the sequel from papers 23, 24 .

3.10
The following Theorem on convergence of difference scheme 3.5 is established.
then the estimate of convergence holds.Here, C 1 δ does not depend on τ.
Proof.Using the formula for the solution of second order difference equation and the definition of c τ kτ and s τ kτ , we can write Using 2.4 and 3.13 , we obtain

3.15
Abstract and Applied Analysis 7 Let us estimate the expected value of J m,k for all m 1, . . ., 6, separately.We start with J 1,k and J 2,k .Using 3.6 , 3.7 , and 3.8 , we obtain

3.16
Estimates for the expected value of J m,k for all m 3, . . ., 6, separately, were also used in paper 17 .Combining these estimates, we obtain 3.12 .Theorem 3.2 is proved.

Applications
First, let Λ be the unit open cube in the n-dimensional Euclidean space R n {x x 1 , . . ., x n : 0 < x i < 1, i 1, . . ., n} with boundary S, Λ Λ ∪ S. In 0, T × Λ, the initial-boundary value problem for the following multidimensional hyperbolic equation: with the Dirichlet condition is considered.Here, a r x , x ∈ Λ , δ ≥ 0 and f t, x t ∈ 0, 1 , x ∈ Λ are given smooth functions with respect to x and a r x ≥ a > 0.
The discretization of 4.1 is carried out in two steps.In the first step, define the grid space Let L 2h denote the Hilbert space as The differential operator A in 4.1 is replaced with where the difference operator A x h is defined on these grid functions u h x 0, for all x ∈ S h .As it is proved in 25 , A x h is a self-adjoint positive definite operator in L 2h .Using 4.1 and 4.3 , we get

4.4
In the second step, we replace 4.4 with the difference scheme 3.5 as • • • h 2 n be sufficiently small numbers.Then, the solution of difference scheme 4.5 satisfies the convergence estimate as where C δ does not depend on τ and |h|.
The proof of Theorem 4.1 is based on the abstract Theorem 3.2 and the symmetry properties of the difference operator A x h defined by 4.3 .Second, in 0, T × Λ, the initial-boundary value problem for the following multidimensional hyperbolic equation: Abstract and Applied Analysis 9 with the Neumann condition is considered.Here, n is the normal vector to Λ, δ > 0, a r x , x ∈ Λ , and f t, x t ∈ 0, 1 , x ∈ Λ are given smooth functions with respect to x and a r x ≥ a > 0.
The discretization of 4.7 is carried out in two steps.In the first step, the differential operator A in 4.7 is replaced with where the difference operator A x h is defined on those grid functions D h u h x 0, for all x ∈ S h , where D h u h x 0 is the second order of approximation of ∂u t, x /∂ n.As it is proved in 25 , A x h is a self-adjoint positive definite operator in L 2h .Using 4.7 and 4.8 , we get 4.9 In the second step, we replace 4.9 with the difference scheme 3.5 as • • • h 2 n be sufficiently small numbers.Then, the solution of difference scheme 4.10 satisfies the convergence estimate as where C δ does not depend on τ and |h|.
The proof of Theorem 4.2 is based on the abstract Theorem 3.2 and the symmetry properties of the difference operator A x h defined by 4.8 .
Third, in 0, T × Λ, the mixed boundary value problem for the following multidimensional hyperbolic equation: with the Dirichlet-Neumann condition is considered.Here, n is the normal vector to Λ, δ > 0, a r x , x ∈ Λ , and f t, x t ∈ 0, 1 , x ∈ Λ are given smooth functions with respect to x and a r x ≥ a > 0.
The discretization of 4.12 is carried out in two steps.In the first step, the differential operator A in 4.12 is replaced with a r x u h x r x r ,j r δu h x , 4.13 where the difference operator A x h is defined on those grid functions u h x 0, for all x ∈ S 1 h and D h u h x 0, for all x ∈ S 2 h , S 1 h ∪ S 2 h S h , where D h u h x 0 is the second order of approximation of ∂u t, x /∂ n.By 25 , we can conclude that A x h is a self-adjoint positive definite operator in L 2h .Using 4.12 and 4.13 , we get

4.14
In the second step, we replace 4.14 with the difference scheme 3.5 as

4.15
Theorem 4.3.Let τ and |h| h 2 1 • • • h 2 n be sufficiently small positive numbers.Then, the solution of difference scheme 4.15 satisfies the convergence estimate as where C δ does not depend on τ and |h|.
The proof of Theorem 4.3 is based on the abstract Theorem 3.2 and the symmetry properties of the difference operator A x h defined by 4.13 .

Numerical Examples
In this section, we apply finite difference scheme 2. Example 5.1.The following initial-boundary value problem: for a stochastic hyperbolic equation is considered.The exact solution of this problem is For the approximate solution of the 5.1 , we apply the finite difference scheme 2.2 and we get The system can be written in the following matrix form:

5.4
Here, and D I N 1 is the identity matrix, This type of system was used by 26 for difference equations.For the solution of matrix equation 5.4 , we will use modified Gauss elimination method.We seek a solution of the matrix equation by the following form: where matrices α 1 is an identity and β 1 is a zero matrices, and

5.10
Example 5.2.The following initial-boundary value problem: for a stochastic hyperbolic equation is considered.We use the same procedure as in the first example.The exact solution of this problem is For the approximate solution of the 5.11 , we can construct the following difference scheme:

5.13
and it can be written in the following matrix form: u 0 u M 0.

5.14
Abstract and Applied Analysis 15 Here, the matrices A, B, C, D are given in the previous example, and

5.15
For the solution of matrix equation 5.14 , we will use modified Gauss elimination method.We seek a solution of the matrix equation in the following form: where u M 0, α j j 1, . . ., M−1 are N 1 × N 1 square matrices, β j j 1, . . ., M−1 are N 1 × 1 column matrices.α 1 and β 1 are zero matrices, and We get the following difference scheme: for the approximate solutions of 5.18 , and we obtain the following matrix equation:

5.21
Here, the matrices A, B, C, D are same as in the first example, and

5.22
For the solution of matrix equation 5.21 , we use the same procedure as in the previous examples.Moreover, u M 0, α 1 is an identity and β 1 is a zero matrices, and

5.23
Example 5.4.The following initial boundary value problem: for a stochastic hyperbolic equation is considered.The exact solution of this problem is

5.25
The following difference scheme: is obtained for the approximate solutions of 5.24 , and we obtain the following matrix equation:

5.27
Here, the matrices A, B, C, D are same as in the first example, and and the results are given in Table 1.
The numerical solutions are recorded for different 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 .To obtain the results, we simulated the 1000 sample paths of Brownian motion for each level of discretization.Thus, results show that the error is stable and decreases in an exponential manner.

Using 5 .
27 that we get α 1 is an identity and β 1 is a zero matrices andu M I − α M −1 β M .The rest are the same as in Example 5.3.For these examples, the errors of the numerical solution derived by difference scheme 2 Nτ 1, x n nh, 1 ≤ n ≤ M − 1, Mh π, 1 ≤ k ≤ N − 1, t k kτ,