A two-step difference scheme for the numerical solution of the initial-boundary value problem for stochastic hyperbolic equations is presented. The convergence
estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of the difference scheme are obtained for different initialboundary value problems. The theoretical statements for the solution of this difference
scheme are supported by numerical examples.
1. 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:
(1.1)dv˙(t)=-Av(t)dt+f(t)dwt,0<t<T,v(0)=0,v˙(0)=0,
for stochastic hyperbolic differential equations. We have the following.
(i) wt 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 Mw2([0,T],H1), where H1 is a subspace of H.
Here, Mw2([0,T],H) [18] denote the space of H-valued measurable processes which satisfy
ϕ(t)isFtmeasurable,a.e.int,
E∫0T∥ϕ(t)∥Hdt<∞.
The convergence estimates for the solution of the difference scheme are established.
In the present work, we consider the following initial value problem:
(1.2)dv˙(t)+Av(t)dt=f(t)dwt,0<t<T,v(0)=φ,v˙(0)=ψ,
for stochastic hyperbolic equation in a Hilbert space H with a self-adjoint positive definite operator A with A≥δI, where δ>δ0>0. In addition to (i) and (ii), we put the following.
(iii) φ and ψ are elements of the space Mw2([0,T],H2) of H2-valued measurable processes, where H2 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:
(1.3)v(t)=c(t)φ+s(t)ψ+∫0ts(t-z)f(z)dwz.
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.
2. The Exact Difference Scheme
We consider the following uniform grid space:
(2.1)[0,T]τ={tk=kτ,k=0,1,…,N,Nτ=T},
with step τ>0. Here, N is a fixed positive integer.
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)}0N is the solution of the initial value problem for the following difference equation:
(2.2)1τ2(v(tk+1)-2v(tk)+v(tk-1))+2τ2(I-c(τ))v(tk)=1τ(f1,k+1+s(τ)f2,k-c(τ)f1,k),f1,k=1τ∫tk-1tks(tk-z)f(z)dwz,f2,k=1τ∫tk-1tkc(tk-z)f(z)dwz,1≤k≤N-1,v(0)=φ,v(τ)=c(τ)φ+s(τ)ψ+τf1,1.
Proof.
Putting t=tk into the formula (1.3), we can write
(2.3)v(tk)=c(tk)φ+s(tk)ψ+∫0tks(tk-z)f(z)dwz.
Using (2.3), the definition of the sine and cosine operator function, we obtain
(2.4)v(tk)=c(tk)φ+s(tk)ψ+∑j=1k∫tj-1tjs(tk-tj+tj-z)f(z)dwz=c(tk)φ+s(tk)ψ+τ∑j=1k(s(tk-tj)f2,j+c(tk-tj)f1,j).
It follows that
(2.5)v(tk+1)+v(tk-1)=[c(tk+1)+c(tk-1)]φ+[s(tk+1)+s(tk-1)]ψ+2c(τ)τ∑j=1k(s(tk-tj)f2,j+c(tk-tj)f1,j)+τ(f1,k+1+s(τ)f2,k-c(τ)f1,k).
Hence, we get the relation between v(tk) and v(tk±1) as
(2.6)v(tk+1)+v(tk-1)-2c(τ)v(tk)=τ(f1,k+1+s(τ)f2,k-c(τ)f1,k).
This relation and equality (2.2) are equivalent. Theorem 2.1 is proved.
3. Convergence of the Difference Scheme
For the approximate solution of problem (1.2), we need to approximate the following expressions:
(3.1)f1,k=1τ∫tk-1tks(tk-z)f(z)dwz,f2,k=1τ∫tk-1tkc(tk-z)f(z)dwz,exp(±iτA1/2).
Using Taylor's formula and Pade approximation of the function exp(-z) at z=0, we get
(3.2)exp(±iτA1/2)≈(I±iτA1/22)(I∓iτA1/22)-1=R(±iτA1/2),f1,k≈-1τ∫tk-1tk(z-tk)f(z)dwz=f~1,k,f2,k≈1τ∫tk-1tkf(z)dwz=f~2,k.
Applying the difference scheme (2.2) and formula (3.2), we can construct the following difference scheme:
(3.3)1τ2(uk+1-2uk+uk-1)+2τ2(I-cτ(τ))uk=1τ(f~1,k+1+sτ(τ)f~2,k-cτ(τ)f~1,k),(3.4)cτ(τ)=R(iτA1/2)+R(-iτA1/2)2,sτ(τ)=A-1/2R(iτA1/2)-R(-iτA1/2)2i,1≤k≤N-1,u0=φ,u1=cτ(τ)φ+sτ(τ)ψ+τf~1,1,
for the approximate solution of the initial value problem (1.2). Using the definition of cτ(τ) and sτ(τ), we can write (3.3) in the following equivalent form:
(3.5)1τ2(uk+1-2uk+uk-1)+12Auk+14Auk+1+Auk-1=1τ((I+14τ2A)f~1,k+1+τf~2,k-(I-14τ2A)f~1,k),tk=kτ,1≤k≤N-1,u0=φ,u1=cτ(τ)φ+sτ(τ)ψ+τf~1,1.
Now, let us give the lemma we need in the sequel from papers [23, 24].
Lemma 3.1.
The following estimates hold:
(3.6)‖c(t)‖H→H≤1,‖A1/2s(t)‖H→H≤1(t≥0),(3.7)‖cτ(kτ)‖H→H≤1,‖A1/2sτ(kτ)‖H→H≤1(k≥0),(3.8)‖A-(1+α)(cτ(kτ)-c(tk))‖H→H≤Cτ(3/2+α),0≤α≤12,(3.9)‖A-(1/2+α)(sτ(kτ)-s(tk))‖H→H≤Cτ(3/2+α),(k≥0),
where
(3.10)cτ(kτ)=Rk(iτA1/2)+Rk(-iτA1/2)2,sτ(kτ)=A-1/2Rk(iτA1/2)-Rk(-iτA1/2)2i.
The following Theorem on convergence of difference scheme (3.5) is established.
Theorem 3.2.
Assume that
(3.11)E(‖Aφ‖H2)≤C,E(‖(A1/2ψ)‖H2)≤C,E∫0T‖Af(t)‖H2dt≤C,
then the estimate of convergence
(3.12)(∑k=1NE‖v(tk)-uk‖H2)1/2≤C1(δ)τ
holds. Here, C1(δ) does not depend on τ.
Proof.
Using the formula for the solution of second order difference equation and the definition of cτ(kτ) and sτ(kτ), we can write
(3.13)uk=cτ(kτ)φ+sτ(kτ)ψ+τ∑j=1k(sτ((k-j)τ)f~2,j+cτ((k-j)τ)f~1,j),1≤k≤N.
Using (2.4) and (3.13), we obtain
(3.14)v(tk)-uk=[c(kτ)-cτ(kτ)]φ+[s(kτ)-sτ(kτ)]ψ+τ∑j=1k(s(tk-tj)f2,j+c(tk-tj)f1,j)-τ∑j=1k(sτ((k-j)τ)f~2,j+cτ((k-j)τ)f~1,j)=[c(kτ)-cτ(kτ)]φ+[s(kτ)-sτ(kτ)]ψ+τ∑j=1k-1sτ((k-j)τ)(f2,j-f~2,j)+τ∑j=1k-1(s(tk-tj)-sτ((k-j)τ))f2,j+τ∑j=1kcτ((k-j)τ)(f1,j-f~1,j)+τ∑j=1k(c(tk-tj)-cτ((k-j)τ))f1,j=J1,k+J2,k+J3,k+J4,k+J5,k+J6,k,1≤k≤N,
where
(3.15)J1,k=[c(kτ)-cτ(kτ)]φ,J2,k=[s(kτ)-sτ(kτ)]ψ,J3,k=τ∑j=1k-1sτ((k-j)τ)(f2,j-f~2,j),J4,k=τ∑j=1k-1(s(tk-tj)-sτ((k-j)τ))f2,j,J5,k=τ∑j=1kcτ((k-j)τ)(f1,j-f~1,j),J6,k=τ∑j=1k(c(tk-tj)-cτ((k-j)τ))f1,j.
Let us estimate the expected value of Jm,k for all m=1,…,6, separately. We start with J1,k and J2,k. Using (3.6), (3.7), and (3.8), we obtain
(3.16)(∑k=1NE‖J1,k‖H2)1/2=(∑k=1NE‖A-1[c(kτ)-cτ(kτ)]Aφ‖H2)1/2≤C(∑k=1Nτ3E‖Aφ‖H2)1/2≤τC(E‖Aφ‖H2)1/2,(∑k=1NE‖J2,k‖H2)1/2=(∑k=1NE‖A-1/2[s(kτ)-sτ(kτ)]A1/2ψ‖H2)1/2≤C(∑k=1Nτ3E‖A1/2ψ‖H2)1/2≤τC(E‖A1/2ψ‖H2)1/2.
Estimates for the expected value of Jm,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.
4. Applications
First, let Λ be the unit open cube in the n-dimensional Euclidean space ℝn={x=(x1,…,xn):0<xi<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:
(4.1)du˙(t,x)-∑r=1n(ar(x)uxr)xrdt=f(t,x)dwt,0<t<T,x=(x1,…,xn)∈Λ,u(0,x)=φ(x),u˙(0,x)=ψ(x),x∈Λ¯;u(t,x)=0,x∈S,0≤t≤T
with the Dirichlet condition is considered. Here, ar(x),(x∈Λ),δ≥0 and f(t,x)(t∈(0,1),x∈Λ) are given smooth functions with respect to x and ar(x)≥a>0.
The discretization of (4.1) is carried out in two steps. In the first step, define the grid space Λ~h={x=xm=(h1m1,…,hnmn);m=(m1,…,mn),0≤mr≤Nr,hrNr=1,r=1,…,n}, Λh=Λ~h∩Λ,Sh=Λ~h∩S.
Let L2h denote the Hilbert space as
(4.2)L2h=L2(Λ~h)={φh(x):(∑x∈Λ~h|φh(x)|2h1⋯hn)1/2<∞}.
The differential operator A in (4.1) is replaced with
(4.3)Ahxuh(x)=-∑r=1n(ar(x)ux¯rh)xr,jr+δuh(x),
where the difference operator Ahx is defined on these grid functions uh(x)=0, for all x∈Sh. As it is proved in [25], Ahx is a self-adjoint positive definite operator in L2h. Using (4.1) and (4.3), we get
(4.4)du˙h(t,x)+Ahxuh(t,x)dt=fh(t,x)dwt,0<t<T,x∈Λh,uh(0,x)=φh(x),u˙h(0,x)=ψh(x),x∈Λ~h.
In the second step, we replace (4.4) with the difference scheme (3.5) as
(4.5)1τ2(uk+1h(x)-2ukh(x)+uk-1h(x))+12Ahxukh(x)+14(Ahxuk+1h(x)+Ahxuk-1h(x))=φkh(x),φkh(x)=1τ[(I+14τ2Ahx)φ1,k+1h(x)+τφ2,kh(x)-(I-14τ2Ahx)φ1,kh(x)],φ1,kh(x)=-1τ∫tk-1tk(z-tk)fh(z,x)dwz,φ2,kh(x)=1τ∫tk-1tkfh(z,x)dwz,tk=kτ,1≤k≤N-1,Nτ=T,x∈Λh,u1h(x)=cτ(τ)φh(x)+sτ(τ)ψh(x)-∫0τ(z-τ)fh(z,x)dwz,x∈Λh,u0h(x)=φh(x).
Theorem 4.1.
Let τ and |h|=h12+⋯+hn2 be sufficiently small numbers. Then, the solution of difference scheme (4.5) satisfies the convergence estimate as
(4.6)(∑k=1NE‖uh(tk)-ukh‖L2h2)1/2≤C(δ)(τ+|h|2),
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 Ahx defined by (4.3).
Second, in [0,T]×Λ, the initial-boundary value problem for the following multidimensional hyperbolic equation:
(4.7)du˙(t,x)-∑r=1n(ar(x)uxr)xrdt+δu(t,x)dt=f(t,x)dwt,0<t<T,x=(x1,…,xn)∈Λ,u(0,x)=φ(x),u˙(0,x)=ψ(x),x∈Λ,¯∂u(t,x)∂n→=0,x∈S,0≤t≤T
with the Neumann condition is considered. Here, n→ is the normal vector to Λ, δ>0, ar(x), (x∈Λ), and f(t,x)(t∈(0,1),x∈Λ) are given smooth functions with respect to x and ar(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
(4.8)Ahxuh(x)=-∑r=1n(ar(x)ux¯rh)xr,jr+δuh(x),
where the difference operator Ahx is defined on those grid functions Dhuh(x)=0, for all x∈Sh, where Dhuh(x)=0 is the second order of approximation of ∂u(t,x)/∂n→. As it is proved in [25], Ahx is a self-adjoint positive definite operator in L2h. Using (4.7) and (4.8), we get
(4.9)du˙h(t,x)+Ahxuh(t,x)dt=fh(t,x)dwt,0<t<T,x∈Λh,uh(0,x)=φh(x),u˙h(0,x)=ψh(x),x∈Λ~h.
In the second step, we replace (4.9) with the difference scheme (3.5) as
(4.10)1τ2(uk+1h(x)-2ukh(x)+uk-1h(x))+12Ahxukh(x)+14(Ahxuk+1h(x)+Ahxuk-1h(x))=φkh(x),φkh(x)=1τ((I+14τ2Ahx)φ1,k+1h(x)+τφ2,kh(x)-(I-14τ2Ahx)φ1,kh(x)),φ1,kh(x)=-1τ∫tk-1tk(z-tk)fh(z,x)dwz,φ2,kh(x)=1τ∫tk-1tkfh(z,x)dwz,tk=kτ,1≤k≤N-1,Nτ=T,x∈Λh,u1h(x)=cτ(τ)φh(x)+sτ(τ)ψh(x)-∫0τ(z-τ)fh(z,x)dwz,u0h(x)=φh(x),x∈Λh.
Theorem 4.2.
Let τ and |h|=h12+⋯+hn2 be sufficiently small numbers. Then, the solution of difference scheme (4.10) satisfies the convergence estimate as
(4.11)(∑k=1NE‖uh(tk)-ukh‖L2h2)1/2≤C(δ)(τ+|h|2),
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 Ahx defined by (4.8).
Third, in [0,T]×Λ, the mixed boundary value problem for the following multidimensional hyperbolic equation:
(4.12)du˙(t,x)-∑r=1n(ar(x)uxr)xrdt+δu(t,x)dt=f(t,x)dwt,0<t<T,x=(x1,…,xn)∈Λ,u(0,x)=φ(x),u˙(0,x)=ψ(x),x∈Λ¯,∂u(t,x)∂n→=0,x∈S2,0≤t≤T,S1∪S2=S,u(t,x)=0,x∈S1
with the Dirichlet-Neumann condition is considered. Here, n→ is the normal vector to Λ,δ>0,ar(x),(x∈Λ), and f(t,x)(t∈(0,1),x∈Λ) are given smooth functions with respect to x and ar(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
(4.13)Ahxuh(x)=-∑r=1n(ar(x)ux¯rh)xr,jr+δuh(x),
where the difference operator Ahx is defined on those grid functions uh(x)=0, for all x∈Sh1 and Dhuh(x)=0, for all x∈Sh2, Sh1∪Sh2=Sh, where Dhuh(x)=0 is the second order of approximation of ∂u(t,x)/∂n→. By [25], we can conclude that Ahx is a self-adjoint positive definite operator in L2h. Using (4.12) and (4.13), we get
(4.14)du˙h(t,x)+Ahxuh(t,x)dt=fh(t,x)dwt,0<t<T,x∈Λh,uh(0,x)=φ(x),u˙h(0,x)=ψ(x),x∈Λ~h.
In the second step, we replace (4.14) with the difference scheme (3.5) as
(4.15)1τ2(uk+1h(x)-2ukh(x)+uk-1h(x))+12Ahxukh(x)+14(Ahxuk+1h(x)+Ahxuk-1h(x))=φkh(x),φkh(x)=1τ((I+14τ2Ahx)φ1,k+1h(x)+τφ2,kh(x)-(I-14τ2Ahx)φ1,kh(x)),φ1,kh(x)=-1τ∫tk-1tk(z-tk)fh(z,x)dwz,φ2,kh(x)=1τ∫tk-1tkfh(z,x)dwz,tk=kτ,1≤k≤N-1,Nτ=T,x∈Λh,u1h(x)=cτ(τ)φh(x)+sτ(τ)ψh(x)-∫0τ(z-τ)fh(z,x)dwz,u0h(x)=φh(x),x∈Λh.
Theorem 4.3.
Let τ and |h|=h12+⋯+hn2 be sufficiently small positive numbers. Then, the solution of difference scheme (4.15) satisfies the convergence estimate as
(4.16)(∑k=1NE‖uh(tk)-ukh‖L2h2)1/2≤C(δ)(τ+|h|2),
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 Ahx defined by (4.13).
5. Numerical Examples
In this section, we apply finite difference scheme (2.2) to four examples which are stochastic hyperbolic equation with Neumann, Dirichlet, Dirichlet-Neumann, and Neumann-Dirichlet conditions.
Example 5.1.
The following initial-boundary value problem:
(5.1)du˙(t,x)-∂2u(t,x)∂x2dt+u(t,x)dt=f(t,x)dwt,f(t,x)=2cosx,wt=tξ,0<t<1,0<x<π,u(0,x)=cosx,u˙(0,x)=0,0≤x≤π,ux(t,0)=ux(t,π)=0,0≤t≤1
for a stochastic hyperbolic equation is considered. The exact solution of this problem is
(5.2)u(t,x)=∫0tsin(2(t-s))cosxdws+cos(2t)cosx.
For the approximate solution of the (5.1), we apply the finite difference scheme (2.2) and we get
(5.3)unk+1-2unk+unk-1τ+τ2[-un+1k-2unk+un-1kh2+unk]+τ4[-un+1k+1-2unk+1+un-1k+1h2+unk+1-un+1k-1-2unk-1+un-1k-1h2+unk-1]=fnk,fnk=2ξcosxn[tk+1-tk-1+τ2-22τ[τ(tk+1-tk-1)-23(tk+13-tk3-tk-13)]],Nτ=1,xn=nh,1≤n≤M-1,Mh=π,1≤k≤N-1,tk=kτ,un0=cosxn,un1-un0=4+τ262τ3ξcosxn,1≤n≤M-1,u0k=u1k,uMk=uM-1k,1≤k≤N.
The system can be written in the following matrix form:
(5.4)Aun+1+Bun+Cun-1=Dφn,1≤n≤M-1,u0=u1,uM=uM-1.
Here,
(5.5)A=[0000⋅⋅⋅0000a2aa0⋅⋅⋅00000a2aa⋅⋅⋅0000⋮⋮⋮⋮⋱⋮⋮⋮⋮0000⋅⋅⋅a2aa00000⋅⋅⋅0a2aasa-sa00⋅⋅⋅0000](N+1)×(N+1),B=[1000⋅⋅⋅0000bcb0⋅⋅⋅00000bcb⋅⋅⋅0000⋮⋮⋮⋮⋱⋮⋮⋮⋮0000⋅⋅⋅bcb00000⋅⋅⋅0bcbsb-sb00⋅⋅⋅0000](N+1)×(N+1),
the matrix is C=A,
(5.6)a=-τ4h2,b=1τ+τ2h2+τ4,c=-2τ+τh2+τ2,sa=τ24h2,sb=1+τ22h2+τ24,(5.7)fn=[fn0fn1fn2⋮fnN](N+1)×1,fnk=2ξcosxn[tk+13tk+1-tk-1+τ2-22τ2ξcosxn×[τ(tk+1-tk-1)-23(tk+13-tk3-tk-13)]],1≤k≤N-1,fn0=cosxn,0≤n≤M,fnN=4+τ262τ3ξcosxn,0≤n≤M,
and D=IN+1 is the identity matrix,
(5.8)Us=[us0us1us2⋮usN](N+1)×1,s=n-1,n,n+1.
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:
(5.9)un=αn+1un+1+βn+1,n=M-1,…,2,1,
where uM=(I-αM)-1βM,αj(j=1,…,M-1) are (N+1)×(N+1) square matrices, βj(j=1,…,M-1) are (N+1)×1column matricesα1 is an identity and β1 is a zero matrices, and
(5.10)αn+1=-(B+Cαn)-1A,βn+1=(B+Cαn)-1(Dφn-Cβn),n=1,2,3,…,M-1.
Example 5.2.
The following initial-boundary value problem:
(5.11)du˙(t,x)-∂2u(t,x)∂x2dt+u(t,x)dt=f(t,x)dwt,f(t,x)=2sinx,wt=tξ,0<t<1,0<x<π,u(0,x)=sinx,u˙(0,x)=0,0≤x≤π,u(t,0)=u(t,π)=0,0≤t≤1
for a stochastic hyperbolic equation is considered. We use the same procedure as in the first example. The exact solution of this problem is
(5.12)u(t,x)=∫0tsin(2(t-s))sinxdws+cos(2t)sinx.
For the approximate solution of the (5.11), we can construct the following difference scheme:
(5.13)unk+1-2unk+unk-1τ+τ2[-un+1k-2unk+un-1kh2+unk]×τ4[-un+1k+1-2unk+1+un-1k+1h2+unk+1-un+1k-1-2unk-1+un-1k-1h2+unk-1]=fnk,fnk=2ξsinxn[tk+13tk+1-tk-12ξsinxn+τ2-22τ[τ(tk+1-tk-1)-23(tk+13-tk3-tk-13)]],Nτ=1,xn=nh,1≤n≤M-1,Mh=π,1≤k≤N-1,tk=kτ,un0=sinxn,un1-un0=4+τ262τ3ξsinxn,1≤n≤M-1,u0k=uMk=0,1≤k≤N,
and it can be written in the following matrix form:
(5.14)Aun+1+Bun+Cun-1=Dfn,1≤n≤M-1,u0=uM=0→.
Here, the matrices A, B, C, D are given in the previous example, and
(5.15)fn=[fn0fn1fn2⋮fnN](N+1)×1,fnk=2ξsinxn[tk-13tk+13tk+1-tk-1+τ2-22τ2ξsinxn×[τ(tk+1-tk-1)-23(tk+13-tk3-tk-13)]],1≤k≤N-1,fn0=sinxn,0≤n≤M,fnN=4+τ262τ3ξsinxn,0≤n≤M.
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:
(5.16)un=αn+1un+1+βn+1,n=M-1,…,2,1,
where uM=0,αj(j=1,…,M-1) are (N+1)×(N+1) square matrices,βj
(j=1,…,M-1) are (N+1)×1 column matrices.α1 and β1 are zero matrices, and
(5.17)αn+1=-(B+Cαn)-1A,βn+1=(B+Cαn)-1(Dφn-Cβn),n=1,2,3,…,M-1.
Example 5.3.
The following initial-boundary value problem:
(5.18)du˙(t,x)-∂2u(t,x)∂x2dt+u(t,x)dt=f(t,x)dwt,f(t,x)=52sin(x2),wt=tξ,0<t<1,0<x<π,u(0,x)=sin(x2),u˙(0,x)=0,0≤x≤π,u(t,0)=ux(t,π)=0,0≤t≤1
for a stochastic hyperbolic equation is considered. The exact solution of this problem is
(5.19)u(t,x)=[∫0tsin(52(t-s))dws+cos(52t)]sin(x2).
We get the following difference scheme:
(5.20)unk+1-2unk+unk-1τ+τ2[-un+1k-2unk+un-1kh2+unk]τ4[-un+1k+1-2unk+1+un-1k+1h2unk+1-2unk+unk-1τ+τ2[-un+1k-2unk+un-1kh2+unk]τ4+unk+1-un+1k-1-2unk-1+un-1k-1h2+unk-1]=fnk,fnk=52ξsinxn2[τ2-22τtk+1-tk-1+τ2-22τ52ξsinxn2×[tk3τ(tk+1-tk-1)-23(tk+13-tk3-tk-13)]],Nτ=1,xn=nh,1≤n≤M-1,Mh=π,1≤k≤N-1,tk=kτ,un0=sinxn2,un1-un0=4+τ2125τ3ξsinxn2,1≤n≤M-1,u0k=0,uMk=uM-1k,1≤k≤N,
for the approximate solutions of (5.18), and we obtain the following matrix equation:
(5.21)Aun+1+Bun+Cun-1=Dfn,1≤n≤M-1,u0k=0,uMk=uM-1k,1≤k≤N.
Here, the matrices A, B, C, D are same as in the first example, and
(5.22)fn=[fn0fn1fn2⋮fnN](N+1)×1,fnk=52ξsinxn2[tk+13tk+1-tk-1+τ2-22τ52ξsinxn2×[tk+13τ(tk+1-tk-1)-23(tk+13-tk3-tk-13)]],1≤k≤N-1,fn0=sinxn2,0≤n≤M,fnN=4+τ2125τ3ξsinxn2,0≤n≤M.
For the solution of matrix equation (5.21), we use the same procedure as in the previous examples. Moreover, uM=0→, α1 is an identity and β1 is a zero matrices, and
(5.23)αn+1=-(B+Cαn)-1A,βn+1=(B+Cαn)-1(Dφn-Cβn),n=1,2,3,…,M-1.
Example 5.4.
The following initial boundary value problem:
(5.24)du˙(t,x)-∂2u(t,x)∂x2dt+u(t,x)dt=f(t,x)dwt,f(t,x)=52cos(x2),wt=tξ,0<t<1,0<x<π,u(0,x)=cos(x2),u˙(0,x)=0,0≤x≤π,ux(t,0)=u(t,π)=0,0≤t≤1
for a stochastic hyperbolic equation is considered. The exact solution of this problem is
(5.25)u(t,x)=[∫0tsin(52(t-s))dws+cos(52t)]cos(x2).
The following difference scheme:
(5.26)unk+1-2unk+unk-1τ+τ2[-un+1k-2unk+un-1kh2+unk]τ4[-un+1k+1-2unk+1+un-1k+1h2unk+1-2unk+unk-1τ+τ2[-un+1k-2unk+un-1kh2+unk]τ4+unk+1-un+1k-1-2unk-1+un-1k-1h2+unk-1]=fnk,fnk=52ξcosxn2[τ2-22τtk+1-tk-1+τ2-22τ52ξcosxn2×[τ(tk+1-tk-1)-23(tk+13-tk3-tk-13)]],Nτ=1,xn=nh,1≤n≤M-1,Mh=π,1≤k≤N-1,tk=kτ,un0=cosxn2,un1-un0=4+τ2125τ3ξcosxn2,1≤n≤M-1,u0k=u1k,uMk=0,1≤k≤N
is obtained for the approximate solutions of (5.24), and we obtain the following matrix equation:
(5.27)Aun+1+Bun+Cun-1=Dfn,1≤n≤M-1,u1=u2,uM=0→.
Here, the matrices A, B, C, D are same as in the first example, and
(5.28)fn=[fn0fn1fn2⋮fnN](N+1)×1,fnk=52ξcosxn2[tk+13tk+1-tk-1+τ2-22τ52ξcosxn2×[τ(tk+1-tk-1)-23(tk+13-tk3-tk-13)]],1≤k≤N-1,fn0=cosxn2,0≤n≤M,fnN=4+τ2125τ3ξcosxn2,0≤n≤M.
Using (5.27) that we get α1 is an identity and β1 is a zero matrices and uM=(I-αM)-1βM. The rest are the same as in Example 5.3.
For these examples, the errors of the numerical solution derived by difference scheme (2.2) computed by
(5.29)EMN=max1≤k≤N-1,1≤n≤M-1(∑k=1N|u(tk,xn)-unk|2)1/2
and the results are given in Table 1.
N=M=10
N=M=20
N=M=40
Example 5.1 (Neumann)
0.3028
0.1219
0.0554
Example 5.2 (Dirichlet)
0.4004
0.2137
0.1342
Example 5.3 (Dirichlet-Neumann)
0.3040
0.1145
0.0494
Example 5.4 (Neumann-Dirichlet)
0.3439
0.1844
0.0957
The numerical solutions are recorded for different values of N=M, where u(tk,xn) represents the exact solution and unk represents the numerical solution at (tk,xn). 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.
Acknowledgment
The authors would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey) for the helpful suggestions to the improvement of this paper.
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