DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation58471810.1155/2009/584718584718Research ArticleModified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger EquationAshyralyevAllaberen1SirmaAli2BerezanskyLeonid1Department of MathematicsFatih University34500 Büyükcekmece, IstanbulTurkeyfatih.edu.tr2Department of Mathematics and Computer SciencesBahcesehir UniversityBesiktas, 34353 IstanbulTurkeybahcesehir.edu.tr20091407200920092611200830032009190620092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The nonlocal boundary value problem for Schrödinger equation in a Hilbert space is considered. The second-order of accuracy r-modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. A numerical method is proposed for solving a one-dimensional nonlocal boundary value problem for the Schrödinger equation with Dirichlet boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.

1. Introduction

In this article, the nonlocal boundary value problem for the Schrödinger equation iu(t)+Au(t)=f(t),0<t<T,u(0)=m=1pαmu(λm)+φ,0<λ1<λ2<<λpT in a Hilbert space H with the self-adjoint operator A is considered. The Schrödinger equation plays an important role in the modeling of many phenomena. Methods of solutions for the Schrödinger equation have been studied extensively by many researchers (see, e.g.,  and the references given therein).

The idea in this work is inspired from the works [2, 3, 10, 11]. In the articles [2, 3] the existence and the uniqueness of the solution of the nonlocal boundary value problem (1.1) and its general form under some conditions are studied. In the article , to find an approximate solution of the problem (1.1), first-order of accuracy Rothe difference scheme and second-order of accuracy Crank-Nicolson difference scheme are presented. The stability estimates for the solution of this problem and the stability of these difference schemes are established.

The main aim of this paper is to study r modified Crank-Nicolson difference schemes for the approximate solution of problem (1.1). The paper is organized as follows. In Section 2, we establish estimates for the stability of higher order derivatives of the solution of problem (1.1). In Section 3, the second-order of accuracy r modified Crank-Nicolson difference schemes for the approximate solution of problem (1.1) are presented. The stabilities of these difference schemes are established. In Section 4, we study the convergence of these difference schemes. In Section 5, a numerical example is exposed in order to validate the schemes. A procedure involving the modified Gauss elimination method is used for solving these difference schemes.

Throughout this paper, the constants used are not necessarily the same at different occurrences.

2. Nonlocal Boundary Value ProblemTheorem 2.1.

Assume that f(t)C1([0,T],H),φD(A) and m=1p|αm|<1. Then there exists a unique solution u(t) of problem (1.1) and the following inequalities are satisfied: max0tTu(t)HC(α1,,αp)[φH+Tmax0tTf(t)H],max0tTu(t)H+max0tTAu(t)HC(α1,,αp)[AφH+Tmax0tTf(t)H+f(0)H].

Proof.

The proof of the estimate (2.2) is given in . Now we will obtain the estimate (2.3).

It is known that for smooth data of the problem iu(t)+Au(t)=f(t),0<t<T,u(0)=ξ, there exists a unique solution of the problem (1.1), and the following formula holds: u(t)=eiAtξ-0teiA(t-s)if(s)ds. Therefore we have Au(t)=eiAtAξ+f(0)+0tf(s)ds-f(0)eiAt-0teiA(t-s)f(s)ds. So that we get the estimate max0tTu(t)HAξH+2f(0)H+2Tmax0tTf(t)H. Using the condition u(0)=m=1pαmu(λm)+φ and the formula (2.6) we get Aξ=R{m=1pαmf(0)+m=1pαm0λmf(s)ds-f(0)m=1pαmeiAλm-m=1pαm0λmeiA(λm-s)f(s)ds+Aφ}, where R=(I-m=1pαmeiAλm)-1. By using estimates RHH11-m=1p|αm|C(α1,,αp),eiAtHH1, and the assumption m=1p|αm|<1, we get AξHC(α1,,αp){2f(0)H+2Tmax0tTf(t)H+AφH}. By using the estimates (2.7) and (2.11) we obtain an estimate for Au. Then by using the estimate for Au, the relation iu(t)=f(t)-Au=f(0)+0tf(s)ds-Au and the triangle inequality we can obtain estimate (2.3). This completes the proof of Theorem 2.1.

3. Difference Schemes, Stability

In this section, we present r-modified Crank-Nicolson difference schemes for the approximate solutions of problem (1.1) and establish the stabilities of these difference schemes. It is assumed that 2τλm for 1mp. Let us associate the nonlocal boundary value problem (1.1) with the corresponding second-order of accuracy r-modified Crank-Nicolson difference schemes: iuk-uk-1τ+A2(uk+uk-1)=φk,r+1kN,iuk-uk-1τ+Auk=φk,1kr,u0=rτλmαm((I+il0mA)ulm-il0mφlm)+rτ<λmλm/τZ+αmulm+rτ<λmλm/τZ+αm(I+idmA)12(ulm+ulm+1)-irτ<λmλm/τZ+αmdmφlm+φ,0<λ1<λ2<<λpT, for the approximate solutions of this nonlocal boundary value problem. Z+ denotes here the set {2,,n,} and lm=λm/τ,l0m=λm-λm/ττ,dm=λm-λm/ττ-τ/2, φk=f(tk-τ/2), tk=kτ, where x stands for the greatest integer part of the real number x.

By , uk={Rkξ-iτj=1kRk-j+1φj,k=1,,r,Bk-rRrξ-iτj=1rBk-rRr-j+1φj-iτj=r+1kBk-jCφj,k=r+1,,N is the solution of the r-modified Crank-Nicolson difference schemes for the approximate solutions of Cauchy problem iuk-uk-1τ+A2(uk+uk-1)=φk,r+1kN,iuk-uk-1τ+Auk=φk,1kr,u0=ξ. Here R=(I-iτA)-1,C=(I-iA2τ)-1,B=(I+iA2τ)C. For u0, using the formula (3.2) and the condition we obtain ξ=Tτ{(-iτrτλmαm(I+il0mA)j=1lmRlm-j+1φj-irτλmαml0mφlm)-iτrτ<λmλm/τZ+αm(j=1rBlm-rRr-j+1φj+j=r+1lmBlm-jCφj)-iτrτ<λmλm/τZ+αm(I+idmA)12((I+B)(j=1rBlm-rRr-j+1φj+j=r+1lmBlm-jCφj)+Cφlm+1)-irτ<λmλm/τZ+αmdmA+φ}, where Tτ=(I-rτλmαm(I+il0mA)Rlm-rτ<λmλm/τZ+αmBlm-rRr-rτ<λmλm/τZ+αm(I+idmA)12(I+B)Blm-rRr)-1. Note that, here we considered k=r+1lmBlm-jCφj=0 for lm=r. So, for the solution of problem (3.2), we have the following formula: uk={Rku0-iτj=1kRk-j+1φj,k=1,,r,Bk-rRru0-iτj=1rBk-rRr-j+1φj-iτj=r+1kBk-jCφj,k=r+1,,N,Tτ{-iτrτλmαm(I+il0mA)j=1lmRlm-j+1φj-irτλmαml0mφlm-iτrτ<λmλ/τZ+αm(j=1rBlm-rRr-j+1φj+j=r+1lmBlm-jCφj)-iτrτ<λmλ/τZ+αm(I+idmA)×12((I+B)(j=1rBlm-rRr-j+1φj+j=r+1lmBlm-jCφj)+Cφlm+1)-irτ<λmλ/τZ+αmdmφlm+φ},k=0.

Theorem 3.1.

Assume that φD(A) and m=1p|αm|<1. Then the solutions of the difference schemes (3.1) satisfy the stability inequalities max0kNukHC(α1,,αp)[φH+Tmax1kNφkH],max1kNuk-uk-1τH+max1krAukH+maxr+1kNAuk+uk-12HC(α1,,αp)[AφH+φ1HTmax2kNφk-φk-1τH].

Proof.

Using the estimates RHH1,BHH1,CHH1, and the formula (3.2), we can obtain max1kNukH[u0H+Tmax1kNφkH]. Using the spectral representation of the self-adjoint operators one can establish TτHH11-m=1p|αm|C(α1,,αp). Estimate for u0H should also be examined. By using formula (3.7), the triangle inequality, and estimates (3.11), (3.13) the following estimate is obtained: u0HC(α1,,αp)[φH+2Tmax1kNφkH]. The proof of the estimate (3.9) for the difference schemes (3.1) is based on the last estimate and estimate (3.12).

Now, estimate (3.10) will be obtained. Using (3.2), we get Auk={RkAξ-iτj=1kARk-j+1φj,k=1,,r,Bk-rRrAξ-iτj=1rBk-rARr-j+1φj-iτj=r+1kBk-jACφj,k=r+1,,N. So that Auk={RkAξ+(j=2kRk-j+1(φj-1-φj)+φk-Rkφ1),k=1,,r,Bk-rRrAξ+j=2rBk-rRr-j+1(φj-1-φj)-Bk-rRrφ1+j=r+1kBk-j+1(φj-1-φj)+φk,k=r+1,,N. For the estimate (3.10) the two cases should be examined separately: (i) k=1,,r, (ii) k=r+1,,N. Let 1kr. Then, using (3.16) we get max1krAukHRAξH+2Nmax2kNφk-φk-1H+2φ1H. Therefore, max1krAukHRAξH+2Tmax2kNφk-φk-1τH+2φ1H. Estimate for RAξH should also be obtained. Using the formula (3.5) and the formula (3.16) we get ξ=Tτ{rτλmαm(I+il0mA)R(j=2lmRlm-j+1(φj-1-φj)+φlm-Rlmφ1)-irτλmαml0mRAφlm+rτ<λmλm/τZ+αmR(j=2rBlm-rRr-j+1(φj-1-φj)-Rrφ1)+rτ<λmλm/τZ+αmR(j=r+1lmBlm-jC(φj-1-φj)+φlm)+rτ<λmλm/τZ+αm(I+idmA)×12R((j=2rBlm-rRr-j+1(φj-1-φj)-Rrφ1+j=r+1lmBlm-j+1(φj-1-φj)+φlm)+(j=2rBlm+1-rRr-j+1(φj-1-φj)-Rrφ1+j=r+1lm+1Blm-j+2(φj-1-φj)+φlm+1))-irτ<λmλm/τZ+αmdmARφlm+RAφ}. So that RAξHC1(α1,,αp)[AφH+φ1H+Tmax2kNφk-φk-1τH]. Therefore, using the estimates (3.18) and (3.20) we obtain max1krAukHC2(α1,,αp)[AφH+φ1H+Tmax2kNφk-φk-1τH]. Then using the estimate for Auk, the relation i((uk-uk-1)/τ)=φk-Auk=φ1-j=2k(φj-1-φj)-Auk, and the triangle inequality we get the estimate max1kruk-uk-1τH+max1krAukHC2(α1,,αp)[AφH+φ1H+Tmax2kNφk-φk-1τH]. Now, let k=r+1,,N. Then using the formula (3.16) and the identity (1/2)(I+B)=C we get Auk+uk-12=Bk-r-1Cr-1CRAξ+j=2rCBk-1-rRr-j+1(φj-1-φj)-Rrφ1+j=r+1kCBk-j(φj-1-φj)+B(φk-1-φk)+φk+φk-12. So that maxr+1kNAuk+uk-12HRAξH+3Tmax2kNφk-φk-1τH+3φ1H. Therefore, using the estimates (3.20) and (3.24), the estimate maxr+1kNAuk+uk-12HC3(α1,,αp)[AφH+φ1H+Tmax2kNφk-φk-1τH] is obtained. Then, by using the estimate (3.25), the relation i((uk-uk-1)/τ)=φk-A((uk+uk-1)/2)=φ1-j=2k(φj-1-φj)-A((uk+uk-1)/2), and the triangle inequality we get the estimate maxr+1kNuk-uk-1τH+maxr+1kNAuk+uk-12HC4(α1,,αp)[AφH+φ1H+Tmax2kNφk-φk-1τH]. The result (3.10) follows from the estimates (3.22) and (3.26). So the proof is complete.

4. ConvergenceTheorem 4.1.

Assume that m=1p|αm|<1. Assume also that Au′′(t)(0tT) and u(t)(0tT) are continuous, then the solution of the difference scheme (3.1) satisfies the convergence estimate max0kNuk-u(tk)HM*(r)τ2, where M*(r) does not depend on τ but depends on r.

Proof.

If we subtract (1.1) from (3.1) we obtain izk-zk-1τ+A2(zk+zk-1)=Ak,r+1kN,izk-zk-1τ+Azk=Ak,1kr,z0=rτλmαm((I+il0mA)zlm-il0mAlm)+rτ<λmλm/τZ+αmzlm+rτ<λmλm/τZ+αm(I+idmA)12(zlm+zlm+1)-irτ<λmλm/τZ+αmdmAlm+A0,0<λ1<λ2<<λpT, where zk=uk-u(tk) and Ak is defined by the formula Ak={i(ddt(u(tk-1/2))-u(tk)-u(tk-1)τ)+A(u(tk-1/2)-u(tk)),1kr,i(ddt(u(tk-1/2))-u(tk)-u(tk-1)τ)+A(u(tk-1/2)-u(tk)+u(tk-1)2),  r+1rN,rτλmαm(I+il0mA)u(tlm)+rτ<λmλm/τZ+αmu(tlm)+rτ<λmλm/τZ+(I+idmA)12(u(tlm)+u(tlm+1))-m=1pαmu(λm)+irτλmαml0m(Alm-φlm)+irτ<λmλm/τZ+αmdm(Alm-φlm),k=0. Then the difference problem (4.2) has a solution in the form (3.7), but instead of uk, φk, φ we take zk, Ak,A0, k=1,,N, respectively. Using the estimates RHH1,BHH1,CHH1, and the formula obtained for the solution of (4.2), we can obtain max1krzkH[z0H+rτmax1krAkH],maxr+1krzkH[z0H+Tmax1krAkH]. By the estimate (3.14) we have z0HC(α1,,αp)[A0H+2Tmax1kNAkH]. Therefore, in order to obtain the inequality (4.1) we need estimates for Ak for 0kN.

For 0kN, by the use of the triangle inequality, Taylor's formula, continuity of     Au′′(t)(0tT) and u(t)(0tT), the estimates max1krAkHM1τ,maxr+1kNAkHM2τ2,A0HM2τ2 are obtained. From the last estimates the result follows.

5. Numerical Results

In this section, the numerical experiments of the nonlocal boundary value problem iu(t,x)t-((x+1)ux)x=f(t,x),0<t,x<1,  u(0,x)=13u(12,x)+φ(x),0<x<1,u(t,0)=u(t,1)=0,0<t<1,f(t,x)=[π2sinπx-πcosπx+π2(x+1)sinπx]exp(-it),φ(x)=(1-13exp(-i2))(sinπx). by using modified Crank-Nicolson difference scheme (3.1) are investigated. The exact solution of this problem is u(t,x)=(sinπx)exp(-it). For the approximate solution of problem (5.1), the set [0,1]τ×[0,1]h of a family of grid points depending on the small parameters τ and h[0,1]τ×[0,1]h={(tk,xn):tk=kτ,1kN-1,Nτ=1,xn=nh,1nM-1,Mh=1} is defined.

Applying the second-order of accuracy modified Crank-Nicolson difference schemes (3.1) we present following second-order of accuracy difference schemes for the approximate solutions of problem (5.1) iunk-unk-1τ-12(un+1k-un-1k2h+un+1k-1-un-1k-12h)-xn+12(un+1k-2unk+un-1kh2+un+1k-1-2unk-1+un-1k-1h2)=f(tk-1/2,xn),r+1kN-1,1nM-1,iunk-unk-1τ-un+1k-un-1k2h-(xn+1)un+1k-2unk+un-1kh2=f(tk-τ2,xn),1kr,1nM-1,tk-1/2=(k-12)τ,xn=nh,1kN,1nM-1,un0=13un[1/2τ]+φ(xn),1nM-1,u0k=0,uMk=0,0kN. So for each r, we have (N+1)×(N+1) system of linear equations which can be written in the matrix form as AnUn+1+BnUn+CnUn-1=Dnφn,1nM-1,U0=0,UM=0, where φn=[φn0φn1φnN](N+1)×1,φnk={(1-13exp(-i2))(sinπxn),k=0,f(tk-1/2,xn),1kN,    An=[0an00000000an00000000000000000enen00000000enen0000000enen00000000],Bn=[bncn00000000bncn000000000000000000vnsn000000000vnsn00000000vnsn100-1300000],Cn=[0dn00000000dn0000000000000000gngn00000000gngn0000000gngn00000000],D=IN+1((N+1)×(N+1)identitymatrix),Us=[Us0Us1UsN-1UsN],s=n-1,n,n+1. In the above matrices entries are given as an=-12h-nh+1h2,bn=-iτ,cn=iτ+2(nh+1)h2,dn=(12h-nh+1h2)en=-14h-nh+12h2,vn=(-iτ+nh+1h2),sn=iτ+nh+1h2,gn=14h-nh+12h2. Thus, we have the second-order difference equation (5.5) with respect to n with matrix coefficients. To solve this difference equation we have applied the same modified Gauss elimination method for the difference equation with respect to n with matrix coefficients. Hence, we seek a solution of the matrix in the following form: Un=αn+1Un+1+βn+1,n=M-1,,2,1,0, where αj(j=1,,M) are (N+1)×(N+1) square matrices and βj(j=1,,M) are (N+1)×1 column matrices defined by αn+1=-(Bn+Cnαn)-1An,βn+1=(Bn+Cnαn)-1(Dφn-Cnβn),n=1,2,3,,M-1. Note that for obtaining αn+1,βn+1,n=1,,M-1, first we need to find α1,β1. As in , we take α1 is an identity matrix, β1 is the zero column vector.

For their comparison, first the errors computed by EMN=max1kN-1(n=1M-1|u(tk,xn)-unk|2h)1/2 of the numerical solutions of problem (5.1) are recorded for different values of N and M, where u(tk,xn) represents the exact solution and unk represents the numerical solution at (tk,xn). The results are shown in Table 1 for N=M=20,40,80, and 160, respectively.

Comparison of the errors for the approximate solution of problem (5.1).

Method N=M=20N=M=40N=M=80N=M=160
One-modified Crank-Nicholson 0.0137 0.0038 0.0010 0.00025
Two-modified Crank-Nicholson 0.0226 0.0071 0.0019 0.00048
Three-modified Crank-Nicholson 0.0272 0.0099 0.0028 0.00072

Second, for their comparison, the relative errors are computed by relENM=max1kNENM(n=1M|u(tk,xn)|2h)1/2, and Table 2 is constructed for N=M=20,40,80, and 160, respectively.

Relative errors for the approximate solution of problem (5.1).

Method N=M=20N=M=40N=M=80N=M=160
One-modified Crank-Nicholson 0.0194 0.0054 0.0014 0.00035
Two-modified Crank-Nicholson 0.0320 0.0101 0.0027 0.00069
Three-modified Crank-Nicholson 0.0385 0.0141 0.0040 0.00100

In the article  it can also be found, an example that Crank-Nicolson difference scheme is divergent but modified Crank-Nicolson is convergent.

Acknowledgment

The authors are grateful to Mr. Tarkan Aydın (Bahcesehir University, Turkey) for his comments and suggestions on implementation.

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