Modified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger Equation

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.


Introduction
In this article, the nonlocal boundary value problem for the Schr ödinger equation in a Hilbert space H with the self-adjoint operator A is considered.The Schr ödinger equation plays an important role in the modeling of many phenomena.Methods of solutions for the Schr ödinger equation have been studied extensively by many researchers see, e.g., 1-9 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 8 , 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.

2.1
Then there exists a unique solution u t of problem 1.1 and the following inequalities are satisfied:

2.3
Proof.The proof of the estimate 2.2 is given in 8 .Now we will obtain the estimate 2.  α m e iAλ m −1 .

2.9
By using estimates

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 2τ ≤ λ m for 1 ≤ m ≤ p.Let us associate the nonlocal boundary value problem 1.1 with the corresponding second-order of accuracy r-modified Crank-Nicolson difference schemes: where x stands for the greatest integer part of the real number x.
By 10 , is the solution of the r-modified Crank-Nicolson difference schemes for the approximate solutions of Cauchy problem

3.3
Here For u 0 , using the formula 3.2 and the condition we obtain where

3.6
Note that, here we considered l m k r 1 B l m −j Cϕ j 0 for l m r.So, for the solution of problem 3.2 , we have the following formula:

3.8
Then the solutions of the difference schemes 3.1 satisfy the stability inequalities

3.10
Proof.Using the estimates and the formula 3.2 , we can obtain Using the spectral representation of the self-adjoint operators one can establish

3.13
Estimate for u 0 H should also be examined.By using formula 3.7 , the triangle inequality, and estimates 3.11 , 3.13 the following estimate is obtained:

3.14
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

3.15
So that

3.18
Estimate for RAξ H should also be obtained.Using the formula 3.5 and the formula 3.16 we get

3.19
So that

3.20
Therefore, using the estimates 3.18 and 3.20 we obtain

3.21
Then using the estimate for Au k , the relation

3.22
Now, let k r 1, . . ., N. Then using the formula 3.16 and the identity 1/2 I B C we get

3.23
So that max

3.24
Therefore, using the estimates 3.20 and 3.24 , the estimate max is obtained.Then, by using the estimate 3.25 , the relation , and the triangle inequality we get the estimate max

3.26
The result 3.10 follows from the estimates 3.22 and 3.26 .So the proof is complete.Assume also that Au t 0 ≤ t ≤ T and u t 0 ≤ t ≤ T are continuous, then the solution of the difference scheme 3.1 satisfies the convergence estimate

Convergence
where M * r does not depend on τ but depends on r.
Proof.If we subtract 1.1 from 3.1 we obtain where z k u k − u t k and A k is defined by the formula

4.3
Then the difference problem 4.2 has a solution in the form 3.7 , but instead of u k , ϕ k , ϕ we take z k , A k , A 0 , k 1, . . ., N, respectively.Using the estimates and the formula obtained for the solution of 4.2 , we can obtain

4.5
By the estimate 3.14 we have Therefore, in order to obtain the inequality 4.1 we need estimates for A k for 0 ≤ k ≤ N.For 0 ≤ k ≤ N, by the use of the triangle inequality, Taylor's formula, continuity of Au t 0 ≤ t ≤ T and u t 0 ≤ t ≤ T , the estimates are obtained.From the last estimates the result follows.

Numerical Results
In this section, the numerical experiments of the nonlocal boundary value problem 5.1 by using modified Crank-Nicolson difference scheme 3.1 are investigated.The exact solution of this problem is 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 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 So for each r, we have N 1 × N 1 system of linear equations which can be written in the matrix form as where Discrete Dynamics in Nature and Society 13 In the above matrices entries are given as 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: where 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
For their comparison, first the errors computed by   In the article 12 it can also be found, an example that Crank-Nicolson difference scheme is divergent but modified Crank-Nicolson is convergent.
solutions of problem 5.1 are recorded for different values of N and 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 results are shown in and the triangle inequality we get the estimate

Table 1 :
Comparison of the errors for the approximate solution of problem 5.1 .

Table 1 for
N M 20, 40, 80, and 160, respectively.Second, for their comparison, the relative errors are computed by and Table2is constructed for N M 20, 40, 80, and 160, respectively.

Table 2 :
Relative errors for the approximate solution of problem 5.1 .