Schemes for Nonlocal Boundary Value Problem for the Schr ¨odinger Equation

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Volume 2009, Article ID 584718,15pages doi:10.1155/2009/584718

Research Article

Modified Crank-Nicolson Difference

Schemes for Nonlocal Boundary Value Problem for the Schr ¨odinger Equation

Allaberen Ashyralyev

¹

and Ali Sirma

²

1Department of Mathematics, Fatih University, 34500 B ¨uy ¨ukcekmece, Istanbul, Turkey

2Department of Mathematics and Computer Sciences, Bahcesehir University, Besiktas, 34353 Istanbul, Turkey

Correspondence should be addressed to Ali Sirma,[email protected] Received 26 November 2008; Revised 30 March 2009; Accepted 19 June 2009 Recommended by Leonid Berezansky

The nonlocal boundary value problem for Schr ¨odinger equation in a Hilbert space is considered.

The second-order of accuracyr-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 Schr ödinger 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.

Copyrightq2009 A. Ashyralyev and A. Sirma. This 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.

1. Introduction

In this article, the nonlocal boundary value problem for the Schr ¨odinger equation

iut Aut ft, 0< t < T, u0

p m1

αmuλm ϕ, 0< λ1 < λ2<· · ·< λp≤T

1.1

in a Hilbert spaceHwith the self-adjoint operatorAis considered. The Schr ¨odinger equation plays an important role in the modeling of many phenomena. Methods of solutions for the

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Schr ¨odinger equation have been studied extensively by many researcherssee, e.g.,1–9and the references given therein.

The idea in this work is inspired from the works2,3,10,11. In the articles 2,3 the existence and the uniqueness of the solution of the nonlocal boundary value problem 1.1and its general form under some conditions are studied. In the article 8, to find an approximate solution of the problem1.1, 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 studyrmodified Crank-Nicolson difference schemes for the approximate solution of problem1.1. The paper is organized as follows. InSection 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 problem1.1are presented. The stabilities of these difference schemes are established. InSection 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 Problem

Theorem 2.1. Assume thatft∈C¹0, T, H, ϕ∈DAand

p m1

|αm|<1. 2.1

Then there exists a unique solutionutof problem1.1and the following inequalities are satisfied:

max0≤t≤Tut_H ≤C

α1, . . . , αpϕ

HTmax

0≤t≤Tft

H

, 2.2

max0≤t≤Tut

H

max

0≤t≤TAut_H

≤C

α₁, . . . , α_pAϕ

HTmax

0≤t≤Tft

H

f0

H

.

2.3

Proof. The proof of the estimate2.2is given in8. Now we will obtain the estimate2.3.

It is known that for smooth data of the problem

iut Aut ft, 0< t < T, u0 ξ, 2.4

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there exists a unique solution of the problem1.1, and the following formula holds:

ut e^iAtξ− _t

0

e^iAt−sifsds. 2.5

Therefore we have

Aut e^iAtAξf0 _t

0

fsds−f0e^iAt− _t

0

e^iAt−sfsds. 2.6

So that we get the estimate

max0≤t≤Tut_H≤ Aξ_H2f0

H2Tmax

0≤t≤Tft

H. 2.7

Using the conditionu0 ^p_m1α_muλm ϕand the formula2.6we get

AξR _p

m1

αmf0 p m1

αm

_λ_m

0

fsds−f0 p m1

αme^iAλ^m

− p m1

α_m _λ_m

0

e^iAλ^m^−sfsdsAϕ

,

2.8

where

R

I− p m1

α_me^iAλ^m ₋₁

. 2.9

By using estimates

R_H_→_H≤ 1

1− ^p_m1|αm|≤C

α1, . . . , αp

, e^iAt

H→H≤1, 2.10

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and the assumption ^p_m1|αm|<1,we get

Aξ_H≤C

α₁, . . . , α_p

2f0

H2Tmax

0≤t≤Tft

HAϕ

H

. 2.11

By using the estimates 2.7 and 2.11 we obtain an estimate for Au.Then by using the estimate for Au,the relation iut ft−Au f0 _t

0fsds−Auand the triangle inequality we can obtain estimate2.3. This completes the proof ofTheorem 2.1.

3. Difference Schemes, Stability

In this section, we presentr-modified Crank-Nicolson difference schemes for the approximate solutions of problem1.1and establish the stabilities of these difference schemes. It is assumed that 2τ ≤λ_mfor 1≤m≤ p.Let us associate the nonlocal boundary value problem 1.1with the corresponding second-order of accuracyr-modified Crank-Nicolson difference schemes:

iuk−u_k−1

τ A

2uku_k−1 ϕk, r1≤k≤N, iuk−u_k−1

τ Aukϕk, 1≤k≤r,

u₀

rτ≥λm

α_m

Iil_0mAulm−il_0mϕ_lm

rτ<λm

λm/τ∈Z

α_mu_lm

rτ<λm

λm/τ /∈Z

α_mIid_mA1

2ulmu_lm1−i

rτ<λm

λm/τ /∈Z

α_md_mϕ_lmϕ,

0< λ₁< λ₂<· · ·< λ_p≤T,

3.1

for the approximate solutions of this nonlocal boundary value problem.Zdenotes here the set {2, . . . , n, . . .} and l_m λm/τ , l0m λ_m− λm/τ τ, dm λ_m− λm/τ τ −τ/2, ϕ_k ftk−τ/2,tkkτ,wherex stands for the greatest integer part of the real numberx.

By10,

u_k

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

R^kξ−iτ k

j1

R^k−j1ϕj, k1, . . . , r,

B^k−rR^rξ−iτ r

j1

B^k−rR^r−j1ϕ_j−iτ k jr1

B^k−jCϕ_j, kr1, . . . , N

3.2

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is the solution of the r-modified Crank-Nicolson diﬀerence schemes for the approximate solutions of Cauchy problem

iu_k−u_k−1

τ A

2uku_k−1 ϕk, r1≤k≤N, iuk−u_k−1

τ Aukϕk, 1≤k≤r, u0ξ.

3.3

Here

R I−iτA⁻¹, C

I−iA 2τ

₋₁

, B

IiA

2τ

C. 3.4

Foru₀,using the formula3.2and the condition we obtain

ξTτ

⎧⎪

⎪⎨

⎪⎪

⎩

⎛

⎝−iτ

rτ≥λm

αmIil0mA^l^m

j1

R^l^m^−j1ϕj−i

rτ≥λm

αml0mϕlm

⎞

⎠

−iτ

rτ<λm

λm/τ∈Z

α_m

⎛

⎝^r

j1

B^l^m^−rR^r−j1ϕ_j ^l^m

jr1

B^l^m^−jCϕ_j

⎞

⎠

−iτ

rτ<λm

λm/τ /∈Z

αmIidmA1 2

⎛

⎝IB

⎛

⎝^r

j1

B^l^m^−rR^r−j1ϕj ^l^m

jr1

B^l^m^−jCϕj

⎞

⎠Cϕlm1

⎞

⎠

−i

rτ<λm

λm/τ /∈Z

αmdmAϕ

⎫⎪

⎪⎬

⎪⎪

⎭,

3.5 where

Tτ

⎛

⎜⎜

⎝I−

rτ≥λm

αmIil0mAR^l^m−

rτ<λm

λm/τ∈Z

αmB^l^m^−rR^r

−

rτ<λm

λm/τ /∈Z

αmIidmA1

2IBB^l^m^−rR^r

⎞

⎟⎟

⎠

−1

.

3.6

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Note that, here we considered ^l_kr1^m B^l^m^−jCϕj 0 forlm r.So, for the solution of problem 3.2, we have the following formula:

u_k

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

R^ku₀−iτ k j1

R^k−j1ϕ_j, k1, . . . , r,

B^k−rR^ru₀−iτ r

j1

B^k−rR^r−j1ϕ_j−iτ k jr1

B^k−jCϕ_j, kr1, . . . , N,

T_τ

⎧⎪

⎨

⎪⎩−iτ

rτ≥λm

α_mIil_0mA^lm

j1

R^lm−j1ϕ_j−i

rτ≥λm

α_ml_0mϕ_lm

−iτ

rτ<λm

λ/τ∈Z

αm

⎛

⎝^r

j1

B^lm−rR^r−j1ϕj ^l^m

jr1

B^l^m^−jCϕj

⎞

⎠

−iτ

rτ<λm

λ/τ /∈Z

α_mIid_mA

×1 2

⎛

⎝IB

⎛

⎝^r

j1

B^lm−rR^r−j1ϕj ^l^m

jr1

B^lm−jCϕj

⎞

⎠Cϕ_lm1

⎞

⎠

−i

rτ<λm

λ/τ /∈Z

αmdmϕlmϕ

⎫⎪

⎪⎬

⎪⎪

⎭, k0.

3.7

Theorem 3.1. Assume thatϕ∈DAand p m1

|αm|<1. 3.8

Then the solutions of the diﬀerence schemes3.1satisfy the stability inequalities

0≤k≤Nmaxuk_H≤C

α1, . . . , αpϕ

HTmax

1≤k≤Nϕk

H

, 3.9

1≤k≤Nmax

uk−u_k−1 τ

H

max

1≤k≤rAuk_H max

r1≤k≤N

Auku_k−1 2

H

≤C

α₁, . . . , α_pAϕ

Hϕ₁

HTmax

2≤k≤N

ϕ_k−ϕ_k−1 τ

H

.

3.10

Proof. Using the estimates

R_H_→_H≤1, B_H_→_H≤1, C_H_→_H≤1, 3.11

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and the formula3.2, we can obtain

1≤k≤Nmaxuk_H≤

u0_HTmax

1≤k≤Nϕ_k

H

. 3.12

Using the spectral representation of the self-adjoint operators one can establish Tτ_H_→_H≤ 1

1− ^p_m1|αm|≤C α1, . . . , α_p!

. 3.13

Estimate foru0_Hshould also be examined. By using formula3.7, the triangle inequality, and estimates3.11,3.13the following estimate is obtained:

u0_H≤C

α₁, . . . , α_pϕ

H2Tmax

1≤k≤Nϕ_k

H

. 3.14

The proof of the estimate3.9for the diﬀerence schemes3.1is based on the last estimate and estimate3.12.

Now, estimate3.10will be obtained. Using3.2, we get

Auk

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

R^kAξ−iτ k j1

AR^k−j1ϕj, k1, . . . , r,

B^k−rR^rAξ−iτ r j1

B^k−rAR^r−j1ϕj−iτ k jr1

B^k−jACϕj, kr1, . . . , N.

3.15

So that

Auk

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ R^kAξ

⎛

⎝^k

j2

R^k−j1

ϕ_j−1−ϕj

ϕk−R^kϕ1

⎞

⎠, k1, . . . , r,

B^k−rR^rAξ^r

j2

B^k−rR^r−j1

ϕ_j−1−ϕj

−B^k−rR^rϕ1

^k

jr1

B^k−j1

ϕ_j−1−ϕj

ϕk, kr1, . . . , N.

3.16

For the estimate 3.10 the two cases should be examined separately: i k 1, . . . , r, ii kr1, . . . , N.Let 1≤k≤r.Then, using3.16we get

max1≤k≤rAuk_H≤ RAξ_H2Nmax

2≤k≤Nϕk−ϕ_k−1

H2ϕ1

H. 3.17

Therefore,

max1≤k≤rAuk_H≤ RAξ_H2Tmax

2≤k≤N

ϕk−ϕ_k−1 τ

H

2ϕ1

H. 3.18

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Estimate forRAξ_Hshould also be obtained. Using the formula3.5and the formula3.16 we get

ξTτ

⎧⎪

⎪⎨

⎪⎪

⎩

rτ≥λm

αmIil0mAR

⎛

⎝^l^m

j2

R^l^m^−j1

ϕ_j−1−ϕj

ϕlm−R^l^mϕ1

⎞

⎠

−i

rτ≥λm

α_ml_0mRAϕ_lm

rτ<λm

λm/τ∈Z

α_mR

⎛

⎝^r

j2

B^l^m^−rR^r−j1

ϕ_j−1−ϕ_j

−R^rϕ₁

⎞

⎠

rτ<λm

λm/τ∈Z

αmR

⎛

⎝^l^m

jr1

B^l^m^−jC

ϕ_j−1−ϕj

ϕlm

⎞

⎠

rτ<λm

λm/τ /∈Z

αmIidmA

×1 2R

⎛

⎝

⎛

⎝^r

j2

B^l^m^−rR^r−j1

ϕ_j−1−ϕ_j

−R^rϕ₁ ^l^m

jr1

B^l^m^−j1

ϕ_j−1−ϕ_j ϕ_l_m

⎞

⎠

⎛

⎝^r

j2

B^l^m^1−rR^r−j1

ϕ_j−1−ϕ_j

−R^rϕ₁ ^l^m¹

jr1

B^l^m^−j2

ϕ_j−1−ϕ_j ϕ_l_m₁

⎞

⎠

⎞

⎠

−i

rτ<λm

λm/τ /∈Z

αmdmARϕlmRAϕ

⎫⎪

⎪⎬

⎪⎪

⎭.

3.19

So that

RAξ_H≤C₁

α₁, . . . , α_pAϕ

Hϕ₁

HTmax

2≤k≤N

ϕk−ϕ_k−1 τ

H

. 3.20

Therefore, using the estimates3.18and3.20we obtain

max1≤k≤rAuk_H≤C₂

α₁, . . . , α_pAϕ

Hϕ₁

HTmax

2≤k≤N

ϕk−ϕ_k−1 τ

H

. 3.21

Then using the estimate forAu_k,the relationiuk−u_k−1/τ ϕ_k−Au_k ϕ₁− ^k_j2ϕj−1− ϕj−Auk, and the triangle inequality we get the estimate

max1≤k≤r

u_k−u_k−1 τ

H

max

1≤k≤rAuk_H

≤C2

α1, . . . , αpAϕ

Hϕ1

HTmax

2≤k≤N

ϕk−ϕ_k−1 τ

H

.

3.22

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Now, letkr1, . . . , N.Then using the formula3.16and the identity1/2IB Cwe get

Au_ku_k−1

2 B^k−r−1C^r−1CRAξ^r

j2

CB^k−1−rR^r−j1

ϕ_j−1−ϕj

−R^rϕ1

^k

jr1

CB^k−j

ϕ_j−1−ϕ_j B

ϕ_k−1−ϕ_k

ϕ_kϕ_k−1

2 .

3.23

So that

r1≤k≤Nmax

Au_ku_k−1 2

H≤ RAξ_H3Tmax

2≤k≤N

ϕk−ϕ_k−1 τ

H

3ϕ₁

H. 3.24

Therefore, using the estimates3.20and3.24, the estimate

r1≤k≤Nmax

Au_ku_k−1 2

H≤C₃

α₁, . . . , α_pAϕ

Hϕ₁

HTmax

2≤k≤N

ϕ_k−ϕ_k−1 τ

H

3.25

is obtained. Then, by using the estimate3.25, the relationiuk−u_k−1/τ ϕ_k−Auk u_k−1/2 ϕ1− ^k_j2ϕj−1−ϕj−Auku_k−1/2, and the triangle inequality we get the estimate

r1≤k≤Nmax

u_k−u_k−1 τ

H

max

r1≤k≤N

Au_ku_k−1 2

H

≤C4

α1, . . . , αpAϕ

Hϕ1

HTmax

2≤k≤N

ϕk−ϕ_k−1 τ

H

.

3.26

The result3.10follows from the estimates3.22and3.26. So the proof is complete.

4. Convergence

Theorem 4.1. Assume that ^p_m1|αm|<1.Assume also thatAut 0≤t≤Tandut 0≤t≤ Tare continuous, then the solution of the diﬀerence scheme3.1satisfies the convergence estimate

0≤k≤Nmaxuk−utk_H≤M^∗rτ², 4.1

whereM^∗rdoes not depend onτbut depends onr.

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Proof. If we subtract1.1from3.1we obtain

iz_k−z_k−1

τ A

2zkz_k−1 A_k, r1≤k≤N, iz_k−z_k−1

τ AzkAk, 1≤k≤r,

z₀

rτ≥λm

α_mIil_0mAzlm−il_0mA_lm

rτ<λm

λm/τ∈Z

α_mz_lm

rτ<λm

λm/τ /∈Z

αmIidmA1

2zlmz_lm1−i

rτ<λm

λm/τ /∈Z

αmdmAlmA0,

0< λ1 < λ2<· · ·< λp≤T,

4.2

wherezkuk−utkandAkis defined by the formula

A_k

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ i

d

dtutk−1/2−utk−utk−1 τ

Autk−1/2−utk, 1≤k≤r,

i d

dtutk−1/2−utk−utk−1 τ

A

utk−1/2−utk utk−1 2

, r1≤r≤N,

rτ≥λm

α_mIil_0mAutlm

rτ<λm

λm/τ∈Z

α_mutlm

rτ<λm

λm/τ /∈Z

Iid_mA1

2utlm ut_lm1− p m1

α_muλm

i

rτ≥λm

α_ml_0m

A_lm−ϕ_lm

i

rτ<λm

λm/τ /∈Z

α_md_m

A_lm−ϕ_lm

, k0.

4.3 Then the diﬀerence problem4.2has a solution in the form3.7, but instead ofu_k,ϕ_k,ϕwe takezk,Ak, A0,k1, . . . , N, respectively. Using the estimates

R_H_→_H≤1, B_H_→_H≤1, C_H_→_H≤1, 4.4

and the formula obtained for the solution of4.2, we can obtain

max1≤k≤rzk_H ≤

z0_Hrτmax

1≤k≤rAk_H

,

r1≤k≤rmax zk_H≤

z0_HTmax

1≤k≤rAk_H

.

4.5

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By the estimate3.14we have

z0_H≤C

α1, . . . , αp

A0_H2Tmax

1≤k≤NAk_H

. 4.6

Therefore, in order to obtain the inequality4.1we need estimates forA_kfor 0≤k≤N.

For 0 ≤ k ≤ N, by the use of the triangle inequality, Taylor’s formula, continuity of Aut 0≤t≤Tandut 0≤t≤T, the estimates

1≤k≤rmaxAk_H ≤M₁τ, max

r1≤k≤NAk_H≤M₂τ², A0_H≤M₂τ² 4.7 are obtained. From the last estimates the result follows.

5. Numerical Results

In this section, the numerical experiments of the nonlocal boundary value problem

i∂ut, x

∂t −x1ux_x ft, x, 0< t, x <1, u0, x 1

3u 1

2, x

ϕx, 0< x <1, ut,0 ut,1 0, 0< t <1, ft, x "

π²sinπx−πcosπxπ²x1sinπx#

exp−it, ϕx

1−1

3exp

−i 2

sinπx.

5.1

by using modified Crank-Nicolson diﬀerence scheme 3.1 are investigated. The exact solution of this problem is

ut, x sinπxexp−it. 5.2

For the approximate solution of problem5.1, the set0,1_τ×0,1_hof a family of grid points depending on the small parametersτandh

0,1_τ×0,1_h{tk, x_n:t_kkτ,1≤k≤N−1, Nτ 1, x_nnh, 1≤n≤M−1, Mh1}

5.3 is defined.

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Applying the second-order of accuracy modified Crank-Nicolson diﬀerence schemes 3.1we present following second-order of accuracy diﬀerence schemes for the approximate solutions of problem5.1

iu^k_n−u^k−1_n τ −1

2

u^k_n1−u^k_n−1

2h u^k−1_n1−u^k−1_n−1 2h

−x_n1 2

u^k_n1−2u^k_nu^k_n−1

h² u^k−1_n1−2u^k−1_n u^k−1_n−1 h²

ft_k−1/2, x_n, r1≤k≤N−1,1≤n≤M−1,

iu^k_n−u^k−1_n

τ − u^k_n1−u^k_n−1

2h −xn1u^k_n1−2u^k_nu^k_n−1 h² f t_k−τ

2, x_n!

, 1≤k≤r,1≤n≤M−1,

t_k−1/2

k− 1 2

τ, xnnh,1≤k≤N,1≤n≤M−1,

u⁰_n 1

3u^1/2τ_n ϕxn, 1≤n≤M−1, u^k₀ 0, u^k_M0, 0≤k≤N.

5.4

So for eachr, we haveN1×N1system of linear equations which can be written in the matrix form as

AnU_n1BnUnCnU_n−1Dnϕn, 1≤n≤M−1, U₀0, U_M0,

5.5

where

ϕ_n

⎡

⎢⎢

⎣ ϕ⁰_n ϕ¹_n

· · · ϕ^N_n

⎤

⎥⎥

⎦

N1×1

, ϕ^k_n

⎧⎪

⎪⎨

⎪⎪

⎩

1−1 3exp

−i 2

sinπx_n, k0,

ftk−1/2, x_n, 1≤k≤N,

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An

⎡

⎢⎢

⎣

0 an 0 · · · 0 0 0 0 0 0 0 a_n · · · 0 0 0 0 0

0 0 0 · · · 0 0 0 0 0

· · · · 0 0 0 0 en en 0 0 0

· · · · 0 0 0 · · · 0 0 en en 0 0 0 0 · · · 0 0 0 e_n e_n

0 0 0 · · · 0 0 0 0 0

⎤

⎥⎥

⎦ ,

B_n

⎡

⎢⎢

⎣

b_n c_n 0 · · · 0 · · · 0 0 0 0 0 0 b_n c_n · · · 0 · · · 0 0 0 0 0 0 0 0 · · · 0 0 0 0 0

· · · · 0 0 0 · · · 0 · · · vn sn 0 0 0

· · · · 0 0 0 · · · 0 · · · 0 0 v_n s_n 0 0 0 0 · · · 0 · · · 0 0 0 v_n s_n 1 0 0 · · · −1

3 · · · 0 0 0 0 0

⎤

⎥⎥

⎦ ,

Cn

⎡

⎢⎢

⎣

0 d_n 0 · · · 0 0 0 0 0 0 0 d_n · · · 0 0 0 0 0

0 0 0 · · · 0 0 0 0 0

· · · · 0 0 0 · · · gn gn 0 0 0

· · · · 0 0 0 · · · 0 0 g_n g_n 0 0 0 0 · · · 0 0 0 g_n g_n

0 0 0 · · · 0 0 0 0 0

⎤

⎥⎥

⎦ ,

DI_N1

N1×N1identity matrix ,

Us

⎡

⎢⎢

⎣ U_s⁰ U_s¹

· · · U^N−1_s

U^N_s

⎤

⎥⎥

⎦

, sn−1, n, n1. 5.6

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Table 1: Comparison of the errors for the approximate solution of problem5.1.

Method NM20 NM40 NM80 NM160

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

In the above matrices entries are given as

a_n− 1

2h−nh1

h² , b_n−i

τ, c_n i

τ 2nh1

h² , d_n 1

2h− nh1 h²

en− 1

4h−nh1

2h² , vn

−i

τ nh1 h²

, sn i

τ nh1

h² , gn 1

4h−nh1 2h² .

5.7 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 ton with matrix coefficients.

Hence, we seek a solution of the matrix in the following form:

U_nα_n1U_n1β_n1, nM−1, . . . ,2,1,0, 5.8 whereαjj 1, . . . , MareN1×N1square matrices andβjj1, . . . , MareN1×1 column matrices defined by

α_n1−BnC_nα_n⁻¹A_n, β_n1 B_nC_nα_n⁻¹

Dϕ_n−C_nβ_n

, n1,2,3, . . . , M−1.

5.9 Note that for obtainingα_n1, β_n1, n 1, . . . , M−1,first we need to findα₁, β₁.As in8, we takeα₁is an identity matrix,β₁is the zero column vector.

For their comparison, first the errors computed by

E^N_M max

1≤k≤N−1

_M−1

n1

***utk, xn−u^k_n***²h 1/2

5.10

of the numerical solutions of problem5.1are recorded for diﬀerent values ofN and M, where utk, x_n represents the exact solution and u^k_n represents the numerical solution at tk, xn. The results are shown inTable 1forNM20,40,80, and 160, respectively.

Second, for their comparison, the relative errors are computed by

relE^M_N max

1≤k≤N

E_N^M _M

n1|utk, x_n|²h!_1/2, 5.11

andTable 2is constructed forNM20,40,80, and 160, respectively.

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Table 2: Relative errors for the approximate solution of problem5.1.

Method NM20 NM40 NM80 NM160

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 12it 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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