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

Abstract

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)=^𝑝_𝑚=1𝛼_𝑚𝑢(𝜆_𝑚)+𝜑,0<𝜆₁<𝜆₂<⋯<𝜆_𝑝≤𝑇 (1.1) 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., [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. Nonlocal Boundary Value Problem

Theorem 2.1. Assume that 𝑓(𝑡)∈𝐶¹([0,𝑇],𝐻),𝜑∈𝐷(𝐴) and ^𝑝_𝑚=1|𝛼_𝑚|<1. (2.1) Then there exists a unique solution 𝑢(𝑡) of problem (1.1) and the following inequalities are satisfied: max_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖_𝐻≤𝐶(𝛼₁,…,𝛼_𝑝)[‖𝜑‖_𝐻+𝑇max_{0≤𝑡≤𝑇}‖𝑓(𝑡)‖_𝐻],max_{0≤𝑡≤𝑇}‖𝑢^′(𝑡)‖_𝐻+max_{0≤𝑡≤𝑇}‖𝐴𝑢(𝑡)‖_𝐻≤𝐶(𝛼₁,…,𝛼_𝑝)[‖𝐴𝜑‖_𝐻+𝑇max_{0≤𝑡≤𝑇}‖𝑓^′(𝑡)‖_𝐻+‖𝑓(0)‖_𝐻]. (2.2)(2.3)

Proof. The proof of the estimate (2.2) is given in [8]. 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, (2.4) there exists a unique solution of the problem (1.1), and the following formula holds: 𝑢(𝑡)=𝑒^𝑖𝐴𝑡𝜉−^𝑡₀𝑒^{𝑖𝐴(𝑡−𝑠)}𝑖𝑓(𝑠)𝑑𝑠. (2.5) Therefore we have 𝐴𝑢(𝑡)=𝑒^𝑖𝐴𝑡𝐴𝜉+𝑓(0)+^𝑡₀𝑓^′(𝑠)𝑑𝑠−𝑓(0)𝑒^𝑖𝐴𝑡−^𝑡₀𝑒^{𝑖𝐴(𝑡−𝑠)}𝑓^′(𝑠)𝑑𝑠. (2.6) So that we get the estimate max_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖_𝐻≤‖𝐴𝜉‖_𝐻+2‖𝑓(0)‖_𝐻+2𝑇max_{0≤𝑡≤𝑇}‖𝑓^′(𝑡)‖_𝐻. (2.7) Using the condition 𝑢(0)=∑^𝑝_𝑚=1𝛼_𝑚𝑢(𝜆_𝑚)+𝜑 and the formula (2.6) we get 𝐴𝜉=𝑅{^𝑝_𝑚=1𝛼_𝑚𝑓(0)+^𝑝_𝑚=1𝛼_𝑚^𝜆𝑚₀𝑓^′(𝑠)𝑑𝑠−𝑓(0)^𝑝_𝑚=1𝛼_𝑚𝑒^{𝑖𝐴𝜆𝑚}−^𝑝_𝑚=1𝛼_𝑚^𝜆𝑚₀𝑒^{𝑖𝐴(𝜆𝑚−𝑠)}𝑓^′(𝑠)𝑑𝑠+𝐴𝜑}, (2.8) where 𝑅=(𝐼−^𝑝_𝑚=1𝛼_𝑚𝑒^{𝑖𝐴𝜆𝑚})⁻¹. (2.9) By using estimates ‖𝑅‖_𝐻→𝐻≤11−∑^𝑝_𝑚=1|𝛼_𝑚|≤𝐶(𝛼₁,…,𝛼_𝑝),‖𝑒^𝑖𝐴𝑡‖_𝐻→𝐻≤1, (2.10) and the assumption ∑^𝑝_𝑚=1|𝛼_𝑚|<1, we get ‖𝐴𝜉‖_𝐻≤𝐶(𝛼₁,…,𝛼_𝑝){2‖𝑓(0)‖_𝐻+2𝑇max_{0≤𝑡≤𝑇}‖𝑓^′(𝑡)‖_𝐻+‖𝐴𝜑‖_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 𝑖𝑢^′(𝑡)=𝑓(𝑡)−𝐴𝑢=𝑓(0)+∫^𝑡₀𝑓^′(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 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: 𝑖𝑢_𝑘−𝑢_𝑘−1𝜏+𝐴2(𝑢_𝑘+𝑢_𝑘−1)=𝜑_𝑘𝑖𝑢,𝑟+1≤𝑘≤𝑁,_𝑘−𝑢_𝑘−1𝜏+𝐴𝑢_𝑘=𝜑_𝑘𝑢,1≤𝑘≤𝑟,₀=_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚((𝐼+𝑖𝑙_0𝑚𝐴)𝑢_𝑙𝑚−𝑖𝑙_0𝑚𝜑_𝑙𝑚)+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∈𝑍+}𝛼_𝑚𝑢_𝑙𝑚+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚(𝐼+𝑖𝑑_𝑚1𝐴)2(𝑢_𝑙𝑚+𝑢_𝑙𝑚+1)−𝑖_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚𝑑_𝑚𝜑_𝑙𝑚+𝜑,0<𝜆₁<𝜆₂<⋯<𝜆_𝑝≤𝑇, (3.1) for the approximate solutions of this nonlocal boundary value problem. 𝑍⁺ denotes here the set {2,…,𝑛,…} and 𝑙_𝑚=⌊𝜆_𝑚/𝜏⌋,𝑙_0𝑚=𝜆_𝑚−⌊𝜆_𝑚/𝜏⌋𝜏,𝑑_𝑚=𝜆_𝑚−⌊𝜆_𝑚/𝜏⌋𝜏−𝜏/2, 𝜑_𝑘=𝑓(𝑡_𝑘−𝜏/2), 𝑡_𝑘=𝑘𝜏, where ⌊𝑥⌋ stands for the greatest integer part of the real number 𝑥.

By [10], 𝑢_𝑘=𝑅^𝑘𝜉−𝑖𝜏^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝜑_𝑗𝐵,𝑘=1,…,𝑟,_𝑘−𝑟𝑅^𝑟𝜉−𝑖𝜏^𝑟_𝑗=1𝐵^𝑘−𝑟𝑅^{𝑟−𝑗+1}𝜑_𝑗−𝑖𝜏^𝑘_𝑗=𝑟+1𝐵^𝑘−𝑗𝐶𝜑_𝑗,𝑘=𝑟+1,…,𝑁 (3.2) is the solution of the 𝑟-modified Crank-Nicolson difference schemes for the approximate solutions of Cauchy problem 𝑖𝑢_𝑘−𝑢_𝑘−1𝜏+𝐴2(𝑢_𝑘+𝑢_𝑘−1)=𝜑_𝑘𝑖𝑢,𝑟+1≤𝑘≤𝑁,_𝑘−𝑢_𝑘−1𝜏+𝐴𝑢_𝑘=𝜑_𝑘,1≤𝑘≤𝑟,𝑢₀=𝜉. (3.3) Here 𝑅=(𝐼−𝑖𝜏𝐴)⁻¹𝐴,𝐶=(𝐼−𝑖2𝜏)⁻¹𝐴,𝐵=(𝐼+𝑖2𝜏)𝐶. (3.4) For 𝑢₀, using the formula (3.2) and the condition we obtain 𝜉=𝑇_𝜏{(−𝑖𝜏_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚(𝐼+𝑖𝑙_0𝑚𝐴)^𝑙𝑚_𝑗=1𝑅^{𝑙𝑚−𝑗+1}𝜑_𝑗−𝑖_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚𝑙_0𝑚𝜑_𝑙𝑚)−𝑖𝜏_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∈𝑍+}𝛼_𝑚(^𝑟_𝑗=1𝐵^{𝑙𝑚−𝑟}𝑅^{𝑟−𝑗+1}𝜑_𝑗+^𝑙𝑚_𝑗=𝑟+1𝐵^{𝑙𝑚−𝑗}𝐶𝜑_𝑗)−𝑖𝜏_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚(𝐼+𝑖𝑑_𝑚1𝐴)2((𝐼+𝐵)(^𝑟_𝑗=1𝐵^{𝑙𝑚−𝑟}𝑅^{𝑟−𝑗+1}𝜑_𝑗+^𝑙𝑚_𝑗=𝑟+1𝐵^{𝑙𝑚−𝑗}𝐶𝜑_𝑗)+𝐶𝜑_𝑙𝑚+1)−𝑖_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚𝑑_𝑚𝐴+𝜑}, (3.5) where 𝑇_𝜏=(𝐼−_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚(𝐼+𝑖𝑙_0𝑚𝐴)𝑅^𝑙𝑚−_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∈𝑍+}𝛼_𝑚𝐵^{𝑙𝑚−𝑟}𝑅^𝑟−_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚(𝐼+𝑖𝑑_𝑚1𝐴)2(𝐼+𝐵)𝐵^{𝑙𝑚−𝑟}𝑅^𝑟)⁻¹. (3.6) Note that, here we considered ∑^𝑙𝑚_𝑘=𝑟+1𝐵^{𝑙𝑚−𝑗}𝐶𝜑_𝑗=0 for 𝑙_𝑚=𝑟. So, for the solution of problem (3.2), we have the following formula: 𝑢_𝑘=𝑅^𝑘𝑢₀−𝑖𝜏^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝜑_𝑗𝐵,𝑘=1,…,𝑟,_𝑘−𝑟𝑅^𝑟𝑢₀−𝑖𝜏^𝑟_𝑗=1𝐵^𝑘−𝑟𝑅^{𝑟−𝑗+1}𝜑_𝑗−𝑖𝜏^𝑘_𝑗=𝑟+1𝐵^𝑘−𝑗𝐶𝜑_𝑗𝑇,𝑘=𝑟+1,…,𝑁,_𝜏{−𝑖𝜏_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚(𝐼+𝑖𝑙_0𝑚𝐴)^𝑙𝑚_𝑗=1𝑅^{𝑙𝑚−𝑗+1}𝜑_𝑗−𝑖_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚𝑙_0𝑚𝜑_𝑙𝑚−𝑖𝜏_{𝑟𝜏<𝜆𝑚𝜆/𝜏∈𝑍+}𝛼_𝑚(^𝑟_𝑗=1𝐵^{𝑙𝑚−𝑟}𝑅^{𝑟−𝑗+1}𝜑_𝑗+^𝑙𝑚_𝑗=𝑟+1𝐵^{𝑙𝑚−𝑗}𝐶𝜑_𝑗)−𝑖𝜏_{𝑟𝜏<𝜆𝑚𝜆/𝜏∉𝑍+}𝛼_𝑚(𝐼+𝑖𝑑_𝑚×1𝐴)2((𝐼+𝐵)(^𝑟_𝑗=1𝐵^{𝑙𝑚−𝑟}𝑅^{𝑟−𝑗+1}𝜑_𝑗+^𝑙𝑚_𝑗=𝑟+1𝐵^{𝑙𝑚−𝑗}𝐶𝜑_𝑗)+𝐶𝜑_𝑙𝑚+1)−𝑖_{𝑟𝜏<𝜆𝑚𝜆/𝜏∉𝑍+}𝛼_𝑚𝑑_𝑚𝜑_𝑙𝑚+𝜑},𝑘=0. (3.7)

Theorem 3.1. Assume that 𝜑∈𝐷(𝐴) and ^𝑝_𝑚=1|𝛼_𝑚|<1. (3.8) Then the solutions of the difference schemes (3.1) satisfy the stability inequalities max_{0≤𝑘≤𝑁}‖𝑢_𝑘‖_𝐻≤𝐶(𝛼₁,…,𝛼_𝑝)[‖𝜑‖_𝐻+𝑇max_{1≤𝑘≤𝑁}‖𝜑_𝑘‖_𝐻],max_{1≤𝑘≤𝑁}‖𝑢_𝑘−𝑢_𝑘−1𝜏‖_𝐻+max_{1≤𝑘≤𝑟}‖𝐴𝑢_𝑘‖_𝐻+max_{𝑟+1≤𝑘≤𝑁}𝑢‖𝐴^𝑘+𝑢_𝑘−12‖_𝐻≤𝐶(𝛼₁,…,𝛼_𝑝)[‖𝐴𝜑‖_𝐻+‖𝜑₁‖_𝐻𝑇max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1𝜏‖_𝐻]. (3.9)(3.10)

Proof. Using the estimates ‖𝑅‖_𝐻→𝐻≤1,‖𝐵‖_𝐻→𝐻≤1,‖𝐶‖_𝐻→𝐻≤1, (3.11) and the formula (3.2), we can obtain max_{1≤𝑘≤𝑁}‖𝑢_𝑘‖_𝐻≤[‖𝑢₀‖_𝐻+𝑇max_{1≤𝑘≤𝑁}‖𝜑_𝑘‖_𝐻]. (3.12) Using the spectral representation of the self-adjoint operators one can establish ‖𝑇_𝜏‖_𝐻→𝐻≤11−∑^𝑝_𝑚=1|𝛼_𝑚|≤𝐶(𝛼₁,…,𝛼_𝑝). (3.13) Estimate for ‖𝑢₀‖_𝐻 should also be examined. By using formula (3.7), the triangle inequality, and estimates (3.11), (3.13) the following estimate is obtained: ‖𝑢₀‖_𝐻≤𝐶(𝛼₁,…,𝛼_𝑝)[‖𝜑‖_𝐻+2𝑇max_{1≤𝑘≤𝑁}‖𝜑_𝑘‖_𝐻]. (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 𝐴𝑢_𝑘=𝑅^𝑘𝐴𝜉−𝑖𝜏^𝑘_𝑗=1𝐴𝑅^{𝑘−𝑗+1}𝜑_𝑗𝐵,𝑘=1,…,𝑟,_𝑘−𝑟𝑅^𝑟𝐴𝜉−𝑖𝜏^𝑟_𝑗=1𝐵^𝑘−𝑟𝐴𝑅^{𝑟−𝑗+1}𝜑_𝑗−𝑖𝜏^𝑘_𝑗=𝑟+1𝐵^𝑘−𝑗𝐴𝐶𝜑_𝑗,𝑘=𝑟+1,…,𝑁. (3.15) So that 𝐴𝑢_𝑘=𝑅^𝑘𝐴𝜉+(^𝑘_𝑗=2𝑅^{𝑘−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)+𝜑_𝑘−𝑅^𝑘𝜑₁𝐵),𝑘=1,…,𝑟,_𝑘−𝑟𝑅^𝑟𝐴𝜉+^𝑟_𝑗=2𝐵^𝑘−𝑟𝑅^{𝑟−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)−𝐵^𝑘−𝑟𝑅^𝑟𝜑₁+^𝑘_𝑗=𝑟+1𝐵^{𝑘−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)+𝜑_𝑘,𝑘=𝑟+1,…,𝑁. (3.16) For the estimate (3.10) the two cases should be examined separately: (i) 𝑘=1,…,𝑟, (ii) 𝑘=𝑟+1,…,𝑁. Let 1≤𝑘≤𝑟. Then, using (3.16) we get max_{1≤𝑘≤𝑟}‖𝐴𝑢_𝑘‖_𝐻≤‖𝑅𝐴𝜉‖_𝐻+2𝑁max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1‖_𝐻+2‖𝜑₁‖_𝐻. (3.17) Therefore, max_{1≤𝑘≤𝑟}‖𝐴𝑢_𝑘‖_𝐻≤‖𝑅𝐴𝜉‖_𝐻+2𝑇max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1𝜏‖_𝐻+2‖𝜑₁‖_𝐻. (3.18) Estimate for ‖𝑅𝐴𝜉‖_𝐻 should also be obtained. Using the formula (3.5) and the formula (3.16) we get 𝜉=𝑇_𝜏{_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚(𝐼+𝑖𝑙_0𝑚𝐴)𝑅(^𝑙𝑚_𝑗=2𝑅^{𝑙𝑚−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)+𝜑_𝑙𝑚−𝑅^𝑙𝑚𝜑₁)−𝑖_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚𝑙_0𝑚𝑅𝐴𝜑_𝑙𝑚+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∈𝑍+}𝛼_𝑚𝑅(^𝑟_𝑗=2𝐵^{𝑙𝑚−𝑟}𝑅^{𝑟−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)−𝑅^𝑟𝜑₁)+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∈𝑍+}𝛼_𝑚𝑅(^𝑙𝑚_𝑗=𝑟+1𝐵^{𝑙𝑚−𝑗}𝐶(𝜑_𝑗−1−𝜑_𝑗)+𝜑_𝑙𝑚)+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚(𝐼+𝑖𝑑_𝑚×1𝐴)2𝑅((^𝑟_𝑗=2𝐵^{𝑙𝑚−𝑟}𝑅^{𝑟−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)−𝑅^𝑟𝜑₁+^𝑙𝑚_𝑗=𝑟+1𝐵^{𝑙𝑚−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)+𝜑_𝑙𝑚)+(^𝑟_𝑗=2𝐵^{𝑙𝑚+1−𝑟}𝑅^{𝑟−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)−𝑅^𝑟𝜑₁+^𝑙𝑚+1_𝑗=𝑟+1𝐵^{𝑙𝑚−𝑗+2}(𝜑_𝑗−1−𝜑_𝑗)+𝜑_𝑙𝑚+1))−𝑖_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚𝑑_𝑚𝐴𝑅𝜑_𝑙𝑚+𝑅𝐴𝜑}. (3.19) So that ‖𝑅𝐴𝜉‖_𝐻≤𝐶₁(𝛼₁,…,𝛼_𝑝)[‖𝐴𝜑‖_𝐻+‖𝜑₁‖_𝐻+𝑇max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1𝜏‖_𝐻]. (3.20) Therefore, using the estimates (3.18) and (3.20) we obtain max_{1≤𝑘≤𝑟}‖𝐴𝑢_𝑘‖_𝐻≤𝐶₂(𝛼₁,…,𝛼_𝑝)[‖𝐴𝜑‖_𝐻+‖𝜑₁‖_𝐻+𝑇max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1𝜏‖_𝐻]. (3.21) Then using the estimate for 𝐴𝑢_𝑘, the relation 𝑖((𝑢_𝑘−𝑢_𝑘−1)/𝜏)=𝜑_𝑘−𝐴𝑢_𝑘=𝜑₁−∑^𝑘_𝑗=2(𝜑_𝑗−1−𝜑_𝑗)−𝐴𝑢_𝑘, and the triangle inequality we get the estimate max_{1≤𝑘≤𝑟}‖𝑢_𝑘−𝑢_𝑘−1𝜏‖_𝐻+max_{1≤𝑘≤𝑟}‖𝐴𝑢_𝑘‖_𝐻≤𝐶₂(𝛼₁,…,𝛼_𝑝)[‖𝐴𝜑‖_𝐻+‖𝜑₁‖_𝐻+𝑇max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1𝜏‖_𝐻]. (3.22) Now, let 𝑘=𝑟+1,…,𝑁. Then using the formula (3.16) and the identity (1/2)(𝐼+𝐵)=𝐶 we get 𝐴𝑢_𝑘+𝑢_𝑘−12=𝐵^{𝑘−𝑟−1}𝐶^𝑟−1𝐶𝑅𝐴𝜉+^𝑟_𝑗=2𝐶𝐵^{𝑘−1−𝑟}𝑅^{𝑟−𝑗+1}(𝜑_𝑗−1−𝜑_𝑗)−𝑅^𝑟𝜑₁+^𝑘_𝑗=𝑟+1𝐶𝐵^𝑘−𝑗(𝜑_𝑗−1−𝜑_𝑗)+𝐵(𝜑_𝑘−1−𝜑_𝑘𝜑)+^𝑘+𝜑_𝑘−12. (3.23) So that max_{𝑟+1≤𝑘≤𝑁}𝑢‖𝐴^𝑘+𝑢_𝑘−12‖_𝐻≤‖𝑅𝐴𝜉‖_𝐻+3𝑇max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1𝜏‖_𝐻+3‖𝜑₁‖_𝐻. (3.24) Therefore, using the estimates (3.20) and (3.24), the estimate max_{𝑟+1≤𝑘≤𝑁}𝑢‖𝐴^𝑘+𝑢_𝑘−12‖_𝐻≤𝐶₃(𝛼₁,…,𝛼_𝑝)[‖𝐴𝜑‖_𝐻+‖𝜑₁‖_𝐻+𝑇max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1𝜏‖_𝐻] (3.25) is obtained. Then, by using the estimate (3.25), the relation 𝑖((𝑢_𝑘−𝑢_𝑘−1)/𝜏)=𝜑_𝑘−𝐴((𝑢_𝑘+𝑢_𝑘−1)/2)=𝜑₁−∑^𝑘_𝑗=2(𝜑_𝑗−1−𝜑_𝑗)−𝐴((𝑢_𝑘+𝑢_𝑘−1)/2), and the triangle inequality we get the estimate max_{𝑟+1≤𝑘≤𝑁}‖𝑢_𝑘−𝑢_𝑘−1𝜏‖_𝐻+max_{𝑟+1≤𝑘≤𝑁}𝑢‖𝐴^𝑘+𝑢_𝑘−12‖_𝐻≤𝐶₄(𝛼₁,…,𝛼_𝑝)[‖𝐴𝜑‖_𝐻+‖𝜑₁‖_𝐻+𝑇max_{2≤𝑘≤𝑁}‖𝜑_𝑘−𝜑_𝑘−1𝜏‖_𝐻]. (3.26) The result (3.10) follows from the estimates (3.22) and (3.26). So the proof is complete.

4. Convergence

Theorem 4.1. Assume that ∑^𝑝_𝑚=1|𝛼_𝑚|<1. 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 max_{0≤𝑘≤𝑁}‖𝑢_𝑘−𝑢(𝑡_𝑘)‖_𝐻≤𝑀^∗(𝑟)𝜏², (4.1) where 𝑀^∗(𝑟) does not depend on 𝜏 but depends on 𝑟.

Proof. If we subtract (1.1) from (3.1) we obtain 𝑖𝑧_𝑘−𝑧_𝑘−1𝜏+𝐴2(𝑧_𝑘+𝑧_𝑘−1)=𝐴_𝑘𝑖𝑧,𝑟+1≤𝑘≤𝑁,_𝑘−𝑧_𝑘−1𝜏+𝐴𝑧_𝑘=𝐴_𝑘𝑧,1≤𝑘≤𝑟,₀=_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚((𝐼+𝑖𝑙_0𝑚𝐴)𝑧_𝑙𝑚−𝑖𝑙_0𝑚𝐴_𝑙𝑚)+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∈𝑍+}𝛼_𝑚𝑧_𝑙𝑚+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚(𝐼+𝑖𝑑_𝑚1𝐴)2(𝑧_𝑙𝑚+𝑧_𝑙𝑚+1)−𝑖_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚𝑑_𝑚𝐴_𝑙𝑚+𝐴₀,0<𝜆₁<𝜆₂<⋯<𝜆_𝑝≤𝑇, (4.2) where 𝑧_𝑘=𝑢_𝑘−𝑢(𝑡_𝑘) and 𝐴_𝑘 is defined by the formula 𝐴_𝑘=𝑑𝑖(𝑑𝑡(𝑢(𝑡_𝑘−1/2))−𝑢(𝑡_𝑘)−𝑢(𝑡_𝑘−1)𝜏)+𝐴(𝑢(𝑡_𝑘−1/2)−𝑢(𝑡_𝑘𝑑)),1≤𝑘≤𝑟,𝑖(𝑑𝑡(𝑢(𝑡_𝑘−1/2))−𝑢(𝑡_𝑘)−𝑢(𝑡_𝑘−1)𝜏)+𝐴(𝑢(𝑡_𝑘−1/2)−𝑢(𝑡_𝑘)+𝑢(𝑡_𝑘−1)2),𝑟+1≤𝑟≤𝑁,_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚(𝐼+𝑖𝑙_0𝑚𝐴)𝑢(𝑡_𝑙𝑚)+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∈𝑍+}𝛼_𝑚𝑢(𝑡_𝑙𝑚)+_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}(𝐼+𝑖𝑑_𝑚1𝐴)2(𝑢(𝑡_𝑙𝑚)+𝑢(𝑡_𝑙𝑚+1))−^𝑝_𝑚=1𝛼_𝑚𝑢(𝜆_𝑚)+𝑖_{𝑟𝜏≥𝜆𝑚}𝛼_𝑚𝑙_0𝑚(𝐴_𝑙𝑚−𝜑_𝑙𝑚)+𝑖_{𝑟𝜏<𝜆𝑚𝜆𝑚/𝜏∉𝑍+}𝛼_𝑚𝑑_𝑚(𝐴_𝑙𝑚−𝜑_𝑙𝑚),𝑘=0. (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,𝐴₀, 𝑘=1,…,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 of (4.2), we can obtain max_{1≤𝑘≤𝑟}‖𝑧_𝑘‖_𝐻≤[‖𝑧₀‖_𝐻+𝑟𝜏max_{1≤𝑘≤𝑟}‖𝐴_𝑘‖_𝐻],max_{𝑟+1≤𝑘≤𝑟}‖𝑧_𝑘‖_𝐻≤[‖𝑧₀‖_𝐻+𝑇max_{1≤𝑘≤𝑟}‖𝐴_𝑘‖_𝐻]. (4.5) By the estimate (3.14) we have ‖𝑧₀‖_𝐻≤𝐶(𝛼₁,…,𝛼_𝑝)[‖𝐴₀‖_𝐻+2𝑇max_{1≤𝑘≤𝑁}‖𝐴_𝑘‖_𝐻]. (4.6) 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 max_{1≤𝑘≤𝑟}‖𝐴_𝑘‖_𝐻≤𝑀₁𝜏,max_{𝑟+1≤𝑘≤𝑁}‖𝐴_𝑘‖_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𝜕𝑢(𝑡,𝑥)𝜕𝑡−((𝑥+1)𝑢_𝑥)_𝑥1=𝑓(𝑡,𝑥),0<𝑡,𝑥<1,𝑢(0,𝑥)=31𝑢(2,𝑥)+𝜑(𝑥),0<𝑥<1,𝑢(𝑡,0)=𝑢(𝑡,1)=0,0<𝑡<1,𝑓(𝑡,𝑥)=[𝜋²sin𝜋𝑥−𝜋cos𝜋𝑥+𝜋²1(𝑥+1)sin𝜋𝑥]exp(−𝑖𝑡),𝜑(𝑥)=(1−3𝑖exp(−2))(sin𝜋𝑥). (5.1) by using modified Crank-Nicolson difference scheme (3.1) are investigated. The exact solution of this problem is 𝑢(𝑡,𝑥)=(sin𝜋𝑥)exp(−𝑖𝑡). (5.2) 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={(𝑡_𝑘,𝑥_𝑛)∶𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,𝑥_𝑛=𝑛ℎ,1≤𝑛≤𝑀−1,𝑀ℎ=1} (5.3) 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) 𝑖𝑢^𝑘_𝑛−𝑢_𝑛^𝑘−1𝜏−12(𝑢^𝑘_𝑛+1−𝑢^𝑘_𝑛−1+𝑢2ℎ^{𝑘−1𝑛+1}−𝑢^𝑘−1_𝑛−1𝑥2ℎ)−^𝑛+12(𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²+𝑢^𝑘−1_𝑛+1−2𝑢_𝑛^𝑘−1+𝑢^𝑘−1_𝑛−1ℎ²)=𝑓(𝑡_𝑘−1/2,𝑥_𝑛𝑖𝑢),𝑟+1≤𝑘≤𝑁−1,1≤𝑛≤𝑀−1,_𝑘𝑛−𝑢_𝑛^𝑘−1𝜏−𝑢^𝑘_𝑛+1−𝑢^𝑘_𝑛−12ℎ−(𝑥_𝑛𝑢+1)^𝑘𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²=𝑓(𝑡_𝑘−𝜏2,𝑥_𝑛𝑡),1≤𝑘≤𝑟,1≤𝑛≤𝑀−1,_𝑘−1/21=(𝑘−2)𝜏,𝑥_𝑛𝑢=𝑛ℎ,1≤𝑘≤𝑁,1≤𝑛≤𝑀−1,_0𝑛=13𝑢_𝑛^[1/2𝜏]+𝜑(𝑥_𝑛𝑢),1≤𝑛≤𝑀−1,_𝑘0=0,𝑢^𝑘_𝑀=0,0≤𝑘≤𝑁. (5.4) So for each 𝑟, we have (𝑁+1)×(𝑁+1) system of linear equations which can be written in the matrix form as 𝐴_𝑛𝑈_𝑛+1+𝐵_𝑛𝑈_𝑛+𝐶_𝑛𝑈_𝑛−1=𝐷_𝑛𝜑_𝑛𝑈,1≤𝑛≤𝑀−1,₀=0,𝑈_𝑀=0, (5.5) where 𝜑_𝑛=𝜑⁰_𝑛𝜑¹_𝑛⋯𝜑^𝑁_{𝑛(𝑁+1)×1},𝜑^𝑘_𝑛=1(1−3𝑖exp(−2))(sin𝜋𝑥_𝑛),𝑘=0,𝑓(𝑡_𝑘−1/2,𝑥_𝑛𝐴),1≤𝑘≤𝑁,_𝑛=0𝑎_𝑛0⋯0000000𝑎_𝑛⋯00000000⋯00000⋯⋯⋯⋯⋯⋯⋯⋯⋯0000𝑒_𝑛𝑒_𝑛000⋯⋯⋯⋯⋯⋯⋯⋯⋯000⋯00𝑒_𝑛𝑒_𝑛0000⋯000𝑒_𝑛𝑒_𝑛𝐵000⋯00000_𝑛,=𝑏_𝑛𝑐_𝑛0⋯0⋯000000𝑏_𝑛𝑐_𝑛⋯0⋯00000000⋯⋯⋯00000⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯000⋯0⋯𝑣_𝑛𝑠_𝑛000⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯000⋯0⋯00𝑣_𝑛𝑠_𝑛0000⋯0⋯000𝑣_𝑛𝑠_𝑛1100⋯−3𝐶⋯00000_𝑛,=0𝑑_𝑛0⋯0000000𝑑_𝑛⋯00000000⋯00000⋯⋯⋯⋯⋯⋯⋯⋯⋯000⋯𝑔_𝑛𝑔_𝑛000⋯⋯⋯⋯⋯⋯⋯⋯⋯000⋯00𝑔_𝑛𝑔_𝑛0000⋯000𝑔_𝑛𝑔_𝑛000⋯00000,𝐷=𝐼_𝑁+1𝑈((𝑁+1)×(𝑁+1)identitymatrix),_𝑠=𝑈⁰_𝑠𝑈¹_𝑠⋯𝑈_𝑠^𝑁−1𝑈^𝑁_𝑠,𝑠=𝑛−1,𝑛,𝑛+1. (5.6)(1)(2)(3)(4)(5) In the above matrices entries are given as 𝑎_𝑛1=−−2ℎ𝑛ℎ+1ℎ²,𝑏_𝑛𝑖=−𝜏,𝑐_𝑛=𝑖𝜏+2(𝑛ℎ+1)ℎ²,𝑑_𝑛1=(−2ℎ𝑛ℎ+1ℎ²)𝑒_𝑛1=−−4ℎ𝑛ℎ+12ℎ²,𝑣_𝑛𝑖=(−𝜏+𝑛ℎ+1ℎ²),𝑠_𝑛=𝑖𝜏+𝑛ℎ+1ℎ²,𝑔_𝑛=1−4ℎ𝑛ℎ+12ℎ². (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 to 𝑛 with matrix coefficients. Hence, we seek a solution of the matrix in the following form: 𝑈_𝑛=𝛼_𝑛+1𝑈_𝑛+1+𝛽_𝑛+1,𝑛=𝑀−1,…,2,1,0, (5.8) 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 𝛼_𝑛+1=−(𝐵_𝑛+𝐶_𝑛𝛼_𝑛)⁻¹𝐴_𝑛,𝛽_𝑛+1=(𝐵_𝑛+𝐶_𝑛𝛼_𝑛)⁻¹(𝐷𝜑_𝑛−𝐶_𝑛𝛽_𝑛),𝑛=1,2,3,…,𝑀−1. (5.9) Note that for obtaining 𝛼_𝑛+1,𝛽_𝑛+1,𝑛=1,…,𝑀−1, first we need to find 𝛼₁,𝛽₁. As in [8], we take 𝛼₁ is an identity matrix, 𝛽₁ is the zero column vector.

For their comparison, first the errors computed by 𝐸^𝑁_𝑀=max_{1≤𝑘≤𝑁−1}(^𝑀−1_𝑛=1|𝑢(𝑡_𝑘,𝑥_𝑛)−𝑢^𝑘_𝑛|²ℎ)^1/2 (5.10) of the numerical 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 Table 1 for 𝑁=𝑀=20,40,80, and 160, respectively.

Table 1 

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

Second, for their comparison, the relative errors are computed by rel𝐸^𝑀_𝑁=max_{1≤𝑘≤𝑁}𝐸^𝑀_𝑁(∑^𝑀_𝑛=1|𝑢(𝑡_𝑘,𝑥_𝑛)|²ℎ)^1/2, (5.11) and Table 2 is constructed for 𝑁=𝑀=20,40,80, and 160, respectively.

Table 2 

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

In the article [12] 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.