Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 584718, 15 pages
doi:10.1155/2009/584718
Research Article

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

Allaberen Ashyralyev1 and Ali Sirma2

1Department of Mathematics, Fatih University, 34500 Büyükcekmece, Istanbul, Turkey
2Department of Mathematics and Computer Sciences, Bahcesehir University, Besiktas, 34353 Istanbul, Turkey

Received 26 November 2008; Revised 30 March 2009; Accepted 19 June 2009

Academic Editor: Leonid Berezansky

Copyright © 2009 Allaberen Ashyralyev and Ali 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.

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 𝑖 𝑢 ( 𝑡 ) + 𝐴 𝑢 ( 𝑡 ) = 𝑓 ( 𝑡 ) , 0 < 𝑡 < 𝑇 , 𝑢 ( 0 ) = 𝑝 𝑚 = 1 𝛼 𝑚 𝑢 𝜆 𝑚 + 𝜑 , 0 < 𝜆 1 < 𝜆 2 < ⋯ < 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 ) ∈ 𝐶 1 ( [ 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: m a x 0 ≤ 𝑡 ≤ 𝑇 ‖ 𝑢 ( 𝑡 ) ‖ 𝐻 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 ‖ 𝜑 ‖ 𝐻 + 𝑇 m a x 0 ≤ 𝑡 ≤ 𝑇 ‖ 𝑓 ( 𝑡 ) ‖ 𝐻 , ( 2 . 2 ) m a x 0 ≤ 𝑡 ≤ 𝑇 ‖ ‖ 𝑢 ‖ ‖ ( 𝑡 ) 𝐻 + m a x 0 ≤ 𝑡 ≤ 𝑇 ‖ 𝐴 𝑢 ( 𝑡 ) ‖ 𝐻 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 ‖ 𝐴 𝜑 ‖ 𝐻 + 𝑇 m a x 0 ≤ 𝑡 ≤ 𝑇 ‖ ‖ 𝑓 ‖ ‖ ( 𝑡 ) 𝐻 + ‖ 𝑓 ( 0 ) ‖ 𝐻 . ( 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 𝑖 𝑢 ( 𝑡 ) + 𝐴 𝑢 ( 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 𝑖 𝐴 𝑡 𝜉 − 𝑡 0 𝑒 𝑖 𝐴 ( 𝑡 − 𝑠 ) 𝑖 𝑓 ( 𝑠 ) 𝑑 𝑠 . ( 2 . 5 ) Therefore we have 𝐴 𝑢 ( 𝑡 ) = 𝑒 𝑖 𝐴 𝑡 𝐴 𝜉 + 𝑓 ( 0 ) + 𝑡 0 𝑓 ( 𝑠 ) 𝑑 𝑠 − 𝑓 ( 0 ) 𝑒 𝑖 𝐴 𝑡 − 𝑡 0 𝑒 𝑖 𝐴 ( 𝑡 − 𝑠 ) 𝑓 ( 𝑠 ) 𝑑 𝑠 . ( 2 . 6 ) So that we get the estimate m a x 0 ≤ 𝑡 ≤ 𝑇 ‖ 𝑢 ( 𝑡 ) ‖ 𝐻 ≤ ‖ 𝐴 𝜉 ‖ 𝐻 + 2 ‖ 𝑓 ( 0 ) ‖ 𝐻 + 2 𝑇 m a x 0 ≤ 𝑡 ≤ 𝑇 ‖ ‖ 𝑓 ‖ ‖ ( 𝑡 ) 𝐻 . ( 2 . 7 ) Using the condition ∑ 𝑢 ( 0 ) = 𝑝 𝑚 = 1 𝛼 𝑚 𝑢 ( 𝜆 𝑚 ) + 𝜑 and the formula (2.6) we get 𝐴 𝜉 = 𝑅 𝑝 𝑚 = 1 𝛼 𝑚 𝑓 ( 0 ) + 𝑝 𝑚 = 1 𝛼 𝑚 𝜆 𝑚 0 𝑓 ( 𝑠 ) 𝑑 𝑠 − 𝑓 ( 0 ) 𝑝 𝑚 = 1 𝛼 𝑚 𝑒 𝑖 𝐴 𝜆 𝑚 − 𝑝 𝑚 = 1 𝛼 𝑚 𝜆 𝑚 0 𝑒 𝑖 𝐴 ( 𝜆 𝑚 − 𝑠 ) 𝑓 , ( 𝑠 ) 𝑑 𝑠 + 𝐴 𝜑 ( 2 . 8 ) where 𝑅 = 𝐼 − 𝑝 𝑚 = 1 𝛼 𝑚 𝑒 𝑖 𝐴 𝜆 𝑚 − 1 . ( 2 . 9 ) By using estimates ‖ 𝑅 ‖ 𝐻 → 𝐻 ≤ 1 ∑ 1 − 𝑝 𝑚 = 1 | | 𝛼 𝑚 | | 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 , ‖ ‖ 𝑒 𝑖 𝐴 𝑡 ‖ ‖ 𝐻 → 𝐻 ≤ 1 , ( 2 . 1 0 ) and the assumption ∑ 𝑝 𝑚 = 1 | 𝛼 𝑚 | < 1 , we get ‖ 𝐴 𝜉 ‖ 𝐻 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 2 ‖ 𝑓 ( 0 ) ‖ 𝐻 + 2 𝑇 m a x 0 ≤ 𝑡 ≤ 𝑇 ‖ ‖ 𝑓 ‖ ‖ ( 𝑡 ) 𝐻 + ‖ 𝐴 𝜑 ‖ 𝐻 . ( 2 . 1 1 ) By using the estimates (2.7) and (2.11) we obtain an estimate for 𝐴 𝑢 . Then by using the estimate for 𝐴 𝑢 , the relation 𝑖 𝑢 ∫ ( 𝑡 ) = 𝑓 ( 𝑡 ) − 𝐴 𝑢 = 𝑓 ( 0 ) + 𝑡 0 𝑓 ( 𝑠 ) 𝑑 𝑠 − 𝐴 𝑢 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 2 𝜏 ≤ 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: 𝑖 𝑢 𝑘 − 𝑢 𝑘 − 1 𝜏 + 𝐴 2 𝑢 𝑘 + 𝑢 𝑘 − 1 = 𝜑 𝑘 𝑖 𝑢 , 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 , 𝑘 − 𝑢 𝑘 − 1 𝜏 + 𝐴 𝑢 𝑘 = 𝜑 𝑘 𝑢 , 1 ≤ 𝑘 ≤ 𝑟 , 0 = 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝐼 + 𝑖 𝑙 0 𝑚 𝐴 𝑢 𝑙 𝑚 − 𝑖 𝑙 0 𝑚 𝜑 𝑙 𝑚 + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∈ 𝑍 + 𝛼 𝑚 𝑢 𝑙 𝑚 + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝐼 + 𝑖 𝑑 𝑚 𝐴 1 2 𝑢 𝑙 𝑚 + 𝑢 𝑙 𝑚 + 1 − 𝑖 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝑑 𝑚 𝜑 𝑙 𝑚 + 𝜑 , 0 < 𝜆 1 < 𝜆 2 < ⋯ < 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 0 𝑚 = 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 = 𝑘 𝜏 , where ⌊ 𝑥 ⌋ stands for the greatest integer part of the real number 𝑥 .

By [10], 𝑢 𝑘 = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑅 𝑘 𝜉 − 𝑖 𝜏 𝑘 𝑗 = 1 𝑅 𝑘 − 𝑗 + 1 𝜑 𝑗 𝐵 , 𝑘 = 1 , … , 𝑟 , 𝑘 − 𝑟 𝑅 𝑟 𝜉 − 𝑖 𝜏 𝑟 𝑗 = 1 𝐵 𝑘 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 − 𝑖 𝜏 𝑘 𝑗 = 𝑟 + 1 𝐵 𝑘 − 𝑗 𝐶 𝜑 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 𝑖 𝑢 𝑘 − 𝑢 𝑘 − 1 𝜏 + 𝐴 2 𝑢 𝑘 + 𝑢 𝑘 − 1 = 𝜑 𝑘 𝑖 𝑢 , 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 , 𝑘 − 𝑢 𝑘 − 1 𝜏 + 𝐴 𝑢 𝑘 = 𝜑 𝑘 , 1 ≤ 𝑘 ≤ 𝑟 , 𝑢 0 = 𝜉 . ( 3 . 3 ) Here 𝑅 = ( 𝐼 − 𝑖 𝜏 𝐴 ) − 1 𝐴 , 𝐶 = 𝐼 − 𝑖 2 𝜏 − 1 𝐴 , 𝐵 = 𝐼 + 𝑖 2 𝜏 𝐶 . ( 3 . 4 ) For 𝑢 0 , using the formula (3.2) and the condition we obtain 𝜉 = 𝑇 𝜏 ⎧ ⎪ ⎨ ⎪ ⎩ ⎛ ⎜ ⎜ ⎝ − 𝑖 𝜏 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝐼 + 𝑖 𝑙 0 𝑚 𝐴 𝑙 𝑚 𝑗 = 1 𝑅 𝑙 𝑚 − 𝑗 + 1 𝜑 𝑗 − 𝑖 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝑙 0 𝑚 𝜑 𝑙 𝑚 ⎞ ⎟ ⎟ ⎠ − 𝑖 𝜏 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∈ 𝑍 + 𝛼 𝑚 ⎛ ⎜ ⎜ ⎝ 𝑟 𝑗 = 1 𝐵 𝑙 𝑚 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 + 𝑙 𝑚 𝑗 = 𝑟 + 1 𝐵 𝑙 𝑚 − 𝑗 𝐶 𝜑 𝑗 ⎞ ⎟ ⎟ ⎠ − 𝑖 𝜏 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝐼 + 𝑖 𝑑 𝑚 𝐴 1 2 ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ ( 𝐼 + 𝐵 ) 𝑟 𝑗 = 1 𝐵 𝑙 𝑚 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 + 𝑙 𝑚 𝑗 = 𝑟 + 1 𝐵 𝑙 𝑚 − 𝑗 𝐶 𝜑 𝑗 ⎞ ⎟ ⎟ ⎠ + 𝐶 𝜑 𝑙 𝑚 + 1 ⎞ ⎟ ⎟ ⎠ − 𝑖 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝑑 𝑚 ⎫ ⎪ ⎬ ⎪ ⎭ , 𝐴 + 𝜑 ( 3 . 5 ) where 𝑇 𝜏 = ⎛ ⎜ ⎜ ⎜ ⎝ 𝐼 − 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝐼 + 𝑖 𝑙 0 𝑚 𝐴 𝑅 𝑙 𝑚 − 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∈ 𝑍 + 𝛼 𝑚 𝐵 𝑙 𝑚 − 𝑟 𝑅 𝑟 − 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 ( 𝐼 + 𝑖 𝑑 𝑚 1 𝐴 ) 2 ( 𝐼 + 𝐵 ) 𝐵 𝑙 𝑚 − 𝑟 𝑅 𝑟 ⎞ ⎟ ⎟ ⎟ ⎠ − 1 . ( 3 . 6 ) Note that, here we considered ∑ 𝑙 𝑚 𝑘 = 𝑟 + 1 𝐵 𝑙 𝑚 − 𝑗 𝐶 𝜑 𝑗 = 0 for 𝑙 𝑚 = 𝑟 . So, for the solution of problem (3.2), we have the following formula: 𝑢 𝑘 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑅 𝑘 𝑢 0 − 𝑖 𝜏 𝑘 𝑗 = 1 𝑅 𝑘 − 𝑗 + 1 𝜑 𝑗 𝐵 , 𝑘 = 1 , … , 𝑟 , 𝑘 − 𝑟 𝑅 𝑟 𝑢 0 − 𝑖 𝜏 𝑟 𝑗 = 1 𝐵 𝑘 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 − 𝑖 𝜏 𝑘 𝑗 = 𝑟 + 1 𝐵 𝑘 − 𝑗 𝐶 𝜑 𝑗 𝑇 , 𝑘 = 𝑟 + 1 , … , 𝑁 , 𝜏 ⎧ ⎪ ⎨ ⎪ ⎩ − 𝑖 𝜏 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝐼 + 𝑖 𝑙 0 𝑚 𝐴 𝑙 𝑚 𝑗 = 1 𝑅 𝑙 𝑚 − 𝑗 + 1 𝜑 𝑗 − 𝑖 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝑙 0 𝑚 𝜑 𝑙 𝑚 − 𝑖 𝜏 𝑟 𝜏 < 𝜆 𝑚 𝜆 / 𝜏 ∈ 𝑍 + 𝛼 𝑚 ⎛ ⎜ ⎜ ⎝ 𝑟 𝑗 = 1 𝐵 𝑙 𝑚 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 + 𝑙 𝑚 𝑗 = 𝑟 + 1 𝐵 𝑙 𝑚 − 𝑗 𝐶 𝜑 𝑗 ⎞ ⎟ ⎟ ⎠ − 𝑖 𝜏 𝑟 𝜏 < 𝜆 𝑚 𝜆 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝐼 + 𝑖 𝑑 𝑚 𝐴 × 1 2 ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ ( 𝐼 + 𝐵 ) 𝑟 𝑗 = 1 𝐵 𝑙 𝑚 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 + 𝑙 𝑚 𝑗 = 𝑟 + 1 𝐵 𝑙 𝑚 − 𝑗 𝐶 𝜑 𝑗 ⎞ ⎟ ⎟ ⎠ + 𝐶 𝜑 𝑙 𝑚 + 1 ⎞ ⎟ ⎟ ⎠ − 𝑖 𝑟 𝜏 < 𝜆 𝑚 𝜆 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝑑 𝑚 𝜑 𝑙 𝑚 ⎫ ⎪ ⎬ ⎪ ⎭ + 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 m a x 0 ≤ 𝑘 ≤ 𝑁 ‖ ‖ 𝑢 𝑘 ‖ ‖ 𝐻 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 ‖ 𝜑 ‖ 𝐻 + 𝑇 m a x 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ 𝜑 𝑘 ‖ ‖ 𝐻 , ( 3 . 9 ) m a x 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝑢 𝑘 − 𝑢 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 + m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝐴 𝑢 𝑘 ‖ ‖ 𝐻 + m a x 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝐴 𝑢 𝑘 + 𝑢 𝑘 − 1 2 ‖ ‖ ‖ 𝐻 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 ‖ 𝐴 𝜑 ‖ 𝐻 + ‖ ‖ 𝜑 1 ‖ ‖ 𝐻 𝑇 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 . ( 3 . 1 0 )

Proof. Using the estimates ‖ 𝑅 ‖ 𝐻 → 𝐻 ≤ 1 , ‖ 𝐵 ‖ 𝐻 → 𝐻 ≤ 1 , ‖ 𝐶 ‖ 𝐻 → 𝐻 ≤ 1 , ( 3 . 1 1 ) and the formula (3.2), we can obtain m a x 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ 𝑢 𝑘 ‖ ‖ 𝐻 ≤ ‖ ‖ 𝑢 0 ‖ ‖ 𝐻 + 𝑇 m a x 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ 𝜑 𝑘 ‖ ‖ 𝐻 . ( 3 . 1 2 ) Using the spectral representation of the self-adjoint operators one can establish ‖ ‖ 𝑇 𝜏 ‖ ‖ 𝐻 → 𝐻 ≤ 1 ∑ 1 − 𝑝 𝑚 = 1 | | 𝛼 𝑚 | | 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 . ( 3 . 1 3 ) 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: ‖ ‖ 𝑢 0 ‖ ‖ 𝐻 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 ‖ 𝜑 ‖ 𝐻 + 2 𝑇 m a x 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ 𝜑 𝑘 ‖ ‖ 𝐻 . ( 3 . 1 4 ) 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 𝐴 𝑢 𝑘 = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑅 𝑘 𝐴 𝜉 − 𝑖 𝜏 𝑘 𝑗 = 1 𝐴 𝑅 𝑘 − 𝑗 + 1 𝜑 𝑗 𝐵 , 𝑘 = 1 , … , 𝑟 , 𝑘 − 𝑟 𝑅 𝑟 𝐴 𝜉 − 𝑖 𝜏 𝑟 𝑗 = 1 𝐵 𝑘 − 𝑟 𝐴 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 − 𝑖 𝜏 𝑘 𝑗 = 𝑟 + 1 𝐵 𝑘 − 𝑗 𝐴 𝐶 𝜑 𝑗 , 𝑘 = 𝑟 + 1 , … , 𝑁 . ( 3 . 1 5 ) So that 𝐴 𝑢 𝑘 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑅 𝑘 𝐴 𝜉 + 𝑘 𝑗 = 2 𝑅 𝑘 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 + 𝜑 𝑘 − 𝑅 𝑘 𝜑 1 𝐵 , 𝑘 = 1 , … , 𝑟 , 𝑘 − 𝑟 𝑅 𝑟 𝐴 𝜉 + 𝑟 𝑗 = 2 𝐵 𝑘 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 − 𝐵 𝑘 − 𝑟 𝑅 𝑟 𝜑 1 + 𝑘 𝑗 = 𝑟 + 1 𝐵 𝑘 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 + 𝜑 𝑘 , 𝑘 = 𝑟 + 1 , … , 𝑁 . ( 3 . 1 6 ) 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 m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝐴 𝑢 𝑘 ‖ ‖ 𝐻 ≤ ‖ 𝑅 𝐴 𝜉 ‖ 𝐻 + 2 𝑁 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 ‖ ‖ 𝐻 ‖ ‖ 𝜑 + 2 1 ‖ ‖ 𝐻 . ( 3 . 1 7 ) Therefore, m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝐴 𝑢 𝑘 ‖ ‖ 𝐻 ≤ ‖ 𝑅 𝐴 𝜉 ‖ 𝐻 + 2 𝑇 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 ‖ ‖ 𝜑 + 2 1 ‖ ‖ 𝐻 . ( 3 . 1 8 ) Estimate for ‖ 𝑅 𝐴 𝜉 ‖ 𝐻 should also be obtained. Using the formula (3.5) and the formula (3.16) we get 𝜉 = 𝑇 𝜏 ⎧ ⎪ ⎨ ⎪ ⎩ 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝐼 + 𝑖 𝑙 0 𝑚 𝐴 𝑅 ⎛ ⎜ ⎜ ⎝ 𝑙 𝑚 𝑗 = 2 𝑅 𝑙 𝑚 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 + 𝜑 𝑙 𝑚 − 𝑅 𝑙 𝑚 𝜑 1 ⎞ ⎟ ⎟ ⎠ − 𝑖 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝑙 0 𝑚 𝑅 𝐴 𝜑 𝑙 𝑚 + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∈ 𝑍 + 𝛼 𝑚 𝑅 𝑟 𝑗 = 2 𝐵 𝑙 𝑚 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 − 𝑅 𝑟 𝜑 1 + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∈ 𝑍 + 𝛼 𝑚 𝑅 ⎛ ⎜ ⎜ ⎝ 𝑙 𝑚 𝑗 = 𝑟 + 1 𝐵 𝑙 𝑚 − 𝑗 𝐶 𝜑 𝑗 − 1 − 𝜑 𝑗 + 𝜑 𝑙 𝑚 ⎞ ⎟ ⎟ ⎠ + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝐼 + 𝑖 𝑑 𝑚 𝐴 × 1 2 𝑅 ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ 𝑟 𝑗 = 2 𝐵 𝑙 𝑚 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 − 𝑅 𝑟 𝜑 1 + 𝑙 𝑚 𝑗 = 𝑟 + 1 𝐵 𝑙 𝑚 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 + 𝜑 𝑙 𝑚 ⎞ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ 𝑟 𝑗 = 2 𝐵 𝑙 𝑚 + 1 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 − 𝑅 𝑟 𝜑 1 + 𝑙 𝑚 + 1 𝑗 = 𝑟 + 1 𝐵 𝑙 𝑚 − 𝑗 + 2 𝜑 𝑗 − 1 − 𝜑 𝑗 + 𝜑 𝑙 𝑚 + 1 ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ − 𝑖 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝑑 𝑚 𝐴 𝑅 𝜑 𝑙 𝑚 ⎫ ⎪ ⎬ ⎪ ⎭ . + 𝑅 𝐴 𝜑 ( 3 . 1 9 ) So that ‖ 𝑅 𝐴 𝜉 ‖ 𝐻 ≤ 𝐶 1 𝛼 1 , … , 𝛼 𝑝 ‖ 𝐴 𝜑 ‖ 𝐻 + ‖ ‖ 𝜑 1 ‖ ‖ 𝐻 + 𝑇 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 . ( 3 . 2 0 ) Therefore, using the estimates (3.18) and (3.20) we obtain m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝐴 𝑢 𝑘 ‖ ‖ 𝐻 ≤ 𝐶 2 𝛼 1 , … , 𝛼 𝑝 ‖ 𝐴 𝜑 ‖ 𝐻 + ‖ ‖ 𝜑 1 ‖ ‖ 𝐻 + 𝑇 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 . ( 3 . 2 1 ) Then using the estimate for 𝐴 𝑢 𝑘 , the relation 𝑖 ( ( 𝑢 𝑘 − 𝑢 𝑘 − 1 ) / 𝜏 ) = 𝜑 𝑘 − 𝐴 𝑢 𝑘 = 𝜑 1 − ∑ 𝑘 𝑗 = 2 ( 𝜑 𝑗 − 1 − 𝜑 𝑗 ) − 𝐴 𝑢 𝑘 , and the triangle inequality we get the estimate m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ ‖ 𝑢 𝑘 − 𝑢 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 + m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝐴 𝑢 𝑘 ‖ ‖ 𝐻 ≤ 𝐶 2 𝛼 1 , … , 𝛼 𝑝 ‖ 𝐴 𝜑 ‖ 𝐻 + ‖ ‖ 𝜑 1 ‖ ‖ 𝐻 + 𝑇 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 . ( 3 . 2 2 ) Now, let 𝑘 = 𝑟 + 1 , … , 𝑁 . Then using the formula (3.16) and the identity ( 1 / 2 ) ( 𝐼 + 𝐵 ) = 𝐶 we get 𝐴 𝑢 𝑘 + 𝑢 𝑘 − 1 2 = 𝐵 𝑘 − 𝑟 − 1 𝐶 𝑟 − 1 𝐶 𝑅 𝐴 𝜉 + 𝑟 𝑗 = 2 𝐶 𝐵 𝑘 − 1 − 𝑟 𝑅 𝑟 − 𝑗 + 1 𝜑 𝑗 − 1 − 𝜑 𝑗 − 𝑅 𝑟 𝜑 1 + 𝑘 𝑗 = 𝑟 + 1 𝐶 𝐵 𝑘 − 𝑗 𝜑 𝑗 − 1 − 𝜑 𝑗 𝜑 + 𝐵 𝑘 − 1 − 𝜑 𝑘 + 𝜑 𝑘 + 𝜑 𝑘 − 1 2 . ( 3 . 2 3 ) So that m a x 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝐴 𝑢 𝑘 + 𝑢 𝑘 − 1 2 ‖ ‖ ‖ 𝐻 ≤ ‖ 𝑅 𝐴 𝜉 ‖ 𝐻 + 3 𝑇 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 ‖ ‖ 𝜑 + 3 1 ‖ ‖ 𝐻 . ( 3 . 2 4 ) Therefore, using the estimates (3.20) and (3.24), the estimate m a x 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝐴 𝑢 𝑘 + 𝑢 𝑘 − 1 2 ‖ ‖ ‖ 𝐻 ≤ 𝐶 3 𝛼 1 , … , 𝛼 𝑝 ‖ 𝐴 𝜑 ‖ 𝐻 + ‖ ‖ 𝜑 1 ‖ ‖ 𝐻 + 𝑇 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 ( 3 . 2 5 ) 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 ) = 𝜑 1 − ∑ 𝑘 𝑗 = 2 ( 𝜑 𝑗 − 1 − 𝜑 𝑗 ) − 𝐴 ( ( 𝑢 𝑘 + 𝑢 𝑘 − 1 ) / 2 ) , and the triangle inequality we get the estimate m a x 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝑢 𝑘 − 𝑢 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 + m a x 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝐴 𝑢 𝑘 + 𝑢 𝑘 − 1 2 ‖ ‖ ‖ 𝐻 ≤ 𝐶 4 𝛼 1 , … , 𝛼 𝑝 ‖ 𝐴 𝜑 ‖ 𝐻 + ‖ ‖ 𝜑 1 ‖ ‖ 𝐻 + 𝑇 m a x 2 ≤ 𝑘 ≤ 𝑁 ‖ ‖ ‖ 𝜑 𝑘 − 𝜑 𝑘 − 1 𝜏 ‖ ‖ ‖ 𝐻 . ( 3 . 2 6 ) 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 𝐴 𝑢 ( 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 m a x 0 ≤ 𝑘 ≤ 𝑁 ‖ ‖ 𝑢 𝑘 − 𝑢 ( 𝑡 𝑘 ) ‖ ‖ 𝐻 ≤ 𝑀 ∗ ( 𝑟 ) 𝜏 2 , ( 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 𝑖 𝑧 𝑘 − 𝑧 𝑘 − 1 𝜏 + 𝐴 2 𝑧 𝑘 + 𝑧 𝑘 − 1 = 𝐴 𝑘 𝑖 𝑧 , 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 , 𝑘 − 𝑧 𝑘 − 1 𝜏 + 𝐴 𝑧 𝑘 = 𝐴 𝑘 𝑧 , 1 ≤ 𝑘 ≤ 𝑟 , 0 = 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝐼 + 𝑖 𝑙 0 𝑚 𝐴 𝑧 𝑙 𝑚 − 𝑖 𝑙 0 𝑚 𝐴 𝑙 𝑚 + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∈ 𝑍 + 𝛼 𝑚 𝑧 𝑙 𝑚 + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝐼 + 𝑖 𝑑 𝑚 𝐴 1 2 𝑧 𝑙 𝑚 + 𝑧 𝑙 𝑚 + 1 − 𝑖 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝑑 𝑚 𝐴 𝑙 𝑚 + 𝐴 0 , 0 < 𝜆 1 < 𝜆 2 < ⋯ < 𝜆 𝑝 ≤ 𝑇 , ( 4 . 2 ) where 𝑧 𝑘 = 𝑢 𝑘 − 𝑢 ( 𝑡 𝑘 ) and 𝐴 𝑘 is defined by the formula 𝐴 𝑘 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑖 𝑑 𝑢 𝑡 𝑑 𝑡 𝑘 − 1 / 2 − 𝑢 𝑡 𝑘 𝑡 − 𝑢 𝑘 − 1 𝜏 𝑢 𝑡 + 𝐴 𝑘 − 1 / 2 𝑡 − 𝑢 𝑘 𝑖 𝑑 , 1 ≤ 𝑘 ≤ 𝑟 , 𝑢 𝑡 𝑑 𝑡 𝑘 − 1 / 2 − 𝑢 𝑡 𝑘 𝑡 − 𝑢 𝑘 − 1 𝜏 𝑢 𝑡 + 𝐴 𝑘 − 1 / 2 − 𝑢 𝑡 𝑘 𝑡 + 𝑢 𝑘 − 1 2 , 𝑟 + 1 ≤ 𝑟 ≤ 𝑁 , 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝐼 + 𝑖 𝑙 0 𝑚 𝐴 𝑢 𝑡 𝑙 𝑚 + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∈ 𝑍 + 𝛼 𝑚 𝑢 𝑡 𝑙 𝑚 + 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝐼 + 𝑖 𝑑 𝑚 𝐴 1 2 𝑢 𝑡 𝑙 𝑚 𝑡 + 𝑢 𝑙 𝑚 + 1 − 𝑝 𝑚 = 1 𝛼 𝑚 𝑢 𝜆 𝑚 + 𝑖 𝑟 𝜏 ≥ 𝜆 𝑚 𝛼 𝑚 𝑙 0 𝑚 𝐴 𝑙 𝑚 − 𝜑 𝑙 𝑚 + 𝑖 𝑟 𝜏 < 𝜆 𝑚 𝜆 𝑚 / 𝜏 ∉ 𝑍 + 𝛼 𝑚 𝑑 𝑚 𝐴 𝑙 𝑚 − 𝜑 𝑙 𝑚 , 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 , 𝐴 0 , 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 m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝑧 𝑘 ‖ ‖ 𝐻 ≤ ‖ ‖ 𝑧 0 ‖ ‖ 𝐻 + 𝑟 𝜏 m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝐴 𝑘 ‖ ‖ 𝐻 , m a x 𝑟 + 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝑧 𝑘 ‖ ‖ 𝐻 ≤ ‖ ‖ 𝑧 0 ‖ ‖ 𝐻 + 𝑇 m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝐴 𝑘 ‖ ‖ 𝐻 . ( 4 . 5 ) By the estimate (3.14) we have ‖ ‖ 𝑧 0 ‖ ‖ 𝐻 𝛼 ≤ 𝐶 1 , … , 𝛼 𝑝 ‖ ‖ 𝐴 0 ‖ ‖ 𝐻 + 2 𝑇 m a x 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 𝐴 𝑢 ( 𝑡 ) ( 0 ≤ 𝑡 ≤ 𝑇 ) and 𝑢 ( 𝑡 ) ( 0 ≤ 𝑡 ≤ 𝑇 ) , the estimates m a x 1 ≤ 𝑘 ≤ 𝑟 ‖ ‖ 𝐴 𝑘 ‖ ‖ 𝐻 ≤ 𝑀 1 𝜏 , m a x 𝑟 + 1 ≤ 𝑘 ≤ 𝑁 ‖ ‖ 𝐴 𝑘 ‖ ‖ 𝐻 ≤ 𝑀 2 𝜏 2 , ‖ ‖ 𝐴 0 ‖ ‖ 𝐻 ≤ 𝑀 2 𝜏 2 ( 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 𝑥 1 = 𝑓 ( 𝑡 , 𝑥 ) , 0 < 𝑡 , 𝑥 < 1 , 𝑢 ( 0 , 𝑥 ) = 3 𝑢 1