# On Stability of a Third Order of Accuracy Difference Scheme for Hyperbolic Nonlocal BVP with Self-Adjoint Operator

Authors: Allaberen Ashyralyev; Ozgur Yildirim
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A third order of accuracy absolutely stable difference schemes is presented for nonlocal boundary value hyperbolic problem of the differential equations in a Hilbert spaceH with self-adjoint positive definite operator A. Stability estimates for solution of the difference scheme are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions is considered.


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## Body

## 1. Introduction

In modeling several phenomena of physics, biology, and ecology mathematically, there often arise problems with nonlocal boundary conditions (see [1–5] and the references given therein). Nonlocal boundary value problems have been a major research area in the case when it is impossible to determine the boundary conditions of the unknown function. Over the last few decades, the study of nonlocal boundary value problems is of substantial contemporary interest (see, e.g., [6–14] and the references given therein).We consider the nonlocal boundary value problem(1)d2u(t)dt2+Au(t)=f(t),0<t<1,u(0)=αu(1)+φ,u′(0)=βu′(1)+ψ,
for hyperbolic equations in a Hilbert space H with self-adjoint positive definite linear operator A with domain D(A).A functionu(t) is called a solution of problem (1) if the following conditions are satisfied.(i)
u(t) is twice continuously differentiable on the segment [0,1]. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii)
The elementu(t) belongs to D(A) for all t∈[0,1] and the function Au(t) is continuous on the segment [0,1].(iii)
u(t) satisfies the equations and the nonlocal boundary conditions (1).Here,φ(x), ψ(x)(x∈[0,1]) and f(t,x)(t,x∈[0,1]) are smooth functions.In the study of numerical methods for solving PDEs, stability is an important research area (see [6–27]). Many scientists work on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitudes of the grid steps τ and h with respect to the time and space variables are connected. This particularly means that τ∥Ah∥→0 when τ→0.We are interested in studying high order of accuracy unconditionally stable difference schemes for hyperbolic PDEs.In the present paper, third order of accuracy difference scheme generated by integer power ofA for approximately solving nonlocal boundary value problem (1) is presented. The stability estimates for solution of the difference scheme are established.In [8], some results of this paper, without proof, were presented.The well posedness of nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [11–14, 19–32] and the references therein).


## 2. Third Order of Accuracy Difference Scheme Subject to Nonlocal Conditions

In this section, we obtain stability estimates for the solution of third order of accuracy difference scheme(2)τ-2(uk+1-2uk+uk-1)+23Auk+16A(uk+1+uk-1)+112τ2A2uk+1=fk,fk=23f(tk)+16(f(tk+1)+f(tk-1))1111-112τ2(-Af(tk+1)+f′′(tk+1)),1111tk=kτ,1≤k≤N-1,Nτ=1,u0=αuN+φ,(I+τ212A+τ4144A2)τ-1(u1-u0)+τ2Au0-τf1,11111=β(I-τ2A12)11111×(7uN-8uN-1+uN-26τ+τ3(fN-AuN))11111+(I-τ2A12)ψ
for numerical solution of nonlocal boundary value problem (1). Here,
(3)f1,1=f(0)+(-f(0)+τf′(0))12-2f′(0)τ6.
We study the stability of solutions of difference scheme (2) under the following assumption:
(4)|α|+2|β|+2|α||β|<1.
We give a lemma that will be needed in the sequel which was presented in [18]. First, let us present the following operators:
(5)R=(I-13τ2A+iτA1/2I+172τ4A2)R=×(I+16τ2A+112τ4A2)-1,
and its conjugate R~,
(6)R1=(I+172τ4A2-5τ4144A2+7τ6216A3-iτA1/2×(I+τ212A+τ4144A2)I+172τ4A2)×(-iτA1/2(I+172τ4A2)×(I+τ212A+τ4144A2)(I+172τ4A2))-1,
and its conjugate R~1,
(7)R2=(I-τ212A)(I+τ26A+τ412A2)×(-iA1/2(I+τ212A+τ4144A2)I+172τ4A2)-1,R3=(I+τ26A+τ412A2)×((I+τ212A+τ4144A2)(-iτA1/2I+172τ4A2))-1,R4=(I+τ23A+τ49A2+τ672A3)×(-iA1/2(I+172τ4A2)(I+τ26A+τ412A2)×(I+τ26A)(I+172τ4A2))-1,R5=(-τ22A-τ412A2+iτA1/2I+172τ4A2)×(I+τ26A+τ412A2)-1,
and its conjugate R~5, and
(8)R6=(I-13τ2A+iτA1/2I+172τ4A2)R6=×(τ22A+τ412A2-iτA1/2I+172τ4A2)-1,
and its conjugate R~6.We consider the following operators:(9)R7=(7R-I)6τR7=(I-512τ2A+172τ4A2+76iτA1/2I+172τ4A2)R7=×τ-1(I+16τ2A+112τ4A2)-1,
and its conjugate R~7,
(10)R~7=(7R~-I)6τR~7=(I-512τ2A+172τ4A2-76iτA1/2I+172τ4A2)R~7=×τ-1(I+16τ2A+112τ4A2)-1,R8=(7I-2τ2A6τ)(I+τ2A3+τ4A29+τ6A372)R8=×τ-1(I+τ2A6)-1(I+τ26A+τ412A2)-2,R9=(I-53τ2A+τ4A29)(I+τ2A3+τ4A29+τ6A372)R9=×τ-1(I+τ2A6)-1(I+16τ2A+112τ4A2)-3,R10=I+(5144τ4A2-9288τ6A3+91728τ8A4)R10=×(iτA1/2I+172τ4A2(I+τ212A+τ4144A2))-1,
and its conjugate R~10.Lemma 1.
The following estimates hold:(11)∥R∥H→H≤1,∥R~∥H→H≤1,∥R1∥H→H≤1,∥R~1∥H→H≤1,∥A1/2R2∥H→H≤1,∥τA1/2R3∥H→H≤1,∥A1/2R4∥H→H≤1,∥A-1/2R5∥H→H≤τ,∥A-1/2R~5∥H→H≤τ,∥τA1/2R6∥H→H≤1,∥τA1/2R~6∥H→H≤1.Now let us give, without proof, the second lemma.Lemma 2.
The following estimates hold:(12)∥(I+iτA1/2)R∥H→H≤2,∥(I+iτA1/2)R~∥H→H≤2,∥τR7∥H→H≤1,∥τR~7∥H→H≤1,∥13τA1/2R2∥H→H≤1,∥13τA1/2R~2∥H→H≤1,∥τR8∥H→H≤76,∥τR9∥H→H≤1,∥R10(I+iτA1/2)-1∥H→H≤2,∥R~10(I+iτA1/2)-1∥H→H≤2.Throughout the section, for simplicity, we denote(13)Bτ=β12R2(R~7R~5-τA3R~2)R~N-2Bτ=+β12R2(R7R5-τA3R2)RN-2Bτ=-α12[R~1RN-R1R~N]Bτ=+αβ14R~1R2(R~7R~5-τA3R~2)RNR~N-2Bτ=+αβ14R1R2(R7R5-τA3R2)R~NRN-2Bτ=-αβ14R~1R2(R7R5-τA3R2)R~NRN-2Bτ=-αβ14R1R2(R~7R~5-τA3R~2)RNR~N-2.Lemma 3.
Suppose that assumption (4) holds. Then, the operator I-Bτ has an inverse Tτ=(I-Bτ)-1. From symmetry and positivity properties of operator A, the following estimate is satisfied:
(14)∥Tτ∥H↦H≤11-|α|-2|β|-2|α||β|.Proof.
Using the definitions ofBτ,R,R~, estimates (11), and the following simple estimates,
(15)∥τA1/2(I-τ212A)(I+τ212A+τ4144A2)-1∥H→H≤12,∥τA1/2(I+112τ2A+1144τ4A2)-1∥H→H≤121112+11,
and the triangle inequality, we get
(16)Bτ≤|β|12∥A1/2R2∥H→H∥R~N-2∥H→HBτ≤×(∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)Bτ≤+|β|12∥A1/2R2∥H→H×∥RN-2∥H→HBτ≤×(∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)Bτ≤+α12[∥R~1∥H→H∥RN∥H→H+∥R1∥H→H∥R~N∥H→H]Bτ≤+|α||β|14∥R~1∥H→H∥A1/2R2∥H→HBτ≤×∥RN∥H→H∥R~N-2∥H→HBτ≤×(∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)Bτ≤+|α||β|14∥R1∥H→H∥A1/2R2∥H→HBτ≤×∥R~N∥H→H∥RN-2∥H→HBτ≤×(∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)Bτ≤+|α||β|14∥R~1∥H→H∥A1/2R2∥H→HBτ≤×∥R~N∥H→H∥RN-2∥H→HBτ≤×(∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)Bτ≤+|α||β|14∥R1∥H→H∥A1/2R2∥H→HBτ≤×∥RN∥H→H∥R~N-2∥H→HBτ≤×(∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)Bτ≤q,
where
(17)q=|α|+2|β|+2|α||β|.
Since q<1, the operator I-Bτ has a bounded inverse and
(18)∥(I-Bτ)-1∥H→H≤11-q=11-|α|-2|β|-2|α||β|.
Lemma 3 is proved.Now, let us obtain formula for the solution of problem (2). Using the results of [18], one can obtain the following formula:
(19)u0=μ,u1=(I+τ212A+τ4144A2)-1u0=×((I-512τ2A+τ4144A2)μ+τ(I-τ212A)ω+τ2f1,1),uk=12[R~10Rk-R10R~k]μ+12[R~k-Rk]R2ωuk=+12[R~k-Rk]R3τ2f1,1+12R4∑s=1k-1‍[R~k-s-Rk-s]fsτ2
for the solution of difference scheme
(20)τ-2(uk+1-2uk+uk-1)+23Auk+16A(uk+1+uk-1)+112τ2A2uk+1=fk,fk=23f(tk)+16(f(tk+1)+f(tk-1))-112τ2(-Af(tk+1)+f′′(tk+1)),tk=kτ,1≤k≤N-1,Nτ=1,u0=μ,(I+τ212A+τ4144A2)τ-1(u1-u0)+τ2Au0-τf1,1=(I-τ2A12)ω.
Applying formula (19) and nonlocal boundary conditions
(21)u0=αuN+φ,ω=β(7uN-8uN-1+uN-26τ+τ3(fN-AuN))+ψ,
one can write
(22)μ=α{∑s=1N-1‍12[R~10RN-R10R~N]μ+12[R~N-RN]R2ω+12[R~N-RN]R3τ2f1,1+12R4∑s=1N-1‍[R~N-s-RN-s]fsτ2}+φ,ω=β{12[(R7R5-τA3R2)R~10RN-2-(R~7R~5-τA3R~2)R10R~N-2]μ+12[(R~7R~5-τA3R~2)R~N-2-(R7R5-τA3R2)RN-2]R2ω+12[(R~7R~5-τA3R~2)R~N-2-(R7R5-τA3R2)RN-2]R3τ2f1,1+τ3fN+12R8fN-1τ2+12R9fN-2τ2+12R4τ∑s=1N-3‍[(R~7R~5-τA3R~2)R~N-2-s-(R7R5-τA3R2)RN-2-s]fsτ}+ψ.
Using formulas in (22), we obtain
(23)μ=Tτ{∑s=1N-1‍[∑s=1N-1‍α(∑s=1N-1‍12(R~N-RN)R3τ2f1,1μ=+12R4τ∑s=1N-1‍(R~N-s-RN-s)fsτ)+φ]μ=×[I-12((R~7R~5-τA3R~2)R~N-2μ=-(R7R5-τA3R2)RN-2(R~7R~5-τA3R~2))R212((R~7R~5-τA3R~2)R~N-212]+[α12(R~N-RN)R2]μ=×[-(R7R5-τA3R2)RN-2-s]fsτ((R~7R~5-τA3R~2)R~N-2}β12{((R~7R~5-τA3R~2)R~N-2μ=-(R7R5-τA3R2)RN-2(R~7R~5-τA3R~2))×R3τ2f1,1+2τ3fN+R8fN-1τ2μ=+R9fN-2τ2+R4τ∑s=1N-3‍[(R~7R~5-τA3R~2)R~N-2-sμ=-(R7R5-τA3R2)×RN-2-s(R~7R~5-τA3R~2)]fsτ((R~7R~5-τA3R~2)R~N-2}+ψ]∑s=1N-1‍},ω=Tτ{+12R4τ∑s=1N-1‍(R~N-s-RN-s)fsτ)+φ][I-α12(R~10RN-R10R~N)]ω=×[β12{((R~7R~5-τA3R~2)R~N-2ω=-(R7R5-τA3R2)RN-2(R~7R~5-τA3R~2))×R3τ2f1,1+2τ3fN+R8fN-1τ2ω=+R9fN-2τ2+R4τ×∑s=1N-3‍((R~7R~5-τA3R~2)R~N-2-sω=-(R7R5-τA3R2)×RN-2-s(R~7R~5-τA3R~2))fsτ}+ψ]ω=+12[(R7R5-τA3R2)R~10RN-2+(R~7R~5-τA3R~2)R10R~N-2]ω=×[α(∑s=1N-1‍12(R~N-RN)R3τ2f1,1+12R4τ∑s=1N-1‍(R~N-s-RN-s)fsτ)+φ]}.So, formulas (19) and (23) give a solution of problem (2).Unfortunately, the estimates formax1≤k≤N∥uk∥H, max1≤k≤N∥A1/2uk∥H, and max1≤k≤N∥Auk∥H cannot be obtained under the conditions
(24)max1≤k≤N∥uk∥H≤M{∑s=1N-1‍∥A-1/2fs∥Hτ+∥φ∥H+∥A-1/2ψ∥H1111111+τ∥A-1/2f1,1∥H∑s=1N-1‍},max1≤k≤N∥A1/2uk∥H≤M{∑s=1N-1‍∥fs∥Hτ+∥A1/2φ∥H+∥ψ∥H+τ∥f1,1∥H},max1≤k≤N∥Auk∥H≤M{∑s=2N-1‍∥fs-fs-1∥H+∥f1∥H+∥Aφ∥H+∥A1/2ψ∥H+τ∥A1/2f1,1∥H∑s=2N-1‍}.
Nevertheless, we have the following theorem.Theorem 4.
Suppose that assumption (4) holds and φ∈D(A3/2), ψ∈D(A1/2). Then, for solution of difference scheme (2), the following stability estimates hold:
(25)max1≤k≤N∥uk∥H≤M{∑s=1N-1‍∥A-1/2fs∥Hτ+∥(I+iτA1/2)φ∥H11111111111+∥A-1/2ψ∥H+τ∥A-1/2f1,1∥H∑s=1N-1‍},max1≤k≤N∥A1/2uk∥H≤M{∑s=1N-1‍∥fs∥Hτ+∥A1/2(I+iτA1/2)φ∥H11111111111+∥ψ∥H+τ∥f1,1∥H∑s=1N-1‍},max1≤k≤N∥Auk∥H≤M{∑s=2N-1‍∥fs-fs-1∥H+∥f1∥H+∥A(I+iτA1/2)φ∥H11111111111+∥A1/2ψ∥H+τ∥A1/2f1,1∥H∑s=2N-1‍},
where M does not depend on τ, φ, ψ, f1,1(x), and fs(x),  1≤s≤N-1.Proof.
Using formulas in (23) and estimates (11), (12), and (14), we obtain
(26)∥(I+iτA1/2)μ∥H≤∥Tτ∥H→H×{|β|12{((∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)∥R~N-2∥H→H[|α|(12(∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H∥(I+iτA1/2)R~N∥H→H)×∥τA1/2R3∥H→Hτ∥A-1/2f1,1∥H+12∥τA1/2R4∥H→H×∑s=1N-1‍(∥(I+iτA1/2)R~N-s∥H→H+∥(I+iτA1/2)RN-s∥H→H)×∥A-1/2fs∥H→Hτ12(∥(I+iτA1/2)R~N∥H→H)+∥(I+iτA1/2)φ∥H(12(∥(I+iτA1/2)R~N∥H→H]×[1+12((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2∥H→H+(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2∥H→H(∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H(∥13τA1/2R~2∥H→H)×∥A1/2R2∥H→H12((∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)]+|α|12(∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H)×∥A1/2R2∥H→H×[|β|12{((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2∥H→H+(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2∥H→H+∥13τA1/2R~2∥H→H))×∥τA1/2R3∥H→Hτ∥A-1/2f1,1∥H+2τ3∥A-1/2fN∥H+∥τR8∥H→H∥A-1/2fN-1∥Hτ+∥τR9∥H→H∥A-1/2fN-2∥Hτ+∥τA1/2R4∥H→H×∑s=1N-3‍((2τ3∥A-1/2fN∥H∥τR~7∥H→H×∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2-s∥H→H+(∥13τA1/2R~2∥H→H∥τR7∥H→H×∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2-s∥H→H(∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H))×∥A-1/2fs∥Hτ((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H}+∥A-1/2ψ∥H|β|12{((∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)∥R~N-2∥H→H]}≤M{∑s=1k-1‍∥A-1/2fs∥Hτ+∥(I+iτA1/2)φ∥H+∥A-1/2ψ∥H+τ∥A-1/2f1,1∥H∑s=1k-1‍},∥A-1/2ω∥H≤∥Tτ∥H→H×{[1+|α|12(∥(I+iτA1/2)-1R~10∥H→H×∥(I+iτA1/2)RN∥H→H+∥(I+iτA1/2)-1R10∥H→H×∥(I+iτA1/2)R~N∥H→H∥(I+iτA1/2)-1R~10∥H→H)12(∥(I+iτA1/2)-1R~10∥H→H]×[|β|12{((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2∥H→H+(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2∥H→H(∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H)×∥τA1/2R3∥H→H∥A-1/2f1,1∥Hτ+23∥A-1/2fN∥Hτ+∥τR8∥H→H∥A-1/2fN-1∥Hτ+∥τR9∥H→H∥A-1/2fN-2∥Hτ+∥τA1/2R4∥H→H×∑s=1N-3‍((∥13τA1/2R2∥H→H∥τR~7∥H→H×∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2-s∥H→H+(∥13τA1/2R2∥H→H∥τR7∥H→H×∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2-s∥H→H(∥13τA1/2R2∥H→H∥τR~7∥H→H(∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H)×∥A-1/2fs∥Hτ((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H}+∥A-1/2ψ∥H12{((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H]+12[(∥13τA1/2R~2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥(I+iτA1/2)-1R~10∥H→H∥RN-2∥H→H+(∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥(I+iτA1/2)-1R10∥H→H∥R~N-2∥H→H]×[|α|12((∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H)×∥τA1/2R3∥H→Hτ∥A-1/2f1,1∥H+∥τA1/2R4∥H→H×∑s=1N-1‍(∥(I+iτA1/2)R~N-s∥H→H+∥(I+iτA1/2)RN-s∥H→H∥(I+iτA1/2)R~N-s∥H→H)×∥A-1/2fs∥Hτ)+∥(I+iτA1/2)φ∥H|α|12((∥(I+iτA1/2)R~N∥H→H((∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H)][1+|α|12(∥(I+iτA1/2)-1R~10∥H→H}≤M{∑s=1k-1‍∥A-1/2fs∥Hτ+∥(I+iτA1/2)φ∥H+∥A-1/2ψ∥H+τ∥A-1/2f1,1∥H∑s=1k-1‍}.
ApplyingA1/2 to formulas in (23), we get
(27)∥A1/2(I+iτA1/2)μ∥H≤∥Tτ∥H→H×{[|α|(∑s=1N-1‍12(∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H∥(I+iτA1/2)R~N∥H→H)×∥τA1/2R3∥H→Hτ∥f1,1∥H+12∥τA1/2R4∥H→H×∑s=1N-1‍(∥(I+iτA1/2)R~N-s∥H→H+∥(I+iτA1/2)RN-s∥H→H∥(I+iτA1/2)R~N-s∥H→H)×∥fs∥H→Hτ∑s=1N-1‍)+∥A1/2(I+iτA1/2)φ∥H∑s=1N-1‍]×[1+12((13∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2∥H→H+(12∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)∥RN-2∥H→H)×∥A1/2R2∥H→H(12)]+|α|12(12∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H12)×∥A1/2R2∥H→H×[|β|12{((12∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2∥H→H+(13∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)∥RN-2∥H→H)×∥τA1/2R3∥H→Hτ∥f1,1∥H+2τ3∥fN∥H+∥τR8∥H→H∥fN-1∥Hτ+∥τR9∥H→H×∥fN-2∥Hτ+∥τA1/2R4∥H→H×∑s=1N-3‍((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2-s∥H→H+(13∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2-s∥H→H13)×∥fs∥Hτ(13)}+∥ψ∥H]∑s=1N-3‍}≤M{∑s=1k-1‍∥fs∥Hτ+∥A1/2(I+iτA1/2)φ∥H+∥ψ∥H+τ∥f1,1∥H∑s=1k-1‍},∥ω∥H≤∥Tτ∥H→H×{∑s=1N-1‍[1+|α|12(∥(I+iτA1/2)-1R~10∥H→H×∥(I+iτA1/2)RN∥H→H+∥(I+iτA1/2)-1R10∥H→H×∥(I+iτA1/2)R~N∥H→H)12]×[|β|12{((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2∥H→H+(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2∥H→H13)×∥τA1/2R3∥H→H∥f1,1∥Hτ+23∥fN∥Hτ+∥τR8∥H→H∥fN-1∥Hτ+∥τR9∥H→H∥fN-2∥Hτ+∥A1/2R4∥H→H×∑s=1N-3‍((+∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2-s∥H→H+(13∥τR7∥H→H×∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2-s∥H→H13)∥fs∥Hτ13}+∥ψ∥H]+12[(13∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥(I+iτA1/2)-1R~10∥H→H∥RN-2∥H→H+(13∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥(I+iτA1/2)-1R10∥H→H∥R~N-2∥H→H13]×[|α|(∑s=1N-1‍12(∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H∥(I+iτA1/2)R~N∥H→H)×∥τA1/2R3∥H→Hτ∥f1,1∥H+12∥τA1/2R4∥H→H×∑s=1N-1‍(∥(I+iτA1/2)R~N-s∥H→H+∥(I+iτA1/2)RN-s∥H→H∥(I+iτA1/2)R~N∥H→H)×∥fs∥Hτ∑s=1N-1‍)+∥A1/2(I+iτA1/2)φ∥H∑s=1N-1‍]}≤M{∑s=1k-1‍∥fs∥Hτ+∥A1/2(I+iτA1/2)φ∥H+∥ψ∥H+τ∥f1,1∥H∑s=1k-1‍}.
Now, applying Abel’s formula to (23), we obtain the following formulas:
(28)μ=Tτ{[α(-(R~6R~N-1-R6RN-1)f1∑s=2N-1‍)12(R~N-RN)R3τ2f1,1+12R4τ2(∑s=2N-1‍(R6RN-s-R~6R~N-s)×(fs-fs-1)+(R~6-R6)fN-1-(R~6R~N-1-R6RN-1)f1∑s=2N-1‍))+φ]×[I-12((R~7R~5-τA3R~2)R~N-2-(R7R5-τA3R2)RN-2)R2]+α12(R~N-RN)R2×[β{∑s=2N-3‍12((R~7R~5-τA3R~2)R~N-2-(R7R5-τA3R2)RN-2)×R3τ2f1,1+τ3fN+12R8fN-1τ2+12R9fN-2τ2+R412τ2(∑s=2N-3‍(R6(R7R5-τA3R2)RN-2-s-R~6(R~7R~5-τA3R~2)R~N-2-s(R7R5-τA3R2))×(fs-fs-1)+(R~6(R~7R~5-τA3R~2)-R6(R7R5-τA3R2))fN-3-(R~6(R~7R~5-τA3R~2)R~N-3-R6(R7R5-τA3R2)RN-3)×f1∑s=2N-3‍)}+ψ]},(29)ω=Tτ{-(R~6R~N-1-R6RN-1)f1∑s=2N-1‍))+φ][I-α12(R~1RN-R1R~N)]×[β{×f1∑s=2N-3‍)12((R~7R~5-τA3R~2)R~N-2-(R7R5-τA3R2)RN-2)×R3τ2f1,1+τ3fN+12R8fN-1τ2+12R9fN-2τ2+R412τ2(∑s=2N-3‍(R6(R7R5τ-τA3R2)RN-2-s-R~6(R~7R~5-τA3R~2)R~N-2-s(R7R5τ-τA3R2))×(fs-fs-1)+(R~6(R~7R~5-τA3R~2)-R6(R7R5-τA3R2))fN-3-(R~6(R~7R~5-τA3R~2)R~N-3-R6(R7R5-τA3R2)×RN-3(R~7R~5-τA3R~2))f1∑s=2N-3‍)}+ψ]+12[(R7R5-τA3R2)R~1RN-2-(R~7R~5-τA3R~2)R1R~N-2]×[-(R~6R~N-1-R6RN-1)f1∑s=2N-1‍))α(∑s=2N-1‍12(R~N-RN)R3τ2f1,1+12R4τ2(∑s=2N-1‍(R6RN-s-R~6R~N-s)×(fs-fs-1)+(R~6-R6)fN-1-(R~6R~N-1-R6RN-1)×f1∑s=2N-1‍)∑s=2N-1‍)+φ]}.
Next, let us obtain the estimates for∥A(I+iτA1/2)μ∥H and ∥A1/2ω∥H. First, applying A to formula (28) and using estimates (11), (12), and (14) and the triangle inequality, one can obtain
(30)∥A(I+iτA1/2)μ∥H≤∥Tτ∥H→H×{[|α|(+∥τA1/2R6∥H→H∥(I+iτA1/2)RN-1∥H→H)∥f1∥H∑s=2N-1‍)12(∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H)×∥τA1/2R3∥H→H∥A1/2f1,1∥Hτ+12∥τA1/2R4∥H→H×(∑s=2N-1‍(∥τA1/2R~6∥H→H∥(I+iτA1/2)R~N-s∥H→H∥τA1/2R6∥H→H×∥(I+iτA1/2)RN-s∥H→H+∥τA1/2R~6∥H→H×∥(I+iτA1/2)R~N-s∥H→H)×∥fs-fs-1∥H+(∥τA1/2(I+iτA1/2)R~6∥H→H+∥τA1/2(I+iτA1/2)R6∥H→H∥τA1/2(I+iτA1/2)R~6∥H→H)×∥fN-1∥H+(∥τA1/2R~6∥H→H×∥(I+iτA1/2)R~N-1∥H→H+∥τA1/2R6∥H→H×∥(I+iτA1/2)RN-1∥H→H∥τA1/2R~6∥H→H∥(I+iτA1/2)R~N-1∥H→H)×∥f1∥H∑s=2N-1‍))+∥A(I+iτA1/2)φ∥H(+∥τA1/2R6∥H→H∥(I+iτA1/2)RN-1∥H→H)∥f1∥H∑s=2N-1‍)12(∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H)]×[1+12((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)∥R~N-2∥H→H+(∥13τA1/2R~2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2∥H→H(∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H(∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H)∥A1/2R2∥H→H12((∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)∥R~N-2∥H→H]+|α|12(∥(I+iτA1/2)R~N∥H→H+∥(I+iτA1/2)RN∥H→H)∥A1/2R2∥H→H×[|β|12{((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)∥R~N-2∥H→H+(∥13τA1/2R~2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2∥H→H(∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H))×∥τA1/2R3∥H→H×τ∥A1/2f1,1∥H+2τ3∥A1/2fN∥H+∥τR8∥H→H∥A1/2fN-1∥Hτ+∥τR9∥H→H∥A1/2fN-2∥Hτ+∥τA1/2R4∥H→H×(∑s=2N-3‍(∥R~N-2-s∥H→H∥τA1/2R6∥H→H×(∥13τA1/2R~2∥H→H∥τR7∥H→H×∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2-s∥H→H+∥τA1/2R~6∥H→H×(∥13τA1/2R~2∥H→H∥τR~7∥H→H×∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2-s∥H→H∥R~N-2-s∥H→H∥τA1/2R6∥H→H)∥fs-fs-1∥H+((∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)∥τA1/2R~6∥H→H×(∥13τA1/2R~2∥H→H∥τR~7∥H→H×∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)+∥τA1/2R6∥H→H×(∥13τA1/2R~2∥H→H∥τR7∥H→H×∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H))×∥fN-3∥H+(∑s=2N-3‍∥τA1/2R~6∥H→H×(∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-3∥H→H+∥τA1/2R6∥H→H×(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-3∥H→H∑s=2N-3‍∥τA1/2R~6∥H→H)∥f1∥H∑s=2N-3‍)}+∥A1/2ψ∥H×∥RN-3∥H→H∥τA1/2R~6∥H→H)∥f1∥H∑s=2N-3‍)}]}≤M{∑s=2N-1‍∥fs-fs-1∥H+∥f1∥H+∥A(I+iτA1/2)φ∥H+∥A1/2ψ∥H+τ∥A1/2f1,1∥H∑s=2N-3‍}.
Second, applyingA1/2 to formula (29) and using estimates (11), (12), and (14) and the triangle inequality, we get
(31)∥A1/2ω∥H≤∥Tτ∥H→H×{[1+|α|12(∥(I+iτA1/2)-1R~10∥H→H×∥(I+iτA1/2)RN∥H→H+∥(I+iτA1/2)-1R10∥H→H×∥(I+iτA1/2)R~N∥H→H∥(I+iτA1/2)-1R~10∥H→H)12(∥(I+iτA1/2)-1R~10∥H→H]×[|β|12{((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)∥R~N-2∥H→H+(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2∥H→H(∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H)×∥τA1/2R3∥H→Hτ∥A1/2f1,1∥H+23τ∥A1/2fN∥H+∥τR8∥H→H∥A1/2fN-1∥Hτ+∥τR9∥H→H∥A1/2fN-2∥Hτ+∥τA1/2R4∥H→Hτ×(∑s=2N-3‍(∑s=2N-3‍∥τA1/2R6∥H→H×(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-2-s∥H→H+∥τA1/2R~6∥H→H×(∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-2-s∥H→H∑s=2N-3‍)∥fs-fs-1∥H+((∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)∥τA1/2R~6∥H→H×(∥13τA1/2R~2∥H→H∥τR~7∥H→H×∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)+∥τA1/2R6∥H→H×(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H))×∥fN-3∥H+(∥τA1/2R~6∥H→H×(∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R~N-3∥H→H+∥τA1/2R6∥H→H×(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥RN-3∥H→H)∥f1∥H∑s=2N-3‍∥τA1/2R~6∥H→H)((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H}+∥A1/2ψ∥H{((∥13τA1/2R~2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H]+12[(∥13τA1/2R2∥H→H∥τR7∥H→H∥τ-1A-1/2R5∥H→H+∥13τA1/2R2∥H→H)×∥R~10(I+iτA1/2)-1∥H→H×∥(I+iτA1/2)RN-2∥H→H+(∥13τA1/2R2∥H→H∥τR~7∥H→H∥τ-1A-1/2R~5∥H→H+∥13τA1/2R~2∥H→H)×∥R10(I+iτA1/2)-1∥H→H×∥(I+iτA1/2)R~N-2∥H→H∥13τA1/2R2∥H→H]×[|α|12(∥R~N∥H→H+∥RN∥H→H)×∥τA1/2R3∥H→Hτ∥A1/2f1,1∥H+∥τA1/2R4∥H→H×(∑s=2N-1‍(∥τA1/2R6∥H→H∥RN-s∥H→H+∥τA1/2R~6∥H→H∥R~N-s∥H→H)×∥fs-fs-1∥H+(∥τA1/2R~6∥H→H+∥τA1/2R6∥H→H)×∥fN-1∥H→H+(∥τA1/2R~6∥H→H∥R~N-1∥H→H+∥τA1/2R6∥H→H∥RN-1∥H→H)×∥f1∥H∑s=2N-1‍(∥R~N∥H→H+∥RN∥H→H))+∥(I+iτA1/2)-1∥H→H∥A(I+iτA1/2)φ∥H((∥R~N∥H→H+∥RN∥H→H)][1+|α|12(∥(I+iτA1/2)-1R~10∥H→H}≤M{∑s=2N-1‍∥fs-fs-1∥H+∥f1∥H+∥A(I+iτA1/2)φ∥H+∥A1/2ψ∥H+τ∥A1/2f1,1∥H∑s=2N-1‍}.
Now, we will prove estimates (25). Using formula (19), estimates (11), (12), (26), and (27), and the triangle inequality, we obtain
(32)∥uk∥H≤12(∥R~10(I+iτA1/2)-1∥H→H∥Rk∥H→H+∥R10(I+iτA1/2)-1∥H→H∥R~k∥H→H)×∥(I+iτA1/2)μ∥H+12(12∥A1/2R2∥H→H∥R~k∥H→H+∥A1/2R2∥H→H∥Rk∥H→H12)×∥A-1/2ω∥H+12(12∥τA1/2R3∥H→H∥R~k∥H→H+∥τA1/2R3∥H→H∥Rk∥H→H12)τ∥A-1/2f1,1∥H+12∥τA1/2R4∥H→H∑s=1k-1‍[∥R~k-s∥H→H+∥Rk-s∥H→H]×∥A-1/2fs∥Hτ≤M{∑s=1N-1‍∥A-1/2fs∥Hτ+∥(I+iτA1/2)φ∥H+∥A-1/2ψ∥H+τ∥A-1/2f1,1∥H∑s=1N-1‍}
for any k≥2. Applying A1/2 to (19), we get
(33)∥A1/2uk∥H≤12(∥R~10(I+iτA1/2)-1∥H→H∥Rk∥H→H+∥R10(I+iτA1/2)-1∥H→H∥R~k∥H→H)×∥A1/2(I+iτA1/2)μ∥H+12(12∥A1/2R2∥H→H∥R~k∥H→H+∥A1/2R2∥H→H∥Rk∥H→H12)×∥ω∥H+12(12∥τA1/2R3∥H→H∥R~k∥H→H+∥τA1/2R3∥H→H∥Rk∥H→H12)×τ∥f1,1∥H+12∥τA1/2R4∥H→H∑s=1k-1‍(∥R~k-s∥H→H+∥Rk-s∥H→H)×∥fs∥Hτ≤M{∑s=1N-1‍∥fs∥Hτ+∥A1/2(I+iτA1/2)φ∥H111111111111111111+∥ψ∥H+τ∥f1,1∥H∑s=1N-1‍}
for k≥2. Now, applying Abel’s formula to (19), we have
(34)uk=12[R~1Rk-R1R~k]μ+12[R~k-Rk]R2ωuk=+12[R~k-Rk]R3τ2f1,1uk=+τ2R412(∑s=2k-1‍[R6Rk-s-R~6R~k-s](fs-fs-1)1111111111111+(R~6-R6)fk-111111111111-[R~6R~k-1-R6Rk-1]f1∑s=2k-1‍),2≤k≤N.
Applying A to formula (34) and using estimates (11) and (12) and the triangle inequality, we obtain
(35)∥Auk∥H≤12(∥R~10(I+iτA1/2)-1∥H→H∥Rk∥H→H+∥R10(I+iτA1/2)-1∥H→H∥R~k∥H→H)×∥A(I+iτA1/2)μ∥H+12(12∥A1/2R2∥H→H∥R~k∥H→H+∥A1/2R2∥H→H∥Rk∥H→H12)×∥A1/2ω∥H+12(12∥τA-1/2R3∥H→H∥R~k∥H→H+∥τA-1/2R3∥H→H∥Rk∥H→H12)×τ∥A1/2f1,1∥H+12∥τA1/2R4∥H→H×(∑s=2k-1‍[12∥τA1/2R6∥H→H∥Rk-s∥H→H+∥τA1/2R~6∥H→H∥R~k-s∥H→H12]×∥fs-fs-1∥H+(∥τA1/2R~6∥H→H+∥τA1/2R6∥H→H)∥fk-1∥H+[∥τA1/2R~6∥H→H∥R~k-1∥H→H+∥τA1/2R6∥H→H∥Rk-1∥H→H]∥f1∥H∑s=2k-1‍)≤M{∑s=2N-1‍∥fs-fs-1∥H+∥f1∥H+∥A(I+iτA1/2)φ∥H+∥A1/2ψ∥H+τ∥A1/2f1,1∥H∑s=2N-1‍}
for k≥2. Theorem 4 is proved.Note that the stability estimates obtained previously permit us to get the convergence estimate of difference scheme (2) under the smoothness property of solution (1). Actually, under the condition u(t)∈C([0,1],H), we can obtain the third order of accuracy for the error of difference scheme (2). Since u(6)(t)=-A3u(t)+A2f(t)-Af′′(t)+f(4)(t), this condition is satisfied under the given data φ∈D(A3), ψ∈D(A5/2), f′(t)∈D(A2), and f(0)∈D(A3).Now, let us give application of this abstract result for nonlocal boundary value problem(36)utt-(a(x)ux)x+δu=f(t,x),0<t<1,0<x<1,u(0,x)=αu(1,x)+φ(x),0≤x≤1,ut(0,x)=βut(1,x)+ψ(x),0≤x≤1,u(t,0)=u(t,1),ux(t,0)=ux(t,1),0≤t≤1
for hyperbolic equation. Problem (36) has a unique smooth solution u(t,x), δ>0and the smooth functions a(x)≥a>0(a(0)=a(1),x∈(0,1)), φ(x),ψ(x)(x∈[0,1]), and f(t,x)(t,x∈[0,1]). This allows us to reduce mixed problem (36) to nonlocal boundary value problem (1) in a Hilbert space H=L2[0,1] with a self-adjoint positive definite operator Ax defined by (36).The discretization of problem (36) is carried out in two steps. In the first step, let us define the grid space
(37)[0,1]h={x:xr=rh,0≤r≤K,Kh=1}.
We introduce Hilbert space L2h=L2([0,1]h), W2h1=W2h1([0,1]h), and W2h2=W2h2([0,1]h) of the grid functions φh(x)={φr}1K-1 defined on [0,1]h, and we assign the difference operator Ahx by the formula
(38)Ahxφh(x)={-(a(x)φx¯)x,r+δφr}1K-1,
acting in the space of grid functions φh(x)={φr}0K satisfying the conditions φ0=φK, φ1-φ0=φK-φK-1.With the help ofAhx, we arrive at the nonlocal boundary value problem
(2)d2vh(t,x)dt2+Ahxvh(t,x)=fh(t,x),11111111110<t<1,x∈[0,1]h,vh(0,x)=αvh(1,x)+φh(x),x∈[0,1]h,vth(0,x)=βvth(1,x)+ψh(x),x∈[0,1]h
for a system of ordinary differential equations.In the second step, we replace problem (2) with difference scheme (40)
(40)τ-2(uk+1h(x)-2ukh(x)+uk-1h(x))+23Ahxukh(x)+16Ahx(uk+1h(x)+uk-1h(x))+112τ2(Ahx)2uk+1h(x)=fkh(x),fkh(x)=23fh(tk,x)+16(fh(tk+1,x)+fh(tk-1,x))-112τ2(-Afh(tk+1,x)+ftth(tk+1,x)),x∈[0,1]h,tk=kτ,Nτ=1,1≤k≤N-1,u0h(x)=αuNh(x)+φh(x),x∈[0,1]h,(I+τ212(Ahx)+τ4144(Ahx)2)τ-1(u1h(x)-u0h(x))+τ2(Ahx)φh(x)-τf1,1h(x)=β(I-τ212(Ahx))×(16τ(7uNh(x)-8uN-1h(x)+uN-2h(x))+τ3(fNh(x)-AuNh(x)))+(I-τ212(Ahx))ψh(x),x∈[0,1]h,f1,1h(x)=12fh(0,x)+τ6fth(0,x).Theorem 5.
Letτ and h be sufficiently small numbers. Then, the solution of difference scheme (40) satisfies the following stability estimates:
(41)max0≤k≤N∥ukh∥L2h+max0≤k≤N∥ukh∥W2h1≤M1[max1≤k≤N-1∥fkh∥L2h+∥ψh∥L2h+∥φh∥W2h1+τ∥φh∥W2h2+τ∥f1,1h∥L2hmax1≤k≤N-1∥fkh∥L2h+∥ψh∥L2h],max1≤k≤N-1∥τ-2(uk+1h-2ukh+uk-1h)∥L2h+max0≤k≤N∥ukh∥W2h2≤M1[∥f1h∥L2h+max2≤k≤N-1∥τ-1(fkh-fk-1h)∥L2h+∥ψh∥W2h1+∥φh∥W2h2+τ∥φh∥W2h3+τ∥f1,1h∥W2h1∥f1h∥L2h+max2≤k≤N-1∥τ-1(fkh-fk-1h)∥L2h].
Here, M1 does not depend on τ,h,φh(x),ψh(x),f1,1h(x), and fkh(x),1≤k<N.The proof of Theorem5 is based on the proof of abstract Theorem 4 and the symmetry property of operator Ahx defined by (38).



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*Source: 959216-2013-12-18.xml*