# On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-Parabolic Equations

Authors: Allaberen Ashyralyev; Okan Gercek
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We are interested in studying a second order of accuracy implicit difference scheme for the solution of the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of this difference scheme is established. In an application, coercivity estimates in Hölder norms for approximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolic differential equations are obtained.


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## Body

## 1.  Introduction

Methods of solutions of nonlocal boundary value problems for mixed-type differential equations have been studied extensively by various researchers (see, e.g., [1–19] and the references therein).In [20], we considered the well-posedness of the following multipoint nonlocal boundary value problem:
(1.1)-d2u(t)dt2+Au(t)=g(t),(0≤t≤1),du(t)dt-Au(t)=f(t),(-1≤t≤0),u(1)=∑i=1J‍αiu(λi)+φ,-1≤λ1<λ2<⋯<λi<⋯<λJ≤0,
in a Hilbert space H with the self-adjoint positive definite operator A under assumption
(1.2)∑i=1J‍|αi|≤1.The well-posedness of multipoint nonlocal boundary value problem (1.1) in Hölder spaces with a weight was established. Moreover, coercivity estimates in Hölder norms for the solutions of nonlocal boundary value problems for elliptic-parabolic equations were obtained.In [21], we studied the well-posedness of the first order of accuracy difference scheme for the approximate solution of boundary value problem (1.1) under assumption (1.2).Throughout this work, we consider the following second order of accuracy difference scheme:(1.3)-τ-2(uk+1-2uk+uk-1)+Auk=gk,gk=g(tk),tk=kτ,1≤k≤N-1,Nτ=1,τ-1(uk-uk-1)-(I+τ2A)Auk-1=(I+τ2A)fk,fk=f(tk-1/2),tk-1/2=(k-12)τ,-(N-1)≤k≤0,u2-4u1+3u0=-3u0+4u-1-u-2,uN=∑k=1J‍αi(u[λi/τ]+(λi-[λiτ]τ)(f[λi/τ]+Au[λi/τ]))+φ,
for the approximate solution of boundary value problem (1.1) under assumption (1.2).The well-posedness of difference scheme (1.3) in Hölder spaces with a weight is established. As an application, the stability, almost coercivity stability, and coercivity stability estimates for solutions of second order of accuracy difference scheme for the approximate solution of the nonlocal boundary elliptic-parabolic problem are obtained.


## 2. Main Theorems

Throughout the paper,H is a Hilbert space and we denote B=(1/2)(τA+A(4+τ2A)), where A is a self-adjoint positive definite operator. Then, it is clear that B is the self-adjoint positive definite operator and B≥δ1/2I  where δ>δ0>0, and R=(I+τB)-1, which is defined on the whole space H, is a bounded operator. Here, I is the identity operator. The following operators
(2.1)D=(I+τA+(τA)22),G=(I-τ2A2),P=(I+τ2A),R=(I+τB)-1,Tτ=(∑i=1nαi‍I+B-1A(I+τA+τ2P-2)K(I-R2N-1)+GKP-2R2N-1I+I-GKP-2(2I+τB)RN[∑i=1nαi‍(I+(λi-[λiτ]τ)A)D-[λi/τ]u0])-1
exist and are bounded for a self-adjoint positive operator A. Here,
(2.2)B=12(τA+A(4+τ2A)),K=(I+2τA+54(τA)2)-1.Furthermore, positive constants will be indicated byM which can differ in time. On the other hand Mi(α,β,…) is used to focus on the fact that the constant depends only on α,β,… and the subindex i is used to indicate a different constant.First of all, let us start with some auxiliary lemmas from [16, 22–24] that are essential below.Lemma 2.1.
For a self-adjoint positive operator A, the following estimates are satisfied:(2.3)‖Rk‖H→H≤M1(δ)(1+δτ)-k,‖Dk‖H→H≤M1(δ),‖BRk‖H→H≤M1(δ)kτ,‖P-1‖H→H≤M1(δ),‖ADk‖H→H≤M1(δ)kτ,‖Dk-e-kτA‖H→H≤M1(δ)k2,‖(I-R2N)-1‖H→H≤M1(δ),‖Rk-e-kτA1/2‖H→H≤M1(δ)k,k≥1,δ>0.
From these estimates, it follows that
(2.4)‖(×RN[∑i=1n‍αi(I+(λi-[λiτ]τ)A)D-[λi/τ]]I+B-1A(I+τA+τ2P-2)K(I-R2N-1)+GKP-2R2N-1-GKP-2(2I+τB)IIIIIIII×RN[∑i=1n‍αi(I+(λi-[λiτ]τ)A)D-[λi/τ]])-1‖H→H≤M2(δ).Lemma 2.2.
For anygk,1≤k≤N-1 and fk,-N+1≤k≤0, the solution of problem (1.3) exists, and the following formulas hold:
(2.5)uk=(I-R2N)-1×{∑s=1N-1[RN-s-RN+s][Rk-R2N-k]u0+[RN-k-RN+k]Rk×-×[∑i=1n‍αi[(I+(λi-[λiτ]τ)A)(D-[λi/τ]u0-τ∑s=[λi/τ]+10‍PDs-[λi/τ]fs)RkRkR×ssss×k-+(λi-[λiτ]τ)f[λi/τ]∑i=1n2‍αi]+φ∑i=1n2‍αi]Rk×--[RN-k-RN+k](I+τB)(2I+τB)-1B-1∑s=1N-1[RN-s-RN+s]gsτ}+(I+τB)(2I+τB)-1B-1∑s=1N-1[R|k-s|-Rk+s]gsτ,1≤k≤N,uk=D-ku0-τ∑s=k+10‍PDs-kfs,-N≤k≤-1,u0=12TτKP-2×{∑s=1N-1Rs-1‍(2I-τ2A)×{∑s=1N-1Rs-1‍(2+τB)RNRkRRRRRRRRRR-×[∑i=1n‍αi[(I+(λi-[λiτ]τ)A)RkRRRRRRRRRRRRRsssss-×(D-[λi/τ]u0-τ∑s=[λi/τ]+10‍PDs-[λi/τ]fs)RkRRRRRRRRRRRRRsssss-+(λi-[λiτ]τ)f[λi/τ]]+φ∑i=1n‍αi]RkRRRRRRRRRR--RN-1B-1∑s=1N-1‍[RN-s-RN+s]gsτ+(I-R2N)B-1∑s=1N-1Rs-1‍gsτ∑i=1n‍αi}RkR-+(I-R2N)(I+τB)(τB-1g1-4PB-1f0+PDB-1f0+PB-1f-1)∑s=1N-1Rs-1‍},Tτ=(∑i=1nαi‍I+B-1A(I+τA+τ2P-2)K(I-R2N-1)+GKP-2R2N-1I+R-GKP-2(2I+τB)RN[∑i=1nαi‍(I+(λi-[λiτ]τ)A)D-[λi/τ]u0])-1.Now, we study well-posedness of problem (1.3). Let Fτ(H)=F([a,b]τ,H) be the linear space of mesh functions φτ={φk}N~N~~ defined on [a,b]τ={tk=kh,  N~≤k≤N~~,  N~τ=a,  N~~τ=b} with values in the Hilbert space H. Next, on Fτ(H) we denote C([a,b]τ,H),C0,1α([-1,1]τ,H),C0,1α([-1,0]τ,H),  C0α([0,1]τ,H),  C~0,1α([-1,1]τ,H), and C~0α([-1,0]τ,H),0<α<1 Banach spaces with the following norms:
(2.6)‖φτ‖C([a,b]τ,H)=max⁡Na≤k≤Nb‖φk‖H,‖φτ‖C0,1α([-1,1]τ,H)=‖φτ‖C([-1,1]τ,H)+sup⁡-N≤k<k+r≤0‖φk+r-φk‖E(-k)αr-α+sup⁡1≤k<k+r≤N-1‖φk+r-φk‖E((k+r)τ)α(N-k)αr-α,‖φτ‖C0α([-1,0]τ,H)=‖φτ‖C([-1,0]τ,H)+sup⁡-N≤k<k+r≤0‖φk+r-φk‖E(-k)αr-α,‖φτ‖C0,1α([0,1]τ,H)=‖φτ‖C([0,1]τ,H)+sup⁡1≤k<k+r≤N-1∥φk+r-φk∥E((k+r)τ)α(N-k)αr-α,‖φτ‖C~0,1α([-1,1]τ,H)=‖φτ‖C([-1,1]τ,H)+sup⁡-N≤k<k+2r≤0‖φk+2r-φk‖E(-k)α(2r)-α+sup⁡1≤k<k+r≤N-1‖φk+r-φk‖E((k+r)τ)α(N-k)αr-α,‖φτ‖C~0α([-1,0]τ,H)=‖φτ‖C([-1,0]τ,H)+sup⁡-N≤k<k+2r≤0‖φk+2r-φk‖E(-k)α(2r)-α,respectively.Theorem 2.3.
Nonlocal boundary value problem (1.3) is stable in C([-1,1]τ,H) space.Proof.
By [22], we have
(2.7)‖{uk}1N-1‖C([0,1]τ,H)≤M3(δ)[‖gτ‖C([0,1]τ,H)+‖ξ‖H+‖ψ‖H],
for the solution of the following boundary value problem:
(2.8)-τ-2(uk+1-2uk+uk-1)+Auk=gk,gk=g(tk),tk=kτ,1≤k≤N-1,u0=ξ,uN=ψ.
By [24], we have
(2.9)‖{uk}-N0‖C([-1,0]τ,H)≤M4(δ)[‖fτ‖C([-1,0]τ,H)+‖ξ‖H]
for the solution of an inverse Cauchy difference problem:
(2.10)τ-1(uk-uk-1)-(I+τ2A)Auk-1=(I+τ2A)fk,-(N-1)≤k≤0,u0=ξ.
Then, the proof of Theorem 2.3 is based on stability inequalities (2.7) and (2.9) and on the following estimates:
(2.11)‖ξ‖H≤M5(δ)[‖fτ‖C([-1,0]τ,H)+‖gτ‖C([0,1]τ,H)+‖φ‖H],‖ψ‖H≤M6(δ)[‖fτ‖C([-1,0]τ,H)+‖gτ‖C([0,1]τ,H)+‖φ‖H],
for the solution of boundary value problem (1.3). Estimates (2.11) follow from estimates (2.3) and (2.4) and formula (2.5). This finishes the proof of Theorem 2.3.Theorem 2.4.
Assume thatφ∈D(A) and f0,f-1,g1∈D(I+τB). Then, for the solution of difference problem (1.3), the following almost coercivity inequality holds:
(2.12)‖{τ-2(uk+1-2uk+uk-1)}1N-1‖C([0,1]τ,H)+‖{τ-1(uk-uk-1)}-N+10‖C([-1,0]τ,H)uuuu+‖{Auk}1N-1‖C([0,1]τ,H)+‖{(I+τ2A)Auk-1}-N+10‖C([-1,0]τ,H)uu≤M7(δ)[min⁡{ln⁡1τ,1+|ln⁡‖A‖H→H|}[‖fτ‖C([-1,0]τ,H)+‖gτ‖C([0,1]τ,H)]uuuu+‖Aφ‖H+‖(I+τB)f0‖H+‖(I+τB)g1‖H+‖(I+τB)f-1‖H].Proof.
We have(2.13)‖{τ-2(uk+1-2uk+uk-1)}1N-1‖C([0,1]τ,H)+‖{Auk}1N-1‖C([0,1]τ,H)uu≤M8(δ)[min⁡{ln⁡1τ,1+|ln⁡‖A‖H→H|}‖gτ‖C([0,1]τ,H)+‖Aξ‖H+‖Aψ‖H],
for the solution of boundary value problem (2.8) (see [22]), and we get
(2.14)‖{τ-1(uk-uk-1)}-N+10‖C([-1,0]τ,H)+‖{(I+τ2A)Auk-1}-N+10‖C([-1,0]τ,H)uu≤M9(δ)[min⁡{ln⁡1τ,1+|ln⁡‖A‖H→H|}‖fτ‖C([-1,0]τ,H)+‖Aξ‖H],
for the solution of inverse Cauchy difference problem (2.10) (see [24]). Then, the proof of Theorem 2.4 is based on almost coercivity inequalities (2.13) and (2.14) and on the following estimates:
(2.15)‖Aξ‖H≤M10(δ)[‖Aφ‖H+‖(I+τB)f0‖H+min⁡{ln⁡1τ,1+|ln⁡‖A‖H→H|}M10(δ)M×[‖fτ‖C([-1,0]τ,H)+‖gτ‖C([0,1]τ,H)]1τ],‖Aψ‖H≤M11(δ)[‖Aφ‖H+‖(I+τB)f0‖H+min⁡{ln⁡1τ,1+|ln⁡‖A‖H→H|}M10(δ)M×[‖fτ‖C([-1,0]τ,H)+‖gτ‖C([0,1]τ,H)]1τ]
for the solution of boundary value problem (1.3). Proofs of these estimates follow the scheme of the papers [23, 24] and rely on both formula (2.5) and estimates (2.3) and (2.4). Theorem 2.4 is proved.Theorem 2.5.
Let assumptions of Theorem2.5 be satisfied. Then, boundary value problem (1.3) is well-posed in Hölder spaces C0,1α([-1,1]τ,H),  and  C~0,1α([-1,1]τ,H), and the following coercivity inequalities hold:
(2.16)‖{τ-2(uk+1-2uk+uk-1)}1N-1‖C0,1α([0,1]τ,H)+‖{τ-1(uk-uk-1)}-N+10‖C~0α([-1,0]τ,H)uuuu+‖{Auk}1N-1‖C0,1α([0,1]τ,H)+‖{(I+τ2A)Auk-1}-N+10‖C~0α([-1,0]τ,H)uu≤M12(δ)[1α(1-α)[‖fτ‖C0α([-1,0]τ,H)+‖gτ‖C0,1α([0,1]τ,H)]+‖Aφ‖H+‖(I+τB)f0‖HuuMMM12(δ)+‖(I+τB)g1‖H+‖(I+τB)f-1‖H1α(1-α)],‖{τ-2(uk+1-2uk+uk-1)}1N-1‖C0,1α([0,1]τ,H)+‖{τ-1(uk-uk-1)}-N+10‖C~0α([-1,0]τ,H)uuuu+‖{Auk}1N-1‖C0,1α([0,1]τ,H)+‖{(I+τ2A)Auk-1}-N+10‖C~0α([-1,0]τ,H)uu≤M13(δ)[1α(1-α)[‖fτ‖C~0α([-1,0]τ,H)+‖gτ‖C0,1α([0,1]τ,H)]+‖Aφ‖HuuMMM12(δ)+‖(I+τB)f0‖H+‖(I+τB)g1‖H+‖(I+τB)f-1‖H1α(1-α)].Proof.
By [22, 24], we have
(2.17)‖{τ-2(uk+1-2uk+uk-1)}1N-1‖C0,1α([0,1]τ,H)+‖{Auk}1N-1‖C0,1α([0,1]τ,H)uu≤M14(δ)[1α(1-α)‖gτ‖C0,1α([0,1]τ,H)+‖Aξ‖H+‖Aψ‖H],
for the solution of boundary value problem (2.8), and
(2.18)‖{τ-1(uk-uk-1)}-N+10‖C~0α([-1,0]τ,H)+‖{(I+τ2A)Auk-1}-N+10‖C~0α([-1,0]τ,H)uu≤M15(δ)[1α(1-α)‖fτ‖C0α([-1,0]τ,H)+‖Aξ‖H],(2.19)‖{τ-1(uk-uk-1)}-N+10‖C~0α([-1,0]τ,H)+‖{(I+τ2A)Auk-1}-N+10‖C~0α([-1,0]τ,H)uu≤M16(δ)[1α(1-α)‖fτ‖C~0α([-1,0]τ,H)+‖Aξ‖H]
for the solution of inverse Cauchy difference problem (2.10), respectively. Then, the proof of Theorem 2.5 is based on coercivity inequalities (2.17)–(2.19) and the following estimates:
(2.20)‖Aξ‖H≤M17(δ)[1α(1-α)[‖fτ‖C~0α([-1,0]τ,H)+‖gτ‖C0,1α([0,1]τ,H)]uk+1-2uk+‖Aφ‖H+‖(I+τB)f0‖H+‖(I+τB)g1‖H+‖(I+τB)f-1‖H1α(1-α)],‖Aψ‖H≤M18(δ)[1α(1-α)[∥fτ∥C~0α([-1,0]τ,H)+∥gτ∥C0,1α([0,1]τ,H)]uk+1-2uk+∥Aφ∥H+∥(I+τB)f0∥H+∥(I+τB)g1∥H+∥(I+τB)f-1∥H1α(1-α)]
for the solution of difference scheme (1.3). Proofs of these estimates follow the scheme of the papers [22, 24] and rely on both estimates (2.3) and (2.4) and formula (2.5). This concludes the proof of Theorem 2.5.


## 3. An Application

In this section, an application of these abstract Theorems2.3, 2.4, and 2.5 is considered. In [-1,1]×Ω, let us consider the following boundary value problem for multidimensional elliptic-parabolic equation:
(3.1)-utt-∑r=1n‍(ar(x)uxr)xr=g(t,x),0<t<1,x∈Ω,ut+∑r=1n‍(ar(x)uxr)xr=f(t,x),-1<t<0,x∈Ω,u(t,x)=0,x∈S,-1≤t≤1,u(1,x)=∑i=1J‍αiu(λi,x)+φ(x),∑i=1J‍|αi|≤1,-1≤λ1<λ2<⋯<λi<⋯<λJ≤0,u(0+,x)=u(0-,x),ut(0+,x)=ut(0-,x),x∈Ω¯,
where ar(x)(x∈Ω),φ(x)(φ(x)=0,x∈S),g(t,x)(t∈(0,1),x∈Ω¯), and f(t,x)(t∈(-1,0),x∈Ω¯) are given smooth functions. Here, Ω is the unit open cube in the n-dimensional Euclidean space ℝn(0<xk<1,1≤k≤n) with boundary S,Ω¯=Ω∪S, and ar(x)⩾a>0.The discretization of problem (3.1) is carried out in two steps. In the first step, let us define the following grid sets:
(3.2)Ω~h={x=xm=(h1m1,…,hnmn),m=(m1,…,mn),x0≤mr≤Nr,hrNr=1,r=1,…,n},Ωh=Ω~h∩Ω,Sh=Ω~h∩S.We introduce the Hilbert spacesL2h=L2(Ω¯h),W2h1=W21(Ω¯h), and W2h2=W22(Ω¯h) of the grid functions φh(x)={φ(h1m1,…,hnmn)} defined on Ω¯h, equipped with the following norms:
(3.3)‖φh‖L2h=(∑x∈Ωh¯‍|φh(x)|2h1⋯hn)1/2,‖φh‖W2h1=‖φh‖L2h+(∑x∈Ωh¯‍∑r=1n‍|(φh)xr|2h1⋯hn)1/2,‖φh‖W2h2=‖φh‖L2h+(∑x∈Ωh¯‍∑r=1n‍|(φh)xr|2h1⋯hn)1/2+(∑x∈Ωh¯‍∑r=1n‍|(φh)xrxr¯,mr|2h1⋯hn)1/2.To the differential operatorA generated by problem (3.1), we assign the difference operator Ahx by formula
(3.4)Ahxuh=-∑r=1n‍(ar(x)uxr-h)xr,mr
acting in the space of grid functions uh(x), satisfying the conditions uh(x)=0 for all x∈Sh. With the help of Ahx, we arrive at the following nonlocal boundary value problem:
(3.5)-d2uh(t,x)dt2+Ahxuh(t,x)=gh(t,x),0<t<1,x∈Ωh,duh(t,x)dt-Ahxuh(t,x)=fh(t,x),-1<t<0,x∈Ωh,uh(1,x)=∑k=1n‍αkuh(λk,x)+φh(x),∑k=1n‍|αk|≤1,x∈Ω¯h,uh(0+,x)=uh(0-,x),duh(0+,x)dt=duh(0-,x)dt,x∈Ω¯h,
for an infinite system of ordinary differential equations.In the second step, we replace problem (3.5) by difference scheme (1.3) accurate to the following second order (see [22, 24]):
(3.6)-uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)=gkh(x),gkh(x)=gh(tk,x),tk=kτ,1≤k≤N-1,Nτ=1,x∈Ωh,ukh(x)-uk-1h(x)τ-(Ahx+τ2(Ahx)2)uk-1h(x)=(I+τ2Ahx)fkh(x),fkh(x)=fh(tk-1/2,x),tk-1/2=(k-12)τ,-N+1≤k≤0,x∈Ωh,-u2h(x)+4u1h(x)-3u0h(x)=3u0h(x)-4u-1h(x)+u-2h(x),x∈Ω~h,uNh(x)=∑k=1J‍αi(u[λi/τ]h(x)+(λk-[λiτ]τ)(f[λi/τ]h+Ahxu[λi/τ]h(x)))+φh(x),x∈Ω~h.Theorem 3.1.
Letτ and |h|=h12+⋯+hn2 be sufficiently small positive numbers. Then, solutions of difference scheme (3.6) satisfy the following stability and almost coercivity estimates:
(3.7)‖{ukh}-NN-1‖C([-1,1]τ,L2h)≤M19(δ)[‖{fkh}-N+1-1‖C([-1,0]τ,L2h)+‖{gkh}1N-1‖C([0,1]τ,L2h)+‖φh‖L2h],‖{τ-2(uk+1h-2ukh+uk-1h)}1N-1‖C([0,1]τ,L2h)+‖{ukh}1N-1‖C([0,1]τ,W2h2)uuuk+1h+‖{τ-1(ukh-uk-1h)}-N+10‖C([-1,0]τ,L2h)+‖{uk-1h}-N+10‖C([-1,0]τ,W2h2)uk+1h≤M20(δ)[∑ft5s‖f0h‖L2h+‖f-1h‖L2h+‖g1h‖L2h+‖φh‖W2h2+τ‖f0h‖W2h1+τ‖f-1h‖W2h1+τ‖g1h‖W2h1uk+1huk+1husuuk+1h+ln⁡1τ+|h|[‖{fkh}-N+1-1‖C([-1,0]τ,L2h)+‖{gkh}1N-1‖C([0,1]τ,L2h)]∑s].The proof of Theorem3.1 is based on Theorem 2.3, Theorem 2.4, the symmetry property of the difference operator Ahx defined by formula (3.4), the estimate
(3.8)min⁡{ln⁡1τ,1+|ln⁡‖Ahx‖L2h→L2h|}≤M21(δ)ln⁡1τ+|h|,
and the following theorem on the coercivity inequality for the solution of elliptic difference equation in L2h.Theorem 3.2.
For the solution of the following elliptic difference problem:(3.9)Ahxuh(x)=ωh(x),x∈Ωh,uh(x)=0,x∈Sh,
the following coercivity inequality holds [25]:
(3.10)∑r=1n‍‖(uh)xr-xr,mr‖L2h≤M22(δ)‖ωh‖L2h.Theorem 3.3.
Letτ and |h| be sufficiently small positive numbers. Then, solutions of difference scheme (3.6) satisfy the following coercivity stability estimates:
(3.11)‖{τ-2(uk+1h-2ukh+uk-1h)}1N-1‖C0,1α([0,1]τ,L2h)uuuu+‖{τ-1(ukh-uk-1h)}-N+10‖C~0α([-1,0]τ,L2h)+‖{ukh}1N-1‖C0,1α([0,1]τ,W2h2)uuuu+‖{uk-1h}-N+10‖C~0α([-1,0]τ,W2h2)uuuu≤M23(δ)[∑1α(1-α)‖φh‖W2h2+τ‖f0h‖W2h1+τ‖f-1h‖W2h1+τ‖g1h‖W2h1uuuu≤M24(δ)δδ+1α(1-α)[‖{fkh}-N+1-1‖C0α([-1,0]τ,L2h)+‖{gkh}1N-1‖C0,1α([0,1]τ,L2h)]],‖{τ-2(uk+1h-2ukh+uk-1h)}1N-1‖C0,1α([0,1]τ,L2h)+‖{uk-1h}-N+10‖C~0α([-1,0]τ,W2h2)uuuu+‖{τ-1(ukh-uk-1h)}-N+10‖C~0α([-1,0]τ,L2h)+‖{ukh}1N-1‖C0,1α([0,1]τ,W2h2)uuuu≤M24(δ)[∑1α(1-α)‖φh‖W2h2+τ‖f0h‖W2h1+τ‖f-1h‖W2h1+τ‖g1h‖W2h1uuuu≤M24(δ)δδ+1α(1-α)[‖{fkh}-N+1-1‖C~0α([-1,0]τ,L2h)+‖{gkh}1N-1‖C0,1α([0,1]τ,L2h)]].The proof of Theorem3.3 is based on the abstract Theorem 2.5, Theorem 3.2, and the symmetry property of the difference operator Ahx defined by formula (3.4).



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*Source: 230190-2012-07-09.xml*