# Nonlocal Boundary Value Problems for Elliptic-Parabolic Differential and Difference Equations

Authors: Allaberen Ashyralyev; Okan Gercek
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

The abstract nonlocal boundary value problem-d2u(t)/dt2+Au(t)=g(t),0<t<1,du(t)/dt-Au(t)=f(t),1<t<0,u(1)=u(-1)+μ
for differential equations in a Hilbert space H with the self-adjoint positive definite
operator A is considered. The well-posedness of this problem in Hölder spaces with
a weight is established. The coercivity inequalities for the solution of boundary value
problems for elliptic-parabolic equations are obtained. The first order of accuracy difference
scheme for the approximate solution of this nonlocal boundary value problem
is presented. The well-posedness of this difference scheme in Hölder spaces is established.
In applications, coercivity inequalities for the solution of a difference scheme
for elliptic-parabolic equations are obtained.


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## Body

## 1. Introduction

It is known that various problems in fluid mechanics
and other areas of engineering, physics, and biological systems lead to partial
differential equations of variable types. Methods of solutions of nonlocal
boundary value problems for partial differential equations of variable type
have been studied extensively by many researchers (see, e.g.,
[1–4] and the references given
therein).The 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)=u(−1)+μfor differential equations in a
Hilbert space H with the self-adjoint positive definite
operator A is considered.Let us denote byC0,1α([−1,1],H),0<α<1 the Banach space obtained by completion of the
set of all smooth H-valued function φ(t) on [−1,1] in the norm(1.2)∥φ∥C0,1α([−1,1],H)=∥φ∥C([−1,1],H)+sup−1<t<t+τ<0(−t)α∥φ(t+τ)−φ(t)∥Hτα+sup0<t<t+τ<1(1−t)α(t+τ)α∥φ(t+τ)−φ(t)∥Hτα,and denote by C0,1α([0,1],H),0<α<1 the Banach space obtained by completion of the
set of all smooth H-valued function φ(t) on [0,1] in the norm(1.3)∥φ∥C0,1α([0,1],H)=∥φ∥C([0,1],H)+sup0<t<t+τ<1(1−t)α(t+τ)α∥φ(t+τ)−φ(t)∥Hτα,finally denote by C0α([−1,0],H),0<α<1 the Banach space obtained by completion of the
set of all smooth H-valued function φ(t) on [−1,0] in the norm(1.4)∥φ∥C0α([−1,0],H)=∥φ∥C([−1,0],H)+sup−1<t<t+τ<0(−t)α∥φ(t+τ)−φ(t)∥Hτα.Here C([a,b],H) stands for the Banach space of all continuous
functions φ(t) defined on [a,b] with values in H equipped with the norm(1.5)||φ||C([a,b],H)=maxa≤t≤b∥φ(t)∥H.A function u(t) is called a solution of problem (1.1) if the
following conditions are satisfied.(i)
u(t) is twice continuously differentiable on the
segment (0,1] and continuously differentiable on the segment [−1,1];
the derivatives at the endpoints of the segment are understood as the
appropriate unilateral derivatives.(ii)
The elementu(t) belongs to the domain D(A) of A for all t∈[−1,1],
and the function Au(t) is continuous on the segment [−1,1].(iii)
u(t) satisfies the equations and the nonlocal
boundary condition (1.1).A solution of problem (1.1) defined in this manner
will henceforth be referred to as a solution of problem (1.1) in the space C(H)=C([−1,1],H).We say that problem (1.1) is well-posed in C(H), if there exists a unique solution u(t) in C(H) of problem (1.1) for any g(t)∈C([0,1],H),f(t)∈C([−1,0],H), and μ∈D(A), and  the
following coercivity inequality is satisfied:(1.6)∥u′′∥C([0,1],H)+∥u′∥C([−1,0],H)+∥Au∥C(H)≤M[∥g∥C([0,1],H)+∥f∥C([−1,0],H)+∥Aμ∥H],where M is independent of μ,f(t), and g(t).Problem (1.1) is not well-posed in C(H) [5]. The well-posedness of the boundary value problem
(1.1) can be established if one considers this problem in certain spaces F(H) of smooth H-valued functions on [−1,1].A functionu(t) is said to be a solution of problem (1.1) in F(H) if it is a solution of this problem in C(H) and the functions u′′(t)(t∈0,1]),u′(t)(t∈−1,1]) and Au(t)(t∈−1,1]) belong to F(H).As in the case of the spaceC(H), we say that problem (1.1) is well-posed in F(H), if the following coercivity inequality is
satisfied:(1.7)∥u′′∥F([0,1],H)+∥u′∥F([−1,0],H)+∥Au∥F(H)≤M[∥g∥F([0,1],H)+∥f∥F([−1,0],H)+∥Aμ∥H],where M is independent of μ,f(t), and g(t).If we setF(H) equal to C0,1α(H)=C0,1α([−1,1],H)(0<α<1),
then we can establish the following coercivity inequality.Theorem 1.1.
Supposeμ∈D(A). Then the boundary value problem (1.1) is
well-posed in a Hölder space C0,1α(H) and the following coercivity inequality
holds:(1.8)∥u′′∥C0,1α([0,1],H)+∥u′∥C0α([−1,0],H)+∥Au∥C0,1α(H)≤M[1α(1−α)[∥f∥C0α([−1,0],H)+∥g∥C0,1α([0,1],H)]+∥Aμ∥H].Here M is independent of f(t),g(t), and μ.The proof of this assertion follows from the scheme of
the proof of the theorem on well-posedness of paper [5] and is based on the
following formulas:(1.9)u(t)=(I−e−2A1/2)−1[(e−tA1/2−e−(−t+2)A1/2)u0+(e−(1−t)A1/2−e−(t+1)A1/2)u1]+(I−e−2A1/2)−1×(e−(1−t)A1/2−e−(t+1)A1/2)∫01A−1/22−1(e−(1−s)A1/2−e−(s+1)A1/2)g(s)ds−∫01A−1/22−1(e−(t+s)A1/2−e−|t−s|A1/2)g(s)ds,0≤t≤1,u(t)=etAu0+∫0te(t−s)Af(s)ds,−1≤t≤0,u0=(I+e−2A1/2+A1/2(I−e−2A1/2)−2e−(A1/2+A))−1×[e−A1/2[2∫0−1e−(1+s)Af(s)ds+∫01A−1/2(e−(1−s)A1/2−e−(s+1)A1/2)g(s)ds]+2e−A1/2μ]+(I−e−2A1/2)(I+e−2A1/2+A1/2(I−e−2A1/2)−2e−(A1/2+A))−1×[−A−1/2f(0)+∫01A−1/2e−sA1/2g(s)ds]for the solution of problem
(1.1)  and on the
estimates(1.10)∥(I−e−2A1/2)−1∥H→H≤M,∥(I+e−2A1/2+A1/2(I−e−2A1/2)−2e−(A1/2+A))−1∥H→H≤M,∥A1/2(I+e−2A1/2+A1/2(I−e−2A1/2)−2e−(A1/2+A))−1∥H→H≤M,∥(A1/2)αe−tA1/2∥H→H≤t−α,t>0,0≤α≤1,∥Aαe−tA||H→H≤t−α,t>0,0≤α≤1.Remark 1.2.
The nonlocal
boundary value problem for the elliptic-parabolic equation(1.11)du(t)dt+Au(t)=f(t),0<t<1,−d2u(t)dt2+Au(t)=g(t),−1<t<0,u(1)=u(−1)+μin a Hilbert space H with a self-adjoint positive definite
operator A is considered in paper [6]. The well-posedness of this
problem in Hölder spaces Cα(H) without a weight was established under the
strong condition on μ.Now, the applications of this abstract results are
presented.First, the mixed boundary value problem for the
elliptic-parabolic equations(1.12)−utt−(a(x)ux)x+δu=g(t,x),0<t<1,0<x<1,ut+(a(x)ux)x−δu=f(t,x),−1<t<0,0<x<1,u(t,0)=u(t,1),ux(t,0)=ux(t,1),−1≤t≤1,u(1,x)=u(−1,x)+μ(x),0≤x≤1,u(0+,x)=u(0−,x),ut(0+,x)=ut(0−,x),0≤x≤1is considered. Problem (1.12) has
a unique smooth solution u(t,x) for a(x)≥a>0(x∈(0,1)), and g(t,x)(t∈0,1],x∈[0,1]),f(t,x)(t∈[−1,0],x∈0,1]) the smooth functions and δ=const>0. This allows us to reduce the mixed problem
(1.12) to the nonlocal boundary value problem (1.1) in the Hilbert space H=L2[0,1] with a self-adjoint positive definite operator A defined by (1.12).Theorem 1.3.
The
solutions of the nonlocal boundary value problem (1.12) satisfy the coercivity
inequality(1.13)∥utt∥C0,1α([0,1],L2[0,1])+∥ut∥C0α([−1,0],L2[0,1])+∥u∥C0,1α([−1,1],W22[0,1])≤M[1α(1−α)[∥g∥C0,1α([0,1],L2[0,1])+∥f∥C0α([−1,0],L2[0,1])]+∥μ∥W22[0,1]],where M is independent of f(t,x),g(t,x),  and μ(x).The proof of Theorem1.3 is based on the abstract
Theorem 1.1 and the symmetry properties of the space operator
are generated by problem (1.12).Second, letΩ be the unit open cube in the n-dimensional Euclidean space ℝn(0<xk<1,1≤k≤n) with boundary S,Ω¯=Ω∪S.
In [−1,1]×Ω, the boundary value problem for the
multidimensional elliptic-parabolic equation(1.14)−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)=u(−1,x)+μ(x),x∈Ω¯,u(0+,x)=u(0−,x),ut(0+,x)=ut(0−,x),x∈Ω¯is considered. Problem (1.14) has
a unique smooth solution u(t,x) for ar(x)≥a>0(x∈Ω) and g(t,x)(t∈(0,1),x∈Ω¯),f(t,x)(t∈(−1,0),x∈Ω¯), the smooth functions. This allows us to reduce
the mixed problem (1.14) to the nonlocal boundary value problem (1.1) in the
Hilbert space H=L2(Ω¯) of all the integrable functions defined on Ω¯, equipped with
the norm(1.15)∥f∥L2(Ω¯)={∫⋯∫x∈Ω¯|f(x)|2dx1⋯dxn}1/2with a self-adjoint positive
definite operator A defined by (1.14).Theorem 1.4.
The
solution of the nonlocal boundary value problem (1.14) satisfies the coercivity
inequality(1.16)∥utt∥C0,1α([0,1],L2(Ω¯))+∥ut∥C0α([−1,0],L2(Ω¯)])+∥u∥C0,1α([−1,1],W22(Ω¯))≤M[1α(1−α)[∥g∥C0,1α([0,1],L2(Ω¯))+∥f∥C0α([−1,0],L2(Ω¯))]+∥μ∥W22(Ω¯)],where M is independent of f(t,x),g(t,x),  and μ(x).The proof of Theorem1.4 is based on the abstract
Theorem 1.1 and the symmetry properties of the space operator generated by
problem (1.14) and the following theorem on the coercivity inequality for the
solution of the elliptic differential problem in L2(Ω¯).Theorem 1.5.
For
the solution of the elliptic differential problem(1.17)∑r=1n(ar(x)uxr)xr=ω(x),x∈Ω,(1.18)u(x)=0,x∈S,the following coercivity
inequality holds [7]: (1.19)∑r=1n∥uxrxr∥L2(Ω¯)≤M||ω||L2(Ω¯).


## 2. The First Order of Accuracy Difference Scheme

Let us associate the boundary-value problem (1.1) with
the corresponding first order of accuracy difference scheme (2.1)−τ−2(uk+1−2uk+uk−1)+Auk=gk,gk=g(tk),tk=kτ,1≤k≤N−1,τ−1(uk−uk−1)−Auk−1=fk,fk=f(tk−1),tk−1=(k−1)τ,−N+1≤k≤0,uN=u−N+μ,u1−u0=u0−u−1.A study of
discretization, over time only, of the nonlocal boundary value problem also
permits one to include general difference schemes in applications if the
differential operator in space variables, A,
is replaced by the difference operators Ah that act in the Hilbert spaces Hh and are uniformly self-adjoint positive
definite in h for 0<h≤h0.LetP=P(τA)=(I+τA)−1. Then the following estimates are satisfied
[8]:(2.2)∥Pk∥H→H≤M(1+δτ)−k,kτ∥APk∥H→H≤M,k≥1,δ>0,(2.3)∥Aβ(Pk+r−Pk)∥H→H≤M(rτ)α(kτ)α+β,1≤k<k+r≤N,0≤α,β≤1.Furthermore, for a self-adjoint
positive definite operator A it follows that the operator R=(I+τB)−1 is defined on the whole space H, it is a bounded operator, and the following
estimates hold:(2.4)∥Rk∥H→H≤M(1+δτ)−k,kτ∥BRk∥H→H≤M,k≥1,δ>0,(2.5)∥Bβ(Rk+r−Rk)∥H→H≤M(rτ)α(kτ)α+β,1≤k<k+r≤N,0≤α,β≤1.Here B=(1/2)(τA+A(4+τ2A)). From (2.2) and (2.4), it follows that(2.6)||(I−R2N)−1||H→H≤M,(2.7)∥(I+(I+τA)(I+2τA)−1R2N−1+B−1A(I+2τA)−1(I−R2N−1)−(2I+τB)(I+2τA)−1RNPN−1)−1∥H→H≤M.Theorem 2.1.
For
anygk,1≤k≤N−1 and fk,−N+1≤k≤0, the solution of problem (2.1) exists and the
following formulas hold:(2.8)uk=(I−R2N)−1{[Rk−R2N−k]u0+[RN−k−RN+k][PNu0−τ∑s=−N+10Ps+Nfs+μ]−[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,(2.9)uk=P−ku0−τ∑s=k+10Ps−kfs,−N≤k≤0,(2.10)u0=Tτ(I+2τA)−1(I+τA){{(2+τB)RN[−τ∑s=−N+10Ps+Nfs+μ]−RN−1B−1∑s=1N−1[RN−s−RN+s]gsτ}+(I−R2N)B−1∑s=1N−1Rs−1gsτ−(I−R2N)(I+τB)B−1Pf0},where(2.11)Tτ=(I+(I+τA)(I+2τA)−1R2N−1+B−1A(I+2τA)−1(I−R2N−1)−(2I+τB)(I+2τA)−1RNPN−1)−1.Proof.
By
[8, 9],(2.12)uk=(I−R2N)−1{[Rk−R2N−k]ξ+[RN−k−RN+k]ψ−[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,is the solution of the boundary
value difference problem(2.13)−τ−2(uk+1−2uk+uk−1)+Auk=gk,gk=g(tk),tk=kτ,1≤k≤N−1,u0=ξ,uN=ψ,(2.14)uk=P−kξ−τ∑s=k+10Ps−kfs,−N≤k≤0is the solution of the inverse Cauchy
problem(2.15)τ−1(uk−uk−1)−Auk−1=fk,fk=f(tk−1),tk−1=(k−1)τ,−N+1≤k≤0,u0=ξ.Exploiting (2.12), (2.14), and
the formulas(2.16)ψ=u−N+μ,ξ=u0,we obtain formulas (2.8) and
(2.9). For u0, using (2.8), (2.9), and the
formula(2.17)u1−u0=u0−u−1,we obtain the operator
equation(2.18)(I−R2N)−1{[R−R2N−1]u0+[RN−1−RN+1]×[PNu0−τ∑s=−N+10Ps+Nfs+μ]−[RN−1−RN+1](I+τB)(2I+τB)−1B−1∑s=1N−1[RN−s−RN+s]gsτ}+(I+τB)(2I+τB)−1B−1∑s=1N−1[Rs−1−R1+s]gsτ=2u0−Pu0+τPf0.The operator(2.19)I+(I+τA)(I+2τA)−1R2N−1+B−1A(I+2τA)−1(I−R2N−1)−(2I+τB)(I+2τA)−1RNPN−1has an inverse
(2.20)Tτ=(I+(I+τA)(I+2τA)−1R2N−1+B−1A(I+2τA)−1(I−R2N−1)−(2I+τB)(I+2τA)−1RNPN−1)−1,and the following
formula(2.21)u0=Tτ(I+τA)(I+2τA)−1{{(2+τB)RN[−τ∑s=−N+10Ps+Nfs+μ]−RN−1B−1∑s=1N−1[RN−s−RN+s]gsτ}+(I−R2N)B−1∑s=1N−1Rs−1gsτ−(I−R2N)(I+τB)B−1Pf0}is satisfied. This concludes the
proof of Theorem 2.1.LetFτ(H)=F([a,b]τ,H) be the linear space of mesh functions φτ={φk}NaNb defined on [a,b]τ={tk=kh,Na≤k≤Nb,Naτ=a,Nbτ=b} with values in the Hilbert space H. Next on Fτ(H) we denote by C([a,b]τ,H) and C0,1α([−1,1]τ,H),C0,1α([−1,0]τ,H),C0α([0,1]τ,H)(0<α<1) Banach spaces with the norms(2.22)∥φτ∥C([a,b]τ,H)=maxNa≤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)αrα+sup1≤k<k+r≤N−1∥φk+r−φk∥E((k+r)τ)α(N−k)αrα,∥φτ∥C0α([−1,0]τ,H)=∥φτ∥C([−1,0]τ,H)+sup−N≤k<k+r≤0∥φk+r−φk∥E(−k)αrα,∥φτ∥C0,1α([0,1]τ,H)=∥φτ∥C([0,1]τ,H)+sup1≤k<k+r≤N−1∥φk+r−φk∥E((k+r)τ)α(N−k)αrα.The nonlocal boundary value
problem (2.1) is said to be stable in F([−1,1]τ,H) if we have the inequality(2.23)∥uτ∥F([−1,1]τ,H)≤M[∥fτ∥F([−1,0]τ,H)+∥gτ∥F([0,1]τ,H)+∥μ∥H],where M is independent of not only fτ,gτ,μ but also τ.Theorem 2.2.
The  nonlocal boundary  value
problem  (2.1) is  stable in C([−1,1]τ,H) norm.Proof.
By
[9],(2.24)∥{uk}−N0∥C([−1,0]τ,H)≤M[∥fτ∥C([−1,0]τ,H)+∥u0∥H]for the solution of the inverse
Cauchy difference problem (2.15) and(2.25)∥{uk}1N−1∥C([0,1]τ,H)≤M[∥gτ∥C([0,1]τ,H)+∥u0∥H+∥uN∥H]for the solution of the boundary
value problem (2.13). The proof of Theorem 2.2 is based on the stability
inequalities (2.24), (2.25), and on the estimates(2.26)∥u0∥H≤M[∥fτ∥C([−1,0]τ,H)+∥gτ∥C([0,1]τ,H)+∥μ∥H],∥uN∥H≤M[∥fτ∥C([−1,0]τ,H)+∥gτ∥C([0,1]τ,H)+∥μ∥H]for the solution of the boundary
value problem (2.1). Estimates (2.26)  are derived from formula (2.10) and estimates
(2.2), (2.4), (2.7). This concludes the proof of Theorem 2.2.The nonlocal boundary value problem (2.1) is said to
be coercively stable (well-posed) in F([−1,1]τ,H) if we have the coercive
inequality(2.27)∥{τ−2(uk+1−2uk+uk−1)}1N−1∥F([0,1]τ,H)+∥{τ−1(uk−uk−1)}−N+10∥F([−1,0]τ,H)+∥{Auk}−NN−1∥F([−1,1]τ,H)≤M[∥fτ∥F([−1,0]τ,H)+∥gτ∥F([0,1]τ,H)+∥Aμ∥H],where M is independent of not only fτ,gτ,μ but also τ.Since the nonlocal boundary value problem (1.1) in the
space C([0,1],H) of continuous functions defined on [−1,1] and with values in H is not well-posed for the general positive
unbounded operator A and space H,
then the well-posedness of the difference nonlocal boundary value problem (2.1)
in C([−1,1]τ,H) norm does not take place uniformly with
respect to τ>0.
This means that the coercive norm(2.28)∥uτ∥Kτ(E)=∥{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)+∥{Auk}−NN−1∥C([−1,1]τ,H)tends to ∞ as τ→0+. The investigation of the difference problem
(2.1) permits us to establish the order of growth of this norm to ∞.Theorem 2.3.
Assume
thatμ∈D(A) and f0∈D(I+τB). Then for the solution of the difference
problem (2.1) we have the almost coercivity inequality(2.29)∥uτ∥Kτ(E)≤M[∥Aμ∥H+∥(I+τB)f0∥H+min{ln1τ,1+|ln∥A∥H→H|}[∥fτ∥C([−1,0]τ,H)+∥gτ∥C([0,1]τ,H)]],where M is independent of not only fτ,gτ,μ but also τ.Proof.
By
[9],(2.30)∥{τ−1(uk−uk−1)}−N+10∥C([−1,0]τ,H)+∥{Auk}−N0∥C([−1,0]τ,H)≤M[min{ln1τ,1+|ln∥A∥H→H|}∥fτ∥C([−1,0]τ,H)+∥Au0∥H] for the solution of the inverse Cauchy
difference problem (2.15) and(2.31)∥{τ−2(uk+1−2uk+uk−1)}1N−1∥C([0,1]τ,H)+∥{Auk}1N−1∥C([0,1]τ,H)≤M[min{ln1τ,1+|ln∥A∥H→H|}∥gτ∥C([0,1]τ,H)+∥Au0∥H+∥AuN∥H]for the solution of the boundary
value problem (2.13). Then the proof of Theorem 2.3 is based on the almost
coercivity inequalities (2.30), (2.31), and on the estimates(2.32)∥Au0∥H≤M[∥Aμ∥H+∥(I+τB)f0∥H+min{ln1τ,1+|ln∥A∥H→H|}[∥fτ∥C([−1,0]τ,H)+∥gτ∥C([0,1]τ,H)]],∥AuN∥H≤M[∥Aμ∥H+∥(I+τB)f0∥H+min{ln1τ,1+|ln∥A∥H→H|}[∥fτ∥C([−1,0]τ,H)+∥gτ∥C([0,1]τ,H)]]for the solution of the boundary
value problem (2.1). The proof of these estimates follows the scheme of papers
[8, 9] and relies on formula (2.10)
and on estimates (2.2), (2.4), and (2.7). This concludes the proof of Theorem
2.3.Theorem 2.4.
Let
the assumptions of Theorem2.3 be satisfied. Then the boundary value problem
(2.1) is well-posed in a Hölder space C0,1α([−1,1]τ,H) and the following coercivity inequality
holds:(2.33)∥{τ−2(uk+1−2uk+uk−1)}1N−1∥C0,1α([0,1]τ,H)+∥{Auk}−NN−1∥C0,1α([−1,1]τ,H)+∥{τ−1(uk−uk−1)}−N+10∥C0α([−1,0]τ,H)≤M[∥Aμ∥H+∥(I+τB)f0∥H+1α(1−α)[∥fτ∥C0α([−1,0]τ,H)+∥gτ∥C0,1α([0,1]τ,H)]],where M is independent of not only fτ,gτ,μ but also τ and α.Proof.
By
[8, 9],(2.34)∥{τ−1(uk−uk−1)}−N+10∥C0α([−1,0]τ,H)+∥{Auk}−N0∥C0α([−1,0]τ,H)≤M[1α(1−α)∥fτ∥C0α([−1,0]τ,H)+∥Au0∥H]for the solution of the inverse
Cauchy difference problem (2.15) and(2.35)∥{τ−2(uk+1−2uk+uk−1)}1N−1∥C0,1α([0,1]τ,H)+∥{Auk}1N−1∥C0,1α([0,1]τ,H)≤M[1α(1−α)∥gτ∥C0,1α([0,1]τ,H)+∥Au0∥H+∥AuN∥H]for the solution of the boundary
value problem (2.13). Then the proof of Theorem 2.4 is based on the coercivity
inequalities (2.34), (2.35), and on the estimates(2.36)∥Au0∥H≤M[∥Aμ∥H+∥(I+τB)f0∥H+1α(1−α)[∥fτ∥C0α([−1,0]τ,H)+∥gτ∥C0,1α([0,1]τ,H)]],∥AuN∥H≤M[∥Aμ∥H+∥(I+τB)f0∥H+1α(1−α)[∥fτ∥C0α([−1,0]τ,H)+∥gτ∥C0,1α([0,1]τ,H)]]for the solution of the boundary
value problem (2.1). Estimates (2.36)  are derived from the
formulas(2.37)Au0=Tτ(I+2τA)−1(I+τA)×{{(2+τB)RN[−τ∑s=−N+10APs+N(fs−f−N+1)+Aμ]−RN−1AB−2{∑s=1N−1BRN−s(gs−gN−1)τ+∑s=1N−1BRN+s(g1−gs)τ}}+(I−R2N)AB−2∑s=1N−1BRs−1(gs−g1)τ}+Tτ(I+2τA)−1(I+τA){{(2+τB)RN(PN−I)f−N+1−RN−1AB−2{(I−RN−1)gN−1−(RN−2−R2N−1)g1}}+(I−R2N)AB−2(I−RN−1)g1−(I−R2N)(I+τB)B−1APf0},AuN=PN{Tτ(I+2τA)−1(I+τA)×{{(2+τB)RN[−τ∑s=−N+10APs+N(fs−f−N+1)+Aμ]−RN−1AB−2{∑s=1N−1BRN−s(gs−gN−1)τ+∑s=1N−1BRN+s(g1−gs)τ}}+(I−R2N)AB−2∑s=1N−1BRs−1(gs−g1)τ}}−τ∑s=−N+10APs+N(fs−f−N+1)+Aμ+(PN−I)f−N+1+PN{Tτ(I+2τA)−1(I+τA){{(2+τB)RN(PN−I)f−N+1−RN−1AB−2{(I−RN−1)gN−1−(RN−2−R2N−1)g1}}+(I−R2N)AB−2(I−RN−1)g1−(I−R2N)(I+τB)B−1APf0}}for the solution of problem (2.1)
and estimates (2.2), (2.4), and (2.7). This concludes the proof of Theorem 2.4.Now, the applications of this abstract result to the
approximate solution of the mixed boundary value problem for the
elliptic-parabolic equation (1.14) are considered. The discretization of problem
(1.14) is carried out in two steps. In the first step, the grid
sets(2.38)Ω˜h={x=xm=(h1m1,…,hnmn),m=(m1,…,mn),0≤mr≤Nr,hrNr=1,r=1,…,n}Ωh=Ω˜h∩Ω,Sh=Ω˜h∩Sare defined. To the differential
operator A generated by problem (1.14) we assign the
difference operator Ahx by the formula(2.39)Ahxuxh=−∑r=1n(ar(x)ux¯rh)xr,jracting 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 nonlocal boundary-value
problem(2.40)−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)=uh(1,x)+μh(x),x∈Ω˜h,uh(0+,x)=uh(0−,x),duh(0+,x)dt=duh(0−,x)dt,x∈Ω˜hfor an infinite system of
ordinary differential equations.In the second step problem (2) is replaced by the
difference scheme (2.1):(2.41)−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−Ahxuk−1h(x)=fkh(x),fkh(x)=fh(tk,x),tk−1=(k−1)τ,−N+1≤k≤−1,x∈Ωh,u−Nh(x)=uNh(x)+μh(x),x∈Ω˜h,u1h(x)−u0h(x)=u0h(x)−u−1h(x),x∈Ω˜h.Based on the number of
corollaries of the abstract theorems given above, to formulate the result, one
needs to introduce the space L2h=L2(Ω˜h) of all the grid functions φh(x)={φ(h1m1,…,hnmn)} defined on Ω˜h, equipped with the norm(2.42)∥φh∥L2(Ω˜h)=(∑x∈Ωh¯|φh(x)|2h1⋯hn)1/2.Theorem 2.5.
Letτ and |h|=h12+⋯+hn2 be sufficiently small numbers. Then the
solutions of the difference scheme (2.41) satisfy the following stability and
almost coercivity estimates:(2.43)∥{ukh}−NN−1∥C([−1,1]τ,L2h)≤M[∥{fkh}−N+1−1∥C([−1,0]τ,L2h)+∥{gkh}1N−1∥C([0,1]τ,L2h)+∥μh∥L2h],∥{τ−2(uk+1h−2ukh+uk−1h)}1N−1∥C([0,1]τ,L2h)+∥{τ−1(ukh−uk−1h)}−N+10∥C([−1,0]τ,L2h)+∥{ukh}−NN−1∥C([−1,1]τ,W2h2)≤M[∥μh∥W2h2+τ∥f0h∥W2h1+ln1τ+|h|[∥{fkh}−N+1−1∥C([−1,0]τ,L2h)+∥{gkh}1N−1∥C([0,1]τ,L2h)]]where M is independent of τ,h,μh(x), and gkh(x),1≤k≤N−1,fkh,−N+1≤k≤0.The proof of Theorem2.5 is based on the abstract
Theorems 2.2,  2.3, on the estimate(2.44)min{ln1τ,1+|ln∥Ahx∥L2h→L2h|}≤Mln1τ+|h|as well as the symmetry
properties of the difference operator Ahx defined by formula (2.39) in L2h, along with the following theorem on the
coercivity inequality for the solution of the elliptic difference problem in L2h.Theorem 2.6.
For
the solution of the elliptic difference problem,(2.45)Ahxuh(x)=ωh(x),x∈Ωh,(2.46)uh(x)=0,x∈Sh,the following coercivity
inequality holds [7]:(2.47)∑r=1n∥(uh)x¯rxr,jr∥L2h≤M||ωh||L2h.Theorem 2.7.
Letτ and |h| be sufficiently small numbers. Then the
solutions of the difference scheme (2.41) satisfy the following coercivity
stability estimates:(2.48)∥{τ−2(uk+1h−2ukh+uk−1h)}1N−1∥C0,1α([0,1]τ,L2h)+∥{τ−1(ukh−uk−1h)}−N+10∥C0α([−1,0]τ,L2h)+∥{ukh}−NN−1∥C0,1α([−1,1]τ,W2h2)≤M[∥μh∥W2h2+τ∥f0h∥W2h1+1α(1−α)[∥{fkh}−N+1−1∥C0α([−1,0]τ,L2h)+∥{gkh}1N−1∥C0,1α([0,1]τ,L2h)]],where M is independent of τ,h,μh(x), and gkh(x),1≤k≤N−1,fkh,−N+1≤k≤0.The proof of Theorem2.7 is based on the abstract
Theorem 2.4, the symmetry properties of the difference operator Ahx defined by formula (2.39), and on  Theorem
2.6 on the coercivity inequality for the solution of the elliptic difference
equation (2.45) in L2h.Note that in a similar manner the difference schemes
of the first order of accuracy with respect to one variable for approximate
solutions of the boundary value problem (1.12) can be constructed. Abstract
theorems given above permit us to obtain the stability, the almost stability,
and the coercive stability estimates for the solution of these difference
schemes.



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*Source: 904824-2008-11-10.xml*