Nonlocal Boundary Value Problems for

Volume 2008, Article ID 904824,16pages doi:10.1155/2008/904824

Research Article

Nonlocal Boundary Value Problems for

Elliptic-Parabolic Differential and Difference Equations

Allaberen Ashyralyev¹ and Okan Gercek²

1Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

2Vocational School, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

Correspondence should be addressed to Okan Gercek,[email protected] Received 30 June 2008; Accepted 17 September 2008

Recommended by Yong Zhou

The abstract nonlocal boundary value problem−d²ut/dt²Aut gt,0< t <1, dut/dt− Aut ft,1 < t < 0, u1 u−1 μfor differential equations in a Hilbert spaceH with the self-adjoint positive definite operatorAis considered. The well-posedness of this problem in H ¨older 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 H ¨older spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained.

Copyrightq2008 A. Ashyralyev and O. Gercek. 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.

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 researcherssee, e.g.,1–4and the references given therein.

The nonlocal boundary value problem

−d²ut

dt² Aut gt, 0< t <1, dut

dt −Aut ft, −1< t <0, u1 u−1 μ

1.1

for differential equations in a Hilbert spaceHwith the self-adjoint positive definite operator Ais considered.

Let us denote byC^α_0,1−1,1, H, 0< α <1 the Banach space obtained by completion of the set of all smoothH-valued functionϕt on−1,1in the norm

ϕC^α_0,1−1,1,HϕC−1,1,H sup

−1<t<tτ<0

−t^αϕtτ−ϕtH

τ^α sup

0<t<tτ<1

1−t^αtτ^αϕtτ−ϕtH

τ^α ,

1.2

and denote byC^α_0,10,1, H, 0< α < 1 the Banach space obtained by completion of the set of all smoothH-valued functionϕt on0,1in the norm

ϕC^α_0,10,1,Hϕ_C0,1,H sup

0<t<tτ<1

1−t^αtτ^αϕtτ−ϕtH

τ^α , 1.3

finally denote byC^α₀−1,0, H, 0 < α < 1 the Banach space obtained by completion of the set of all smoothH-valued functionϕt on−1,0in the norm

ϕC₀^α−1,0,HϕC−1,0,H sup

−1<t<tτ<0

−t^αϕtτ−ϕtH

τ^α . 1.4

Here Ca, b, H stands for the Banach space of all continuous functions ϕt defined on a, bwith values inHequipped with the norm

||ϕ||_Ca,b,Hmax

a≤t≤bϕtH. 1.5

A functionutis called a solution of problem1.1if the following conditions are satisfied.

iut 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.

iiThe element ut belongs to the domain DA of A for all t ∈ −1,1, and the functionAutis continuous on the segment−1,1.

iiiutsatisfies the equations and the nonlocal boundary condition1.1.

A solution of problem1.1defined in this manner will henceforth be referred to as a solution of problem1.1in the spaceCH C−1,1, H.

We say that problem1.1is well-posed inCH,if there exists a unique solutionut inCHof problem1.1for anygt∈C0,1, H, ft∈C−1,0, H,andμ∈DA, and the following coercivity inequality is satisfied:

u_C0,1,Hu_C−1,0,HAu_CH ≤M

g_C0,1,Hf_C−1,0,HAμ_H

, 1.6 whereMis independent ofμ, ft,andgt.

Problem 1.1 is not well-posed in CH 5. The well-posedness of the boundary value problem1.1can be established if one considers this problem in certain spacesFH of smoothH-valued functions on−1,1.

A functionutis said to be a solution of problem1.1inFHif it is a solution of this problem inCHand the functionsut t∈0,1, ut t∈ −1,1andAut t∈ −1,1 belong toFH.

As in the case of the spaceCH,we say that problem1.1is well-posed inFH,if the following coercivity inequality is satisfied:

u_F0,1,Hu_F−1,0,HAu_FH≤M

g_F0,1,Hf_F−1,0,HAμ_H

, 1.7 whereM is independent ofμ, ft,andgt.

If we setFHequal toC^α_0,1H C^α_0,1−1,1, H 0 < α < 1, then we can establish the following coercivity inequality.

Theorem 1.1. Supposeμ∈DA.Then the boundary value problem1.1is well-posed in a H¨older spaceC^α_0,1Hand the following coercivity inequality holds:

u_C^α

0,10,1,Hu_C^α

0−1,0,HAuC^α_0,1H

≤M 1

α1−α f_C^α

0−1,0,Hg_C^α

0,10,1,H

Aμ_H

. 1.8

HereMis independent offt, gt, andμ.

The proof of this assertion follows from the scheme of the proof of the theorem on well-posedness of paper5and is based on the following formulas:

ut

I−e^{−2A1/2 −1}

e^−tA1/2−e^−−t2A1/2 u₀

e^−1−tA1/2−e^−t1A1/2 u₁

I−e^{−2A1/2 −1}

×

e^−1−tA1/2−e^−t1A1/2

1 0

A^−1/22⁻¹

e^−1−sA1/2−e^−s1A1/2 gsds

− ₁

0

A^−1/22⁻¹

e^−tsA1/2−e^{−|t−s|A1/2} gsds, 0≤t≤1, ut e^tAu₀

_t

0

e^t−sAfsds, −1≤t≤0, u₀

Ie^−2A1/2A^1/2

I−e^−2A1/2 −2e^{−A1/2A −1}

×

e^−A1/2

2 ₋₁

0

e^−1sAfsds ₁

0

A^−1/2

e^−1−sA1/2−e^−s1A1/2 gsds

2e^−A1/2μ

I−e^−2A1/2 Ie^−2A1/2A^1/2

I−e^−2A1/2 −2e^{−A1/2A −1}

×

−A^−1/2f0 ₁

0

A^−1/2e^−sA1/2gsds

1.9

for the solution of problem1.1and on the estimates

I−e^{−2A1/2 −1}

H→H≤M,

Ie^−2A1/2A^1/2

I−e^−2A1/2 −2e^{−A1/2A −1}

H→H≤M, A^1/2

Ie^−2A1/2A^1/2

I−e^−2A1/2 −2e^{−A1/2A −1}

H→H≤M, A^1/2αe^−tA1/2

H→H≤t^−α, t >0, 0≤α≤1, A^αe^−tA||_H→H≤t^−α, t >0, 0≤α≤1.

1.10

Remark 1.2. The nonlocal boundary value problem for the elliptic-parabolic equation

dut

dt Aut ft, 0< t <1,

−d²ut

dt² Aut gt, −1< t <0, u1 u−1 μ

1.11

in a Hilbert spaceHwith a self-adjoint positive definite operatorAis considered in paper 6. The well-posedness of this problem in H ¨older spaces C^αH without a weight was established under the strong condition onμ.

Now, the applications of this abstract results are presented.

First, the mixed boundary value problem for the elliptic-parabolic equations

ga−u_tt−axux_xδugt, x, 0< t <1, 0< x <1, ut axux_x−δuft, x, −1< t <0, 0< x <1,

ut,0 ut,1, uxt,0 uxt,1, −1≤t≤1, u1, x u−1, x μx, 0≤x≤1, u0, x u0−, x, ut0, x u_t0−, x, 0≤x≤1

1.12

is considered. Problem1.12 has a unique smooth solutionut, xfor ax ≥ a > 0 x ∈ 0,1,andgt, x t∈0,1, x∈0,1, ft, x t∈−1,0, x∈0,1the smooth functions and δ const> 0.This allows us to reduce the mixed problem1.12to the nonlocal boundary value problem1.1 in the Hilbert space H L20,1with a self-adjoint positive definite operatorAdefined by1.12.

Theorem 1.3. The solutions of the nonlocal boundary value problem1.12satisfy the coercivity inequality

uttC_0,1^α0,1,L20,1utC^α₀−1,0,L20,1uC^α_0,1−1,1,W₂²0,1

≤M 1

α1−α

gC^α_0,10,1,L20,1fC^α₀−1,0,L20,1

μ_W²

20,1

, 1.13

whereMis independent offt, x, gt, x, andμx.

The proof of Theorem 1.3 is based on the abstract Theorem 1.1 and the symmetry properties of the space operator are generated by problem1.12.

Second, letΩbe the unit open cube in then-dimensional Euclidean spaceRⁿ 0< x_k<

1,1≤k≤nwith boundaryS, Ω Ω∪S. In−1,1×Ω,the boundary value problem for the multidimensional elliptic-parabolic equation

−utt−ⁿ

r1

arxuxr_xr gt, x, 0< t <1, x∈Ω,

utⁿ

r1

arxuxr_xr ft, x, −1< t <0, x∈Ω,

ut, x 0, x∈S, −1≤t≤1; u1, x u−1, x μx, x∈ Ω, u0, x u0−, x, u_t0, x u_t0−, x, x∈ Ω

1.14

is considered. Problem1.14has a unique smooth solutionut, xfora_rx≥a >0x∈Ω andgt, x t∈0,1, x ∈ Ω, ft, x t∈−1,0, x ∈ Ω, the smooth functions. This allows us to reduce the mixed problem1.14to the nonlocal boundary value problem1.1in the Hilbert space H L2Ωof all the integrable functions defined onΩ,equipped with the norm

f_L2Ω

· · ·

x∈Ω|fx|²dx₁· · ·dx_{n 1/2}

1.15

with a self-adjoint positive definite operatorAdefined by1.14.

Theorem 1.4. The solution of the nonlocal boundary value problem 1.14 satisfies the coercivity inequality

utt_Cα

0,10,1,L2Ωut_Cα

0−1,0,L2Ωu_Cα

0,1−1,1,W₂²Ω

≤M 1

α1−α g_Cα

0,10,1,L2Ωf_Cα

0−1,0,L2Ω

μ_W2

2Ω

, 1.16

whereMis independent offt, x, gt, x, andμx.

The proof of Theorem 1.4 is based on the abstract Theorem 1.1 and the symmetry properties of the space operator generated by problem1.14and the following theorem on the coercivity inequality for the solution of the elliptic differential problem inL₂Ω.

Theorem 1.5. For the solution of the elliptic differential problem n

r1

arxuxr_xr ωx, x∈Ω, 1.17

ux 0, x∈S, 1.18

the following coercivity inequality holds [7]:

n r1

uxrxr_L2Ω≤M||ω||_L2Ω. 1.19

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

−τ⁻²u_k1−2u_ku_k−1 Au_kg_k, gkgtk, tkkτ, 1≤k≤N−1, τ⁻¹uk−u_k−1−Au_k−1fk, fkft_k−1,

t_k−1 k−1τ, −N1≤k≤0, uNu_−Nμ, u1−u0u0−u₋₁.

2.1

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 operatorsA_hthat act in the Hilbert spaces Hhand are uniformly self-adjoint positive definite inhfor 0< h≤h0.

LetPPτA IτA⁻¹.Then the following estimates are satisfied8:

P^kH→H≤M1δτ^−k, kτAP^kH→H≤M, k≥1, δ >0, 2.2 A^βP^kr−P^kH→H≤M rτ^α

kτ^αβ, 1≤k < kr≤N, 0≤α, β≤1. 2.3

Furthermore, for a self-adjoint positive definite operatorAit follows that the operatorR I τB⁻¹is defined on the whole spaceH,it is a bounded operator, and the following estimates

hold:

R^kH→H≤M1δτ^−k, kτBR^kH→H≤M, k≥1, δ >0, 2.4 B^βR^kr−R^kH→H ≤M rτ^α

kτ^αβ, 1≤k < kr≤N, 0≤α, β≤1. 2.5

HereB 1/2τA

A4τ²A.From2.2and2.4, it follows that

||I−R^2N−1||_H→H≤M, 2.6 I IτAI2τA⁻¹R^2N−1B⁻¹AI2τA⁻¹I−R^2N−1

−2IτBI2τA⁻¹R^NP^N−1−1

H→H≤M.

2.7

Theorem 2.1. For anygk, 1 ≤k ≤N−1 andfk, −N1≤k≤0,the solution of problem2.1 exists and the following formulas hold:

u_k I−R^2N−1

R^k−R^2N−ku0

R^N−k−R^Nk

P^Nu0−τ 0 s−N1

P^sNfsμ

−R^N−k−R^NkIτB2IτB⁻¹B⁻¹

N−1

s1

R^N−s−R^Nsgsτ

IτB2IτB⁻¹B⁻¹

N−1

s1

R^|k−s|−R^ks

g_sτ, 1≤k≤N,

2.8

u_kP^−ku₀−τ 0 sk1

P^s−kf_s, −N≤k≤0, 2.9

u0TτI2τA⁻¹IτA

2τBR^N

−τ 0 s−N1

P^sNfsμ

−R^N−1B⁻¹

N−1

s1

R^N−s−R^Nsgsτ

I−R^2NB^−1N−1

s1

R^s−1g_sτ−I−R^2NIτBB⁻¹P f₀

, 2.10

where

T_τ

IIτAI2τA⁻¹R^2N−1B⁻¹AI2τA⁻¹I−R^2N−1−2IτBI2τA⁻¹R^NP^N−1−1

. 2.11

Proof. By8,9,

u_k I−R^2N−1

R^k−R^2N−kξ R^N−k−R^Nkψ

−R^N−k−R^NkIτB2IτB⁻¹B⁻¹

N−1

s1

R^N−s−R^Nsgsτ

IτB2IτB⁻¹B⁻¹

N−1

s1

R^|k−s|−R^ksgsτ, 1≤k≤N,

2.12

is the solution of the boundary value difference problem

−τ⁻²u_k1−2u_ku_k−1 Au_kg_k, gkgtk, tkkτ, 1≤k≤N−1,

u₀ξ, u_Nψ,

2.13

ukP^−kξ−τ 0 sk1

P^s−kfs, −N≤k≤0 2.14

is the solution of the inverse Cauchy problem

τ⁻¹uk−u_k−1−Au_k−1fk, fkftk−1,

t_k−1 k−1τ, −N1≤k≤0, u0ξ. 2.15 Exploiting2.12,2.14, and the formulas

ψu_−Nμ, ξu₀, 2.16

we obtain formulas2.8and2.9. Foru0,using2.8,2.9, and the formula

u₁−u₀u₀−u₋₁, 2.17

we obtain the operator equation

I−R^2N−1

R−R^2N−1u0 R^N−1−R^N1

×

P^Nu₀−τ 0 s−N1

P^sNf_sμ

−R^N−1−R^N1IτB2IτB⁻¹B⁻¹

N−1

s1

R^N−s−R^Nsgsτ

IτB2IτB⁻¹B⁻¹

N−1

s1

R^s−1−R^1sgsτ2u0−P u0τP f0.

2.18

The operator

I IτAI2τA⁻¹R^2N−1B⁻¹AI2τA⁻¹I−R^2N−1−2IτBI2τA⁻¹R^NP^N−1 2.19

has an inverse Tτ

IIτAI2τA⁻¹R^2N−1B⁻¹AI2τA⁻¹I−1R^2N−1−2IτBI2τA⁻¹R^NP^{N−1 −1}, 2.20 and the following formula

u₀ T_τIτAI2τA⁻¹

2τBR^N

−τ 0 s−N1

P^sNf_sμ

−R^N−1B⁻¹

N−1

s1

R^N−s−R^Nsgsτ

I−R^2NB^−1N−1

s1

R^s−1gsτ−I−R^2NIτBB⁻¹P f0

2.21

is satisfied. This concludes the proof ofTheorem 2.1.

LetF_τH Fa, b_τ, Hbe the linear space of mesh functionsϕ^τ{ϕk}^N_N^b

adefined on a, b_τ {tk kh, N_a ≤ k ≤ N_b, N_aτ a, N_bτ b}with values in the Hilbert spaceH.

Next onFτHwe denote byCa, b_τ, HandC^α_0,1−1,1_τ, H, C^α_0,1−1,0_τ, H, C^α₀0,1_τ, H0< α <1Banach spaces with the norms

ϕ^τ_Ca,bτ,H max

Na≤k≤Nb

ϕkH,

ϕ^τC^α_0,1−1,1τ,Hϕ^τC−1,1τ,H sup

−N≤k<kr≤0ϕkr−ϕkE−k^α r^α sup

1≤k<kr≤N−1ϕ_kr−ϕkEkrτ^αN−k^α

r^α ,

ϕ^τC^α₀−1,0τ,Hϕ^τ_C−1,0τ,H sup

−N≤k<kr≤0ϕ_kr−ϕkE−k^α r^α , ϕ^τC_0,1^α 0,1τ,Hϕ^τ_C0,1τ,H sup

1≤k<kr≤N−1ϕ_kr−ϕ_kEkrτ^αN−k^α

r^α .

2.22

The nonlocal boundary value problem2.1is said to be stable inF−1,1_τ, Hif we have the inequality

u^τF−1,1τ,H≤Mf^τF−1,0τ,Hg^τF0,1τ,HμH, 2.23 whereMis independent of not onlyf^τ, g^τ, μbut alsoτ.

Theorem 2.2. The nonlocal boundary value problem2.1is stable in C−1,1_τ, Hnorm.

Proof. By9,

{uk}⁰_−N

C−1,0τ,H ≤Mf^τ_C−1,0τ,Hu0_H 2.24 for the solution of the inverse Cauchy difference problem2.15and

{uk}^N−1₁

C0,1τ,H≤Mg^τ_C0,1τ,Hu0_H uN_H 2.25 for the solution of the boundary value problem2.13. The proof ofTheorem 2.2is based on the stability inequalities2.24,2.25, and on the estimates

u0H≤Mf^τ_C−1,0τ,Hg^τ_C0,1τ,HμH,

uNH≤Mf^τC−1,0τ,Hg^τC0,1τ,HμH 2.26

for the solution of the boundary value problem 2.1. Estimates 2.26 are derived from formula2.10and estimates2.2,2.4,2.7. This concludes the proof ofTheorem 2.2.

The nonlocal boundary value problem2.1is said to be coercively stablewell-posed inF−1,1_τ, Hif we have the coercive inequality

{τ⁻²u_k1−2uku_k−1}^N−1₁

F0,1τ,H

{τ⁻¹uk−u_k−1}⁰_−N1

F−1,0τ,H{Auk}^N−1_−N

F−1,1τ,H

≤Mf^τ_F−1,0τ,Hg^τ_F0,1τ,HAμH,

2.27

whereMis independent of not onlyf^τ, g^τ, μbut alsoτ.

Since the nonlocal boundary value problem 1.1 in the space C0,1, H of continuous functions defined on−1,1and with values inHis not well-posed for the general positive unbounded operator A and space H, then the well-posedness of the difference nonlocal boundary value problem2.1inC−1,1_τ, Hnorm does not take place uniformly with respect toτ >0. This means that the coercive norm

u^τKτE{τ⁻²uk1−2uku_k−1}^N−1₁ C0,1τ,H

{τ⁻¹uk−u_k−1}⁰_−N1C−1,0τ,H{Auk}^N−1_{−N C−1,1}

τ,H

2.28

tends to∞asτ→0.The investigation of the difference problem2.1permits us to establish the order of growth of this norm to∞.

Theorem 2.3. Assume thatμ ∈DAandf0 ∈ DIτB.Then for the solution of the difference problem2.1we have the almost coercivity inequality

u^τKτE≤MAμHIτBf0H

min

ln1

τ,1|lnAH→H|

f^τ_C−1,0τ,Hg^τ_C0,1τ,H, 2.29

whereMis independent of not onlyf^τ, g^τ, μbut alsoτ. Proof. By9,

{τ⁻¹uk−u_k−1}⁰_−N1C−1,0τ,H{Auk}⁰_−NC−1,0

τ,H

≤M

min

ln1

τ,1|lnAH→H|

f^τC−1,0τ,HAu0_H

2.30

for the solution of the inverse Cauchy difference problem2.15and {τ⁻²uk1−2uku_k−1}^N−1₁

C0,1τ,H{Auk}^N−1₁

C0,1τ,H

≤M

min

ln1

τ,1|lnAH→H|

g^τC0,1τ,HAu0_HAuN_H

2.31

for the solution of the boundary value problem2.13. Then the proof ofTheorem 2.3is based on the almost coercivity inequalities2.30,2.31, and on the estimates

Au0H≤MAμHIτBf0H

min

ln1

τ,1|lnAH→H|

f^τ_C−1,0τ,Hg^τ_C0,1τ,H, AuN_H≤MAμHIτBf0H

min

ln1

τ,1|lnAH→H|

f^τ_C−1,0τ,Hg^τ_C0,1τ,H

2.32

for the solution of the boundary value problem2.1. The proof of these estimates follows the scheme of papers8,9and relies on formula2.10and on estimates2.2,2.4, and2.7.

This concludes the proof ofTheorem 2.3.

Theorem 2.4. Let the assumptions ofTheorem 2.3be satisfied. Then the boundary value problem2.1 is well-posed in a H¨older spaceC^α_0,1−1,1_τ, Hand the following coercivity inequality holds:

{τ⁻²uk1−2uku_k−1}^N−1₁ C^α_0,10,1τ,H

{Auk}^N−1_−N

C^α_0,1−1,1τ,H{τ⁻¹uk−u_k−1}⁰_−N1C^α₀−1,0τ,H

≤M

AμHIτBf0H 1

α1−αf^τC^α₀−1,0τ,Hg^τC^α_0,10,1τ,H

,

2.33

whereMis independent of not onlyf^τ, g^τ, μbut alsoτandα.

Proof. By8,9,

{τ⁻¹uk−u_k−1}⁰_−N1

C^α₀−1,0τ,H{Auk}⁰_−N

C^α₀−1,0τ,H

≤M 1

α1−αf^τC^α₀−1,0τ,HAu0_H

2.34

for the solution of the inverse Cauchy difference problem2.15and {τ⁻²uk1−2u_ku_k−1}^N−1₁

C^α_0,10,1τ,H{Auk}^N−1₁

C^α_0,10,1τ,H

≤M 1

α1−αg^τC^α_0,10,1τ,HAu0_HAuN_H

2.35

for the solution of the boundary value problem2.13. Then the proof ofTheorem 2.4is based on the coercivity inequalities2.34,2.35, and on the estimates

Au0_H≤M

Aμ_HIτBf0H 1 α1−α

f^τC^α₀−1,0τ,Hg^τC^α_0,10,1τ,H

, AuN_H≤M

Aμ_HIτBf0H 1 α1−α

f^τC^α₀−1,0_τ,Hg^τC^α_0,10,1_τ,H

2.36

for the solution of the boundary value problem2.1. Estimates2.36are derived from the formulas

Au₀T_τI2τA⁻¹IτA

×

2τBR^N

−τ 0 s−N1

AP^sNfs−f_−N1 Aμ

−R^N−1AB⁻² _N−1

s1

BR^N−sgs−g_N−1τ^N−1

s1

BR^Nsg1−g_sτ

I−R^2NAB^−2N−1

s1

BR^s−1gs−g1τ

T_τI2τA⁻¹IτA

2τBR^NP^N−If_−N1

−R^N−1AB⁻²

I−R^N−1g_N−1−R^N−2−R^2N−1g1

I−R^2NAB⁻²I−R^N−1g1−I−R^2NIτBB⁻¹AP f₀

,

AuN P^N

TτI2τA⁻¹IτA

×

2τBR^N

−τ 0 s−N1

AP^sNfs−f_−N1 Aμ

−R^N−1AB⁻² _N−1

s1

BR^N−sgs−g_N−1τ^N−1

s1

BR^Nsg1−gsτ

I−R^2NAB^−2N−1

s1

BR^s−1gs−g₁τ

−τ 0 s−N1

AP^sNfs−f_−N1 Aμ P^N−If−N1

P^N{TτI2τA⁻¹IτA{{2τBR^NP^N−If−N1

−R^N−1AB⁻²{I−R^N−1gN−1−R^N−2−R^2N−1g1}}

I−R^2NAB⁻²I−R^N−1g1−I−R^2NIτBB⁻¹AP f₀}}

2.37 for the solution of problem2.1and estimates2.2,2.4, and2.7. This concludes the proof ofTheorem 2.4.

Now, the applications of this abstract result to the approximate solution of the mixed boundary value problem for the elliptic-parabolic equation1.14are considered. The discretization of problem1.14is carried out in two steps. In the first step, the grid sets

Ωh{xxm h1m1, . . . , hnmn, m m1, . . . , mn,0≤mr ≤Nr, hrNr 1, r1, . . . , n}, ΩhΩh∩Ω, S_hΩh∩S

2.38 are defined. To the differential operator A generated by problem 1.14 we assign the difference operatorA^x_hby the formula

A^x_hu^h_x −ⁿ

r1

a_rxu^h_xr

xr,jr

2.39

acting in the space of grid functionsu^hx,satisfying the conditionsu^hx 0 for allx∈Sh. With the help ofA^x_hwe arrive at the nonlocal boundary-value problem

−d²u^ht, x

dt² A^x_hu^ht, x g^ht, x, 0< t <1, x∈Ωh, du^ht, x

dt −A^x_hu^ht, x f^ht, x, −1< t <0, x∈Ωh,

u^h−1, x u^h1, x μ^hx, x∈Ωh,

u^h0, x u^h0−, x, du^h0, x

dt du^h0−, x

dt , x∈Ωh

2.40 for an infinite system of ordinary differential equations.

In the second step problem2is replaced by the difference scheme2.1:

−u^h_k1x−2u^h_kx u^h_k−1x

τ² A^x_hu^h_kx g_k^hx, g_k^hx g^htk, x, t_kkτ, 1≤k≤N−1, Nτ 1, x∈Ωh,

u^h_kx−u^h_k−1x

τ −A^x_hu^h_k−1x f_k^hx,

f_k^hx f^htk, x, t_k−1 k−1τ, −N1≤k≤ −1, x∈Ωh, u^h_−Nx u^h_Nx μ^hx, x∈Ωh,

u^h₁x−u^h₀x u^h₀x−u^h₋₁x, x∈Ωh.

2.41

Based on the number of corollaries of the abstract theorems given above, to formulate the result, one needs to introduce the space L_2h L₂Ωh of all the grid functions ϕ^hx {ϕh1m1, . . . , hnmn}defined onΩh,equipped with the norm

ϕ^h_L2Ωh

x∈Ωh

|ϕ^hx|²h₁· · ·h_{n 1/2}

. 2.42

Theorem 2.5. Letτ and |h|

h²₁· · ·h²_nbe sufficiently small numbers. Then the solutions of the difference scheme2.41satisfy the following stability and almost coercivity estimates:

{u^h_k}^N−1_−N

C−1,1τ,L2h≤M{f_k^h}⁻¹_−N1

C−1,0τ,L2h{g^h_k}^N−1₁

C0,1τ,L2hμ^h_L2h ,

τ⁻²

u^h_k1−2u^h_ku^h_k−1N−1 1

C0,1τ,L2h

τ⁻¹

u^h_k−u^h_k−10

−N1

C−1,0τ,L2h{u^h_k}^N−1_−N

C−1,1τ,W_2h²

≤M μ^h_W²

2hτf₀^h_W1

2h ln 1

τ|h|{f_k^h}⁻¹_−N1

C−1,0τ,L2h{g_k^h}^N−1₁

C0,1τ,L2h

,

2.43

whereMis independent ofτ, h, μ^hx,andg_k^hx, 1≤k≤N−1, f_k^h,−N1≤k≤0.

The proof ofTheorem 2.5is based on the abstract Theorems2.2,2.3, on the estimate

min

ln1

τ, 1|lnA^x_hL2h→L2h|

≤Mln 1

τ|h| 2.44

as well as the symmetry properties of the difference operatorA^x_h defined by formula2.39 inL2h,along with the following theorem on the coercivity inequality for the solution of the elliptic difference problem inL_2h.

Theorem 2.6. For the solution of the elliptic difference problem,

A^x_hu^hx ω^hx, x∈Ωh, 2.45

u^hx 0, x∈S_h, 2.46

the following coercivity inequality holds [7]:

n r1

u^h_xrxr,jr

L2h

≤M||ω^h||_L2h. 2.47

Theorem 2.7. Letτand|h|be sufficiently small numbers. Then the solutions of the difference scheme 2.41satisfy the following coercivity stability estimates:

τ⁻²

u^h_k1−2u^h_ku^h_k−1N−1

1

C^α_0,10,1τ,L2h

τ⁻¹

u^h_k−u^h_k−10

−N1

C^α₀−1,0τ,L2h{u^h_k}^N−1_−N

C^α_0,1−1,1τ,W_2h²

≤M μ^h_W2

2hτf₀^h_W1

2h 1

α1−α{f_k^h}⁻¹_−N1

C₀^α−1,0τ,L2h{g_k^h}^N−1₁

C_0,1^α 0,1τ,L2h

, 2.48

whereMis independent of τ, h, μ^hx,andg_k^hx, 1≤k≤N−1, f_k^h,−N1≤k≤0.

The proof of Theorem 2.7 is based on the abstract Theorem 2.4, the symmetry properties of the difference operatorA^x_hdefined by formula2.39, and onTheorem 2.6on the coercivity inequality for the solution of the elliptic difference equation2.45inL_2h.

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 problem1.12can 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.

Acknowledgment

The authors would like to thank Professor P. E. SobolevskiiJerusalem, Israelfor his helpful suggestions to the improvement of this paper.

References

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