The structure of fractional spaces generated by multi dimensional elliptic difference operator

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R E S E A R C H

Open Access

The structure of fractional spaces

generated by multi-dimensional elliptic

difference operator

Allaberen Ashyralyev

1,2

_{and Sema Akturk}

3*

*_{Correspondence:}

[email protected]

3_{Department of Mathematics, Fatih}

University, Buyukcekmece, Istanbul, 34500, Turkey

Full list of author information is available at the end of the article

Abstract

In the present paper, the positivity of a multi-dimensional difference operator in the half-space is considered. The structure of fractional spaces generated by this operator is investigated. The equivalence of the norms of these fractional spaces and Hölder spaces is proved. In applications, the stability of difference schemes for elliptic differential equations is presented.

MSC: 35J25; 47E05; 34B27

Keywords: positive operator; fractional spaces; Green’s function; Hölder spaces

1 Introduction

The role played by the positivity of differential and difference operators in Banach spaces in the study of the stability of solutions of boundary value problems for partial differential equations and the stability of difference schemes for partial differential equations, and of summation Fourier series converging inC-norm is well known (see [–]).

Recall that an operatorAdensely deﬁned in a Banach spaceEwith domainD(A) is called positive inE, if its spectrumσA lies in the interior of the sector of angleϕ,  <ϕ<π,

symmetric with respect to the real axis, and, moreover, on the edges of this sectorS(ϕ) =

{ρeiϕ: ≤ρ≤ ∞}andS(ϕ) ={ρe–iϕ: ≤ρ≤ ∞}, and outside of the sector the resolvent (A–λI)–is subject to the bound (see [])

(A–λI)–_E_→_E≤ M  +|λ|,

whereI is the identity operator. The inﬁmum of all such anglesϕ is called the spectral angle of the positive operatorAand is denoted byϕ(A) =ϕ(E,A).

Throughout the present work, we will indicate withMpositive constants which can be diﬀerent from time to time and we are not interested in precise. We will writeM(α,β, . . .) to express the fact that the constant depends only onα,β, . . . .

The theory of diﬀerential and diﬀerence operators in Banach spaces and their related applications have been investigated by many scientists (see, for example, [–]).

Great progress has been made in the study of structure of interpolation spaces generated by positive operators from the view-point of the stability analysis of high-order accuracy

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difference schemes for various partial differential equations. However, the positivity of difference operators and structure of fractional spaces generated by these operators in Banach spaces are not well investigated in general. Therefore, the investigation of structure of fractional spaces generated by positive difference operators in Banach spaces and its applications to the stability of difference schemes for various partial differential equations is an important subject.

For a positive operatorAin the Banach spaceE, let us introduce the fractional spaces Eα=Eα(E,A) ( <α< ) consisting of thosev∈Efor which the norms

vEα=sup

λ>

λαA(A+λI)–v_E

are ﬁnite. Clearly, the positive operator commutesAand its resolvent (A–λI)–_{. By the} deﬁnition of the norm in the fractional spaceEα=Eα(E,A) ( <α< ), we get

(A–λI)–_E

α→Eα≤(A–λI) –

E→E. ()

Thus, from the positivity of operatorAin the Banach spaceEfollows the positivity of this operator in fractional spacesEα=Eα(E,A) ( <α< ).

In [], Simirnitskii and Sobolevskii considered the diﬀerence operatorAx__hwhich is an elliptic diﬀerence operator of an arbitrary high order of accuracy approximating the multi-dimensional elliptic operatorAx_=Bx+δI.

Let us deﬁne the grid spaceRn–

h ( <h≤h) as the set of all points of the Euclidean

spaceRn–whose coordinates are given by

xk=skh, sk= ,±,±, . . . ,k= , . . . ,n– .

The numberhis called the step of the grid space. A function deﬁned onRn–

h will be called

a grid function. To the diﬀerential operatorBwith constant coeﬃcients of the form

Bx=

|r|=m

br

∂r+···+rn– ∂_xr_

 · · ·∂xrnn–– ,

we assign the diﬀerence operator

Bx h=h

–m

m≤|s|≤S

dss–

s +· · ·

sn– (n–)–

sn–

(n–)+, ()

which acts on functions deﬁned on the entire spaceRn–

h . Here,s∈R(n–)is a vector with

nonnegative integer coordinates,

k±fh(x) =±

fh(x±ekh) –fh(x)

,

andekis the unit vector of the axisxk.

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smooth function onRn–_{. Using the Taylor expansion of}_ϕ₍_y_{), one can show that}

sup

x∈Rn–

h

h–k±ϕ(x) – ∂

∂yk

ϕ(x)≤M(ϕ)h.

Here, the grid functionϕ(x) and_∂y∂

kϕ(x) are the traces of the functionsϕ(y) and ∂ ∂ykϕ(y),

respectively. The last inequality means that the diﬀerence operatorh–

k±is a ﬁrst-order

approximation for the diﬀerential operator _∂y∂

k.

We say that the diﬀerence operatorBx

h is aλth-order (λ> ) approximation of the

dif-ferential operatorBx_{if the inequality}

sup

x∈Rn_h–

Bx_hϕ(x) –Bxϕ(x)≤M(ϕ)hλ

holds for any smooth functionϕ(y). We shall assume that the operatorBx_happroximates the diﬀerential operatorBx_{with any prescribed order.}

A function of a continuous [resp., discrete] argument that decays at inﬁnity faster than any negative power of|y|[resp., |x|] is said to be rapidly decreasing. Let us deﬁne the Fourier transform of a grid functionfh₍_x_{) by the formula}

˜

f(ξ) = (π)–(n–)

x∈Rn_h–

exp–i(x,ξ)fh(x)hn–, ξ∈Rn–. ()

This formula deﬁnes a πh–-periodic smooth function of the continuous argumentξ wheneverfh₍_x_{) is a rapidly decreasing grid function. In this last case, () is just a Fourier}

series expansion of the functionf˜(ξ) and the numbersfh₍_x_{) are the Fourier coeﬃcients,}

given by the formula

fh(x) =

|ξ|≤πh–

· · ·

|ξn–|≤πh–

expi(x,ξ)f˜(ξ)dξ· · ·dξn–. ()

The inverse Fourier transform of a πh–_{-periodic function}_ϕ₍_ξ_{) is deﬁned to be the grid} functionϕ˜h(x) given by the formula

˜

ϕh(x) =

|ξ|≤πh–

· · ·

|ξn–|≤πh–

expi(x,ξ)ϕ(ξ)dξ· · ·dξn–. ()

Equations () and () establish a one-to-one correspondence between rapidly decreasing grid functions of a continuous argument. In particular, iffh₍_x_{) is a rapidly decreasing grid}

function, then

[f˜]h(x) =fh(x).

Iffh₍_x_{) is a rapidly decreasing grid function, then the grid function}_Bx

hfh(x) exists and is

given by () and we have the equality

Bx

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The functionB(ξh,h) is obtained by replacing the operatork±in the right-hand side of

equality () with the expression±(exp{±iξkh}– ), respectively, and is called the symbol of

the diﬀerence operator. Sinceexp{±iξkh}is a bounded analytic πh–-periodic function,

the symbolB(ξh,h) is a bounded analytic πh–_{-periodic function. It follows that for large}

|ξ|one has the estimate

B(ξh,h)≤M(h)|ξ|m, |ξ|=|ξ|+· · ·+|ξn–|.

Let us give the diﬀerence operatorAx

hby the formula

Ax__huh(x) = m≤|r|≤S

ax_rDr_huh(x) +δuh(x). ()

The coeﬃcients are chosen in such a way that the operatorAx__happroximates in a speciﬁed way the operator

|r|=m

ar(x)

∂|r|

∂xr  · · ·∂x

rn–

n– +δ.

We shall assume that for|ξkh| ≤πand ﬁxedxthe symbolAx(ξh,h) of the operatorAxh–δ

satisﬁes the inequalities

(–)mAx_(ξh,h)≥M|ξ|m, argAx(ξh,h)≤φ<φ≤ π

.

In [], Danelich considered the diﬀerence elliptic operatorAx

hwhich is to an arbitrary

high order of accuracy approximating the multi-dimensional elliptic operatorAx_deﬁned

by

Ax=

|r|=m

ar(x)

∂|r|

∂xr  · · ·∂x

rn–

n–

+ (–)ma(x)∂ m

∂xm n

+δI ()

with the domain

DAx=

u∈CmR+_n:u(x)|xn== ∂u(x)

∂N

xn=

=· · ·=∂

m–_u₍_x₎

∂Nm–

xn=

= 

.

Here,δ>  is the suﬃciently large number anda(x) is a continuous function deﬁned on

R+₌_{_x_:_x_≥__}_with

 <s≤a(x)≤s<∞. ()

Let us deﬁne the grid spaceR+

hn( <h≤h) as the set of all points of the spaceR+nwhose

coordinates are given by

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The numberhis called the step of the grid space. A functionϕh₍_x_{) deﬁned on}_R+

hnwill be

called a grid function. To the diﬀerential operatorAx_{deﬁned by (), we assign the}

diﬀer-ence operator

Ax_hϕh(x) =Ax__hϕh(x) +Ax__hϕh(x) +δϕh(x), x∈R+_hn,

Ax

hϕh(x) =h–m

m≤|p|≤S

ax r

p –

p +· · ·

pn– (n–)–

pn–

(n–)+ϕh(x), x∈R+hn, ()

Ax__hϕh(x) = (–)mh–m m≤r+s≤S

s≤m

ax_r_,_sr_n₊s_n_–ϕh(x), x∈R+_hn,

which acts on functions deﬁned on the entire spaceR+

hn. Here,p∈R+n– is a vector with nonnegative integers coordinates. The coeﬃcientsax

r are chosen in such a way that the

operatorAx__happroximates in a speciﬁed way the operator

|r|=m

ar(x)

∂|r| ∂xr

 · · ·∂x

rn–

n– .

We shall assume that for|ξkh| ≤π and ﬁxedxthe symbolAx(ξh,h) of the operatorAxh

satisﬁes the inequalities

(–)mAx_ξh,h≥Mξ m

,

argAx_ξh,h≤φ<φ<π, ξ= (ξ, . . . ,ξn–).

()

For the deﬁnition of the operatorAx

h, we will deﬁneϕh(x) which is extended toϕh(x)

deﬁned onR+

hnand additionally on the points

x,xk

=x,kh∈/R+_hn, x∈Rh(n–),k= –, –, . . . , –(m– ),

and

ϕhx, = , h–k –k≤s≤sk

αs,kϕh

x,sh= , k= ,m– ,α–k,k= . ()

The coeﬃcientsαs,kare chosen in such a way that the expression

h–k

–k≤s≤sk

αs,kϕh

x,sh

approximates in a speciﬁed way the expressionϕ(k)₍_x_{, ). The coeﬃcients}_ax

r,sare chosen in

such a way that the operatorh–k

–k≤s≤skαs,kϕ

h₍_x_,_sh_{) approximates in a speciﬁed way the}

operatorAx_. We shall assume that the operatorAx__happroximates the diﬀerential operator Ax_with any prescribed order.

In the present paper, a Green’s function is assigned. The organization of the present paper is as follows. In Section , the main theorem on the structure of fractional spaces Eα(C˙h(Rhn+ ),Axh) generated byAxhis investigated. In Section , applications on theorems on

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2 Green’s function

Theorem [] Suppose that assumptions().Let the function a(x)satisfy the estimate

a(x) –a(y)≤s|x–y|α

 <α≤;x,y∈R+_h_. ()

Then,for suﬃciently large positiveδ andReλ≥,there exists a unique solution of the resolvent equation

Ax_huh(x) +λuh(x) =fh(x), x∈R+_hn ()

and the following formula:

fh(y)hnA_hx+λI–fh(x) =

y∈R+_hn

Gh(x,y;λ) ()

holds.Here,Gh(x,y;λ)is the Green’s function of the resolvent equation().

Lemma  The following identities hold:

where vh₍_x₎_{is the solution of the following problem}_: ⎧

Proof We consider the problem

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Then

y∈R+_hn

Gh(x,y;λ)hn=

 δ+λ

 –vh(x), x∈R+_hn.

Identity () is proved. Sincevh_{() = , we have}

 δ+λ=

y∈R+_hn

Gh(,y;λ)hn+

 δ+λ.

Thus follows (). Lemma  is proved.

For anyx∈R+_hn, we have

Gh(x,y;λ) =Ghx(x,y;λ) +

y∈R+_hn

Gh(x,z;λ)

Ax

h –A z h

Gx

h (z,y;λ)h

n_, _()

whereGx

h (x,y;λ) is the Green’s function of the operatorA x

h .

Moreover, applying (), (), (), and () conditions, one proved the following point-wise estimates of the Green’s functionGh(x,y;λ) of the resolvent equation and its

diﬀer-ence derivatives:

Gh(x,y;λ)≤Mexp

–b|λ+δ|p|x–y||λ+δ|np– ()

for m>n,

–b|λ+δ|p|x–y| +ln|x–y||λ+δ|p–+  ()

for m=n,

–b|λ+δ|p|x–y||x–y|m–n ()

for m<n, and its derivatives respect tox

Dr_hGh(x,y;λ)≤Mexp

–b|λ+δ|p|x–y||λ+δ|(n+r)p– ()

for m–r>n,

_Dr

hGh(x,y;λ)≤Mexp

–b|λ+δ|p|x–y| +ln|x–y||λ+δ|p–+  ()

for m–r=n,

–b|λ+δ|p|x–y||x–y|m–n–r ()

for m–r<n, wherep=__m,a> ,M>  for  <|λ| ≤Lh–m,λ∈Qψ∪ {}, and Gh(x,y;λ)≤Mexp

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–bh–|x–y|hm–n–r|δ+λ|hm+ – ()

for|λ|>Lh–m_{. Here}_p₌ 

m,b> ,M> .

Note that under assumptions () and () there exists a unique solutionvh₍_x_{) of problem}

() and the following estimate holds:

vh(x)≤Mexp

–b|λ+δ|p|x|, ()

for  <|λ| ≤Lh–m_,_λ_∈_Q

ψ∪ {}and _vh₍_x₎_≤_M

exp

–bh–|x||δ+λ|hm+ –, ()

for|λ|>Lh–m_.

Lemma [] |λ|>Lh–m,where L> large enough.Then

–bh–|x–y|hm|δ+λ|hm+ –|x–y|–n, x=y. ()

By()and(),we obtain the estimate

_G_h₍_x_,_y_;λ)

≤Mexp

–bh–|x–y|h–+m(+μ)

×|δ+λ|hm+ –(+mμ)|x–y|–mμ, x=y. ()

Moreover, applying the Green’s function ofAx_hthe following results were proved.

Theorem [] Ax

h is a positive operator in the spaceC˙(R+hn)of all mesh functionsϕh(x)

deﬁned onR+_hnwith the norm

ϕh_C_˙

h= sup x∈R+

hn ϕh(x)

with the spectral angleϕ(Ax

h,C˙h) =π–ψ.

LetCβ_h =Cβ₍_R+

hn) be the Hölder space of all mesh functionsϕh(x) deﬁned onR+hn

satis-fying a Hölder condition with the indicatorβ∈(, ) with the norm

ϕh_Cβ

h= ϕh_C

h+ sup x,y∈R+_hn

x=y

|ϕh₍_x_{) –}_ϕh₍_y₎_| |x–y|β .

One proved the strong positivity ofAx_hin the Banach spaceC˙h=C˙(R+hn) (diﬀerence

ana-log ofC˙(R+

n)) for suﬃciently large positiveδ. Passing to limit whenh→, we can get the

strong positivity of diﬀerential operatorAx_{in the Banach space}_C˙₍_R+

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3 The structure of fractional spacesEα(C˙h(R+hn),Axh)

Theorem  Supposemα∈(, ).Then the norms of the spaces Eα(C˙h,Axh)andC˙mα(R+hn)

are equivalent.Here,

˙

C_hmα=

ϕh:ϕh∈C_hmα,ϕhx, = ,h–k –k≤s≤sk

αs,kϕh

x,sh= ,

k= ,m– ,α–k,k= ,x∈Rh(n–)

.

Proof Assume thatfh_{∈ ˙}_Cmα₍_R+

hn). Letx∈R+hnandλ>  be ﬁxed. Using equation ()

and identity (), we can write

Ax_hAx_h+λ–fh(x)

= δ

λ+δf

h₍_x_{) +}_λ

y∈R+n

Gh(x,y;λ)

fh(x) –fh(y)hn

+ λ

λ+δv

h₍_x₎_fh₍_x_{) –}_fh₍_x_{, . . . ,}_x n–, )

+ λ

λ+δv

h₍_x₎_fh₍_x_{, . . . ,}_x n–, ).

Sincefh₍_x

, . . . ,xn–, ) = , we have

Ax_hAx_h+λ–fh(x) = δ λ+δf

h₍_x_{) +}_λ

y∈R+n

Gh(x,y;λ)

fh(x) –fh(y)hn

+ λ

λ+δv

h₍_x₎_fh₍_x_{) –}_fh₍_x_{, . . . ,}_x n–, )

. ()

Using equation () and the triangle inequality, and the deﬁnition of theC˙mα₍_R+

hn)-norm,

we have

λαAx_hAx_h+λ–fh(x)

≤

λα_δ

λ+δ +λ

α+

y∈R+n

Gh(x,y;λ)|x–y|mαhn+

λα+

λ+δv(x)|x|

mα_fh

˙

Cmα₍_R+

hn)

= [I+I+I]fhC˙mα₍_R+

hn).

We will estimateIi,i= , , , separately. First, let us estimateI. Clearly, using the estimate

λα_δ–α

λ+δ ≤

we get

I≤δα ()

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We will consider two cases:λ>Lh–m_and_λ_≤_Lh–m_{. First, let}_λ_>_Lh–m_{. Using estimate}

(), we have

I≤Mλα+

y∈R+n

e–bh–|x–y|h–n+m(+α)(λ+δ)hm+ –(+mα)hn

≤M

(λhm₎α_λ

((λ+δ)hm₎mα₍_λ₊_δ₎≤M ()

for anyλ>  andx∈R+_hn. From estimate () and the following inequality:

(at)θe–at≤M, a> ,θ∈[, ], ()

it follows that

I≤M λα+ λ+δe

–ah–|x|₍_λ₊_δ₎_hm_{+ }–_| x|mα

≤M

λα+_hmα

(λ+δ)((λ+δ)hm_{+ )}≤M ()

for anyλ>  andx∈R+_hn. Combining estimates ()-(), we get

λαAx_hAx_h+λ–fh(x)≤M(α) ()

for anyλ>  andx∈R+

hn.

Second, letλ≤Lh–m_{. We consider three cases: }_m_>_n_{, }_m₌_n_{, and }_m_<_n_.

In the ﬁrst case, using estimate (), applying inequality (), and the deﬁnition of the

˙

Cmα₍_R+

hn)-norm, we have

I≤Mλα+

y∈R+

hn

e–b(λ+δ)m_|x–y|₍_λ₊_δ₎n

m–

≤M

λα+ (λ+δ)–nm+α+

β m+nm

≤M ()

for anyλ>  andx∈R+_hn.

In the second case, applying estimate (), and inequality (), we have

I≤Mλα+

y∈R+_hn

e–b(λ+δ)m_|x–y|_{ +}_ln_|_x_–_y_|₍_λ₊_δ₎ 

m–_{+ }|_x_–_y|mα+β_hn

≤M

λα+ (λ+δ)α+nm+

β m

≤M ()

for anyλ>  andx∈R+_hn.

In the third case, using estimate () we obtain

I≤Mλα+

y∈R+_hn

e–b(λ+δ)m_|x–y|₍_λ₊_δ₎n

m–|_x_–_y|m–n|_x_–_y|mα+β_hn

≤M

λα+ (λ+δ)m–n+mmα+β–nm

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for anyλ>  andx∈R+_hn. Estimates ()-() result in

I≤M ()

for anyλ>  andx∈R+_hn. From estimate () and inequality () it follows that

I≤M λα+ λ+δe

–b(λ+δ)m|x|_|_x_|mα+β_≤

M ()

hn. Combining estimates (), (), and (), we have

λαAx_hAx_h+λ–fh(x)≤M(α)fh_C_˙mα ()

hn.

Combining estimates () and (), we get

sup

λ> sup

x∈R+_hn

λαAx_hA_hx+λ–fh(x)≤M(α)fh_C_˙mα. ()

Now, we will prove the opposite inequality. Using the deﬁnition ofEα(C˙h,Axh), we obtain

sup

x∈R+

n

fh(x)≤fh_E

α(C˙h,Axh) ()

for anyx∈R+

hn. By Theorem ,Axhis a positive operator in the Banach spaceEα(C˙h,Ax_h).

Hence, forfh∈Eα(C˙h,Axh), we have

fh=

∞



Ax_hAx_h+λ–V dλ. ()

It follows from equation () and equation () that

fh(x) =

∞



Ax_h+λ–Ax_hAx_h+λ–fh(x)dλ

=

∞



y∈R+

hn

Gh(x,y;λ)Axh

Ax_h+λ–fh(y)hndλ. ()

Letx,τ ∈R+_hn be ﬁxed. We will consider two cases:λ>Lh–m andλ≤Lh–m. First, let λ>Lh–m_{. Using equation () and Lemma , we have}

fh₍_x₊_τ_{) –}_fh₍_x₎ |τ|mα =

∞



y∈R+

hn

[Gh(x+τ,y;λ) –Gh(x,y;λ)] |τ|mα_λα A

x h

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From the triangle inequality and the deﬁnition of theEα(C˙h,Axh)-norm it follows that

We will estimateLandL. Let us estimateL. From the Lagrange theorem and estimate (), we have, for somex∗betweenx,x+τ, the inequality () yield

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for anyλ>  andx∈R+_hn. Third, let m<n. Using the triangle inequality, estimate (), we have

L≤M

∞



|τ|m

y∈R+

hn

e–b(λ+δ)m|x+τ–y|

|τ|mα_λα |x+τ–y|

m–n_hn_dλ

+

∞



|τ|m

y∈R+

hn

e–b(λ+δ)m|x–y| |τ|mα_λα |x–y|

m–n_hn_dλ

≤M

∞



|τ|m



|τ|mα_λα+dλ≤M(α). ()

Then, combining estimates ()-(), we have

L≤M(α). ()

Estimates () and () yield

fh(x+τ) –fh₍_x₎ |τ|mα

≤M(α)fh_E

α. ()

Combining estimates () and (), we obtain

sup

x∈R+

hn τ=(,...,)

fh(x+τ) –fh₍_x₎ |τ|mα

≤M(α)fh_E

α. ()

From estimates () and () it follows that

Eα _˙

Ch,Axh

⊂ ˙CmαR+_hn.

This is the end of the proof of Theorem .

From Theorem  and Theorem  follows the following result.

Theorem  Ax

his a positive operator in the spaceC˙ β₍_R+

hn),β∈(, ).

4 Applications

Now, we present some applications of Theorems -.

First, we consider the diﬀerence schemes for the approximate solution of problem

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

–∂_∂tu(t,x)+

|r|=mar(x) ∂ |r|_u₍_t_,_x₎ ∂xr_···∂xrn_n_––

+ (–)ma(x)∂_∂xmum(t,x)+δu(t,x) =f(t,x),  <t<T,x∈R+n,

u(,x) =ϕ(x), u(T,x) =ψ(x), x∈R+

n,

u(t,x)|xn==∂u_∂N(t,x)|xn==· · ·=∂

m–_u₍_t_,_x₎

∂Nm– |xn== , x∈R+n, ≤t≤T.

()

Here,ar(x),a(x),ϕ(x),ψ(x), andf(t,x) are suﬃciently smooth functions and they satisfy all

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Assume that uniform ellipticity holds. Note that problem () can be written in the form of the abstract boundary value problem

⎧ ⎨ ⎩

–u(t) +Au(t) =f(t),  <t<T,

u() =ϕ, u(T) =ψ ()

in a Banach spaceE=C˙hwith positive operatorA=Ax_hdeﬁned by () and (). The

dis-cretization of problem () is carried out in two steps. In the ﬁrst step, let us give the diﬀerence operatorAx

h by equation (). With the help of Axh we arrive at the boundary

value problem

⎧ ⎨ ⎩

–du_dth(t,x)+Ax_huh(t,x) =fh(t,x),  <t<T,x∈R+_hn,

uh(,x) =ϕh(x), uh(T,x) =ψh(x), x∈R+_hn,

()

for an inﬁnite system of ordinary diﬀerential equations.

In the second step, we replace problem () by the diﬀerence scheme

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–_τ(uh_k_+(x) – uh_k(x) +u_kh_–(x)) +Ax_hu_kh(x) =f_kh(x), f_kh(x) =fh₍_t

k,x), tk=kτ, ≤k≤N– ,Nτ =T,x∈R+hn,

uh

(x) =ϕh(x), uhN(x) =ψh(x), x∈R+hn,

()

for the approximate solution of boundary value problem ().

Theorem  Let≤μ< .Then the solution of diﬀerence scheme()satisﬁes the following

stability estimate:

max ≤k≤N

uh_k_C_˙μ

h ≤M(μ)

ϕh_C_˙μ

h + ϕh_C_˙μ

h +≤maxk≤N–

f_kh_C_˙μ

h

.

The proof of Theorem  is based on Theorem  and Theorem , on the positivity of the diﬀerence operatorAx_hin the Banach spaceC˙_hμfor ≤μ< , and on the following abstract theorem on the stability of the diﬀerence scheme

⎧ ⎨ ⎩

–_τ(uk+– uk+uk–) +Auk=fk,

fk=f(tk), tk=kτ, ≤k≤N– ,Nτ=T,u=ϕ,uN=ψ,

()

for the approximate solution of the abstract boundary value problem ().

Theorem [] Let A be a positive operator in a Banach space E.Then,for the solution of diﬀerence scheme(),the following stability inequality holds:

max

≤k≤NukEα≤M(α)

ϕE_α+ψE_α+ max ≤k≤N–fkEα

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Theorem  The solution of diﬀerence scheme()satisﬁes the following almost coercive

The proof of Theorem  is based on Theorem  on the positivity of an elliptic diﬀerence operatorAx

and on the following theorems on the almost coercive stability of diﬀerence scheme () and on the almost coercive stability of the elliptic diﬀerence problem.

Theorem [] Let A be a positive operator in a Banach space E.Then,for the solution of diﬀerence scheme(),the following almost coercive stability inequality holds:

max

Theorem  Under assumptions()and(),the solution of the diﬀerence elliptic problem

Ax_huh(x) =fh(x), x∈R+_hn, ()

satisﬁes the following almost coercive inequality:

h–m

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Theorem  Let < mα< .Then,for the solution of diﬀerence scheme(),the following coercive stability estimate holds:

max

The proof of Theorem  is based on Theorem  on the structure of the fractional spaces Eα(C˙h,Axh), on Theorem  on the positivity of an elliptic diﬀerence operator Axh in the

Banach spaceC˙h, and on the following theorems on the structure of the fractional space

E_α=Eα(E,A/), on the coercive stability of diﬀerence scheme (), and on the coercive

stability of the elliptic diﬀerence problem.

Theorem [] The spaces Eα(E,A)and Eα(A/,E)coincide for any <α<,and their norms are equivalent.

Theorem [] Let A be a strongly positive operator in a Banach space E.Then,for the solution of diﬀerence scheme(),the following coercive stability inequality:

max ence elliptic problem(),we have the following coercive inequality:

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The proof of Theorem  uses the techniques introduced in [] and it is based on esti-mates for the Green’s function of operatorAx_hdeﬁned by ().

Second, we consider the diﬀerence schemes for the approximate solution of problem

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

–∂_∂tu(t,x)+

|r|=mar(x) ∂ |r|_u₍_t_,_x₎ ∂xr_···∂xrn_n_––

+ (–)m_a₍_x₎∂mu(t,x)

∂xm +δu(t,x) =f(t,x),  <t<T,x∈R+n,

u(,x) =u(T,x), ut(,x) =ut(T,x), x∈R+n,

u(t,x)|xn==∂u_∂N(t,x)|xn==· · ·=∂

m–_u₍_t_,_x₎

∂Nm– |xn== , x∈R+n, ≤t≤T.

()

Here,ar(x),a(x), andf(t,x) are suﬃciently smooth functions and they satisfy every

com-patibility conditions which guarantee that problem () has a smooth solutionu(t,x). As-sume that the assumption of the uniform ellipticity holds. The discretization of problem () is carried out in two steps. In the ﬁrst step, let us give the diﬀerence operatorAx

hby

equation (). With the help ofAx_hwe arrive at the boundary value problem

⎧ ⎨ ⎩

–du_dth(t,x)+Axhuh(t,x) =fh(t,x),  <t<T,x∈R+hn,

uh_(,_x_{) =}_uh₍_T_,_x_), _uh

t(,x) =uht(T,x), x∈R+hn,

()

for an inﬁnite system of ordinary diﬀerential equations.

In the second step, we replace problem () by the ﬁrst order of approximation in thet diﬀerence scheme

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–_τ(uh_k_+(x) – uh_k(x) +u_kh_–(x)) +Ax_hu_kh(x) =f_kh(x), f_kh(x) =fh₍_t

k,x), tk=kτ, ≤k≤N– ,Nτ =T,x∈R+hn,

uh

(x) =uhN(x), uh(x) –uh(x) =uhN(x) –uhN–(x), x∈R+hn,

()

and the second order of approximation in thetdiﬀerence scheme

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–_τ(uh_k_+(x) – uh_k(x) +u_kh_–(x)) +Ax_hu_kh(x) =f_kh(x),

f_kh(x) =fh(tk,x), tk=kτ, ≤k≤N– ,Nτ =T,x∈R+hn,uh(x) =uhN(x),

–uh

(x) + uh(x) – uh(x) =uhN–(x) – uhN–(x) + uNh(x), x∈R+hn,

()

for the approximate solution of boundary value problem ().

Theorem  Let≤μ< .Then the solution of the diﬀerence schemes()and()

sat-isﬁes the following stability estimate:

max ≤k≤N

_uh

kC˙μ_h ≤M(μ)

ϕh_C_˙μ

h + ϕh_C_˙μ

h +≤maxk≤N–

_fh kC˙μ_h

.

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Theorem [] Let A be a positive operator in a Banach space E.Then,for the solution of diﬀerence schemes()and(),the following stability inequality holds:

max

Theorem  For the solution of diﬀerence schemes ()and(),the following almost coercive stability estimate holds:

max

The proof of Theorem  is based on Theorem , on the positivity of an elliptic diﬀerence operatorAx

and on the following theorem on the almost coercive stability of diﬀerence schemes () and () and on Theorem  on the almost coercive stability of elliptic diﬀerence problem ().

Theorem [] Let A be a positive operator in a Banach space E.Then,for the solution of

diﬀerence schemes()and(),the following almost coercive stability inequality holds:

max satisﬁes the following coercive stability estimate:

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+h–m m≤r+s≤S

s≤m

r₊s_–uh_C_˙mα

h

≤M(α)

h–m

m≤|p|≤S p

–

p +· · ·

pn– (n–)–

pn–

(n–)+ϕhC˙mα

h

s≤m

r₊s_–ϕh_C_˙mα

h

+h–m m≤|p|≤S

p –

p +· · ·

pn– (n–)–

pn– (n–)+ψ

h

˙

Cmα

h

s≤m

r₊s_–ψh_C_˙mα

h

+M(α) max ≤k≤N–

_fh kC˙mα

h .

The proof of Theorem  is based on Theorem , on the structure of the fractional spaces Eα(C˙h,Axh), on Theorem , on the structure of the fractional spaceEα =Eα(E,A/), on

Theorem , on the positivity of an elliptic diﬀerence operatorAx_hin the Banach spaceC˙h,

and on the following theorem on the coercive stability of diﬀerence schemes () and () and on Theorem  on the coercive stability of elliptic diﬀerence problem ().

Theorem [] Let A be a strongly positive operator in a Banach space E.Then,for the

solution of diﬀerence schemes()and(),the following coercive stability inequality:

max ≤k≤N–

_τ(uk+– uk+uk–)

Eα

+ max

≤k≤NAukEα≤M(α)≤maxk≤N–fkEα is valid.

5 Conclusion

In the present article, the structure of the fractional spacesEα(C˙(R+_hn),Ax_h) generated by

the multi-dimensional elliptic diﬀerence operatorAx_his investigated.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Elementary Mathematics Education, Fatih University, Buyukcekmece, Istanbul, 34500, Turkey.} 2_{Department of Mathematics, ITTU, Ashgabat, Turkmenistan.}3_{Department of Mathematics, Fatih University,}

Buyukcekmece, Istanbul, 34500, Turkey.

Acknowledgements

The authors would like to thank Prof. Pavel Sobolevskii (Universidade Federal do Ceará, Brasil). The second author would also like to thank The Scientiﬁc and Technological Research Council of Turkey (TUB˙ITAK) for the ﬁnancial support.

Received: 24 August 2015 Accepted: 3 December 2015 References

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