• Aucun résultat trouvé

Toeplitz matrices for the study of the fractional Laplacian on an interval ]a, b[

N/A
N/A
Protected

Partager "Toeplitz matrices for the study of the fractional Laplacian on an interval ]a, b["

Copied!
29
0
0

Texte intégral

(1)

HAL

HAL, est

(2)

Toeplitz matrices for the study of the fractional Laplacian on an interval ]a, b[.

November 19, 2018

Abstract Fractionnal Laplacian on a interval]a, b[.

In this work we solve forα∈]0,12[ the fractional equation (−∆)/]a,b[)

/]a,b[ =fwhere (−∆)/]a,b[ is the fractional Laplacian with order α on ]a, b[ and f a locally contracting function on ]a, b[. To do this we introduce a class of fractional operator defined with the Toeplitz matriceTN(fα) wherefα is the functionθ7→ |1−e| for allθ∈[0,2π[.

12

p

+∞

−∞

ixt

1

R

1

R

1+2α

1

2Γ(

1+2α 2 )

π|Γ(−α)|

R

−2α

12

−2α

2Γ(2α) cos(απ)1

+∞

−∞

ψ(t)

|t−x|1−2α

R

p

R

α1

/J

/J

/]a,b[

/J

/J

/J

/J

. It is this operator and the inversion of this operator that we are interested in this article (see 2).

Universit´e de Paris Sud, Bˆatiment 425; F-91405 Orsay Cedex; tel : 01 69 15 57 28 ; fax 01 69 15 60 19 e-mail : philippe.rambour@math.u-psud.fr

Universit´e de Paris Sud, Bˆatiment 425; F-91405 Orsay Cedex; tel : 01 69 15 60 09 ; fax 01 69 15 72 34 e-mail : abdelatif.seghier@math.u-psud.fr

(3)

α

12

12

1

N

N

i+1,j+1

α

α

12

12

α

N→+∞

N

l=0

N

α

k+1,l+1

12

1

α

1

0 ψ(t)

|x−t|2α+1

12

/]0,1[

/[0,1]

1

2

R

12

1

−2α

/]0,1[

/]0,1[

−2α

0,2α+

1

0,µ

0,2µ

[c,d]

0

[c,d]

0

µ

0

(4)

N?

α

Γx2α(α)yα

1 max(x,y)

(t−x)α−1(t−y)α−1

t

1

1

0

α

α−1

α−1

N

12

α

Γ(2α+1) sin(πα)

π

−2α

1

Theorem 1

12

1

α,a,b

α

b a

2α+1

12

0,µ

α,a,b

α

b

a

2α+1

−2α

−2α

α,0,1

α

/]0,1[

/]0,1[

Remark 1

α,a,b

α

b

a

/[a,b]

/[a,b]

2α+1

−2α

−2α

/[a,b]

α

b a

/[a,b]

/[a,b]

2α+1

N

−1

k+1,l+1

Nk

Nl

→ y, and x, y ∈]0, 1[. We find these integral kernels in the statement of Theorem 2 and of the corollary 1 (see also 4 and

??). To get the solution to the equation 2 we use the fine knowledge of the

N

−1

12

N

−1

k+1,1

Nk

(5)

Theorem 2

12

1

−s

+

−s

/]0,1[

0,1−s

R

α

−α,0,1

b

a

α

α

2

α

α

+∞

1

α−1

α−1

+∞

0

α−1

α−1

Remark 2

α

a,b

12

12

α,a,b

α

a,b

u−a b−a

Remark 3

Remark 4

N

α

))

−1[N x]+1,[N y]+1

−α

2α−1

α,0,1

2α−1

1

2

Remark 5

12

−α

α

Remark 6

/[a,b]

[a,b]

[a,b]

/[a,b]

2

(6)

Corollary 1

12

1

/]0,1[

0,1−s

α

1

0

α

α

2

α

α

1

max(x,y)

α−1

α−1

α

Remark 7

α

R

Remark 8

N

α

))

−1[N x]+1,[N y]+1

2α−1

α

2α−1

1

2

Property 1

R

1

2

12

a→−∞,b→+∞

α,a,b

α

+∞

−∞

2α+1

12

a→−∞,b→+∞

α,a,b

α

+∞

−∞

2α+1

α

R→1

α,R

α,R

α

−iθ

α

N→+∞

α

N

l=0

N

α

k+1,l+1

N

l

α

(7)

α

x

0

−α−1

−α

α

R→1

α,R

α,R

α

−iθ

α

1

t

2

2

1

2

1

2

1

Remark

Theorem 3

|x−t|f(t)2α+1

1

12

α

α

1

0

2α+1

12

0,µ

12

α

α

1 0

2α+1

−2α

−2α

12

12

1

2

1

2

α

α

−2α−1

(8)

N

l=0

N

α

k+1,l+1

k−[N δ]−1

l=0

N

α

k+1,l+1

k+[N δ]

l=k−[N δ]

N

α

k+1,l+1

N

l+[N δ]+1

N

α

k+1,l+1

k+[N δ]

l=k−[N δ]

N

α

k+1,l+1

k+[N δ]

l=k−[N δ]

N

α

k+1,l+1

k+[N δ]

l=k−[N δ]

N

α

k+1,l+1

k+[N δ]

l=k−[N δ]

N

α

k+1,l+1

µ

k+[N δ]

l=k−[N δ]

N

α

k+1,l+1

µ

−β

µ

k+[N δ]

l=k−[N δ]

N

α

k+1,l+1

n∈Z

α

k+[N δ]

l=k−[N δ]

N

α

k+1,l+1

l<k−[N δ]

α

l>k+[N δ]

α

l<k−[N δ]

α

k−[N δ]−1

l=0

−2α−1

l>k+[N δ]

α

N

l=k+[N δ]+1

−2α−1

(9)

N→+∞

k−[N δ]−1

l=0

−2α−1

N

l=k+[N δ]+1

−2α−1

1

0

2α+1

α

α

12

u

α−1

N

α

12

α

D´efinition 1

M

N

−1(1,1)

M

u=0

N

−1(u+1,1)

u

M

N

M

M

M

M

Theorem 4

M

\

M

2

M

M

2

M

α

(10)

α

N

α

u+1,1

α−1

α

α−1

N

α

−1k+1,1

k

k

u=0

k−u

α,N

α,N

α,N

+∞

m=0

m,N,α

2m+2

m,N,α

+∞

w0=0

0

1+αN

0

+∞

w1=0

0

1

+∞

w2m−1=0

2m−2

2m−1

+∞

w2m=0

2m−1

2m

wN2m

1+αN

m,N,α

0

+∞

0

0

1+αN

0

+∞

0

0

1

+∞

0

2m−2

2m−1

+∞

0

2m−1

2m

2m

0

1

2m−1

2m

N,α

0

Lemma 1

N

N∈N

1

N

→+∞

N

l=0

N

k+1,l+1

1

0

N

−1l+1,[N t]+1

N→+∞

(11)

N→+∞ k

N

N

l=0

N

k+1,l+1

1

0

N

−1l+1,[N t]+1

N

l=0

N

k+1,l+1

0

N

−1l+1,[N t]+1

N

l=0

N

k+1,l+1

1−

N

−1l+1,[N t]+1

N

l=0

N

k+1,l+1

1

1−

N

−1l+1,[N t]+1

0

N

l=0

N

k+1,l+1

0

N

−1l+1,[N t]+1

N

l=0

N

k+1,l+1

m0

m=0

−1N

l+1,m+1

m+1

m

m0

m=0 N

l=0

N

k+1,l+1

−1N

l+1,m+1

m+1 m

m0

m=0

km

m+1

m

N

l=0

N

k+1,l+1

1−

N

−1l+1,[N t]+1

N

l=0

N

k+1,l+1

N−m0

m−m0

m+1

m

N

−1l+1,[N t]+1

N

l=0

N

k+1,l+1

N−m0

m=m0

N

−1l+1,m+1

N

l=0

N

k+1,l+1

N−m0

m=m0

m+1 m

N

−1l+1,m+1

N

l=0

N

k+1,l+1

N−m0

m−m0

m+1 m

N

−1l+1,m+1

N−m0

m−m0 N

l=0

N

k+1,l+1

N

−1l+1,m+1

m+1

m

N

l=0

N

k+1,l+1

N−1

m=0

m+1 m

N

−1l+1,m+1

N

m=0

km

(12)

N

l=0

N

k+1,l+1

N−m0

m−m0

N

−1l+1,m+1

N

m=0

k,m

2

Property 2

N

N

u=0

u

u

R

N

N

2 −1

k+1,l+1

k

u=0

k−u

l−u

k

u=0

u+N−l

u+N−k

N

α

N

→+∞

N

l=0

N

α

k+1,l+1

1

0

N

α

−1l+1,[N t]+1

N→+∞

u

N,α

α

N

α

−1

u

uΓ(α)α−1

α−1

N

l=0

N

α

k+1,l+1

1

0

N

α

−1l+1,[N t]+1

N

l=0

N

α

k+1,l+1

1

0

3

j=0

j

0

min(l,m)

u=0

u

β

max(l,m)−min(l,m)+u

1

min(l,m)

u=0

u

β

max(l,m)−min(l,m)+u

− γ

max(l,m)−min(l,m)+u

2

min(l,m)

u=0

u

u

) γ

max(l,m)−min(l,m)+u

(13)

3

min(l,m)

u=0

u+N−l

u+N−m

N

l=0

N

α

k+1,l+1

1

0

N

l=0

N

α

k+1,l+1

N−1

m=0

m+1

N m N

N

l=0

N

α

k+1,l+1

N−[N ]

m=[N ]

m+1

N m N

N

l=0

N

α

k+1,l+1

N−[N ]

m=[N ]

N

α

−1l+1,m+1

N

l=0

N

α

k+1,l+1

[N ]−1

m=0

m+1

N m N

N

l=0

N

α

k+1,l+1

N

m=N−[n]+1

m+1

N m N

N

N

l=0

N

α

k+1,l+1

N−1

m=0

N

l=0

N

α

k+1,l+1

N−1

m=0

0

1,N

2,N

1,N

N

l=0

α

l

m=0

m

u=0

u

l−m+u

2,N

N

l=0

α

N−1

m=l+1

l

u=0

u

m−l+u

−α

−α

1,N

N

l=0

α

l

m=0

−α

+∞

u=m+1

u

l−m+u

(14)

2,N

N

l=0

α

N

m=l+1

−α

+∞

u=l

u

m−l+u

N

N

l=0

α

N

m=0

−α

N

l=0

α

l

m=0 +∞

u=m

u

l−m+u

N

l=0

α

N

m=l+1 +∞

u=l

u

m−l+u

N

l=0

α

N

m=0

1,N

N

l=0

α

N

m=0

2,N

N

l=0

α

N

m=0

3,N

1,N

2,N

3,N

N

N

m=0

−α

0,N

N

m=0 +∞

u=min(l,m)

u

β

max(l,m)−min(l,m)+u

j,N

N

m=0

j,N

Lemma 2

1

2

Nl

Nl0

1

2

N

N

0

Nl

Nl0

1−s

j,N

j,N

0

Nl

Nl0

1−s

0

1

2

1−s

N

l=0

α

N

m=0

−α

α

k−N δ−1

l=0

−2α−1

N

N

α

N

k+N δ+1

−2α−1

N

N

α

N

l<0

−2α−1

l>N

−2α−1

(15)

N

l=0

α

j,N

α

k−N δ−1

l=0

−2α−1

j,N

j,N

α N

k+N δ+1

−2α−1

j,N

j,N

α

j,N

l<0

−2α−1

l>N

−2α−1

−α

0,1−s

N

l=0

α

N

m=0

−α

α

1 0

−α

−α

2α+1

−2{α

−2α

−α

N→+∞ k

N

−α

α

N

m=0

−α

α

−α

α

1

0

α

0,1−s

N

l=0

α

3

j=0

j,N

α

1

0

1

0

α

1

0

α

2α+1

−2{α

−2α

1

0

α

N

l=0

α

3

j=0

j,N

α

α

N→+∞

N

α

−α

α

that ends the demonstration.

Références

Documents relatifs

Key words: spectral fractional Laplacian, Dirichlet problem, boundary blow-up solutions, large solutions MSC 2010: Primary: 35B40; Secondary: 35B30, 45P05, 35C15... Laplacian, as

The first testcase corresponds to the loop detection problem in the trajectory of a mobile robot using proprioceptive measurements only.. In this situation, we have to find the

This result is accom- plished by the use of a nonlocal version of the notion of viscosity solution with generalized boundary conditions, see [21, 2, 8] for an introduction of

Keywords: random matrices, norm of random matrices, approximation of co- variance matrices, compressed sensing, restricted isometry property, log-concave random vectors,

This last result is deduced from a more general remark of independent interest on the lack of controllability of any finite linear combination of eigenfunctions of systems with

1 In [8, Theorem 1.1] the local L ∞ estimate is proved for solutions of the homogeneous equation, while [14, Theorem 3.2] contains the global L ∞ bound for eigenfunctions.. Both

Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the moving

For Brownian motion, we derive an analytic expression for the probability that the span reaches 1 for the first time, then generalized to fBm.. Using large-scale numerical