Stable methods for solving the standard Abel integral equation by means of orthogonal polynomials.

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Journ´ ees d’Analyse Math´ ematique et Applications.

(JAMA 2011) Hammamet, 22-24 March 2011

Stable methods for solving the standard Abel integral equation by means of orthogonal polynomials.

Ammari Amara

^a

and Abderrazek Karoui

^b

a

Institut Pr´ eparatoire aux Etudes d’Ing´ enieurs de Bizerte.

b

D´ epartement de Math´ ematiques, Facult´ e des Sciences de Bizerte.

March 23, 2011

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Introduction Stable solution of Abel integral equation by means of orthogonal polynomials Another Stable method for solving the Abel deconvolution problem

Outline

1

Introduction

2

Stable solution of Abel integral equation

A Jacobi based method for Stable Solution of Abel equation A Legendre based method for Stable Solution

Numerical Results

3

Another Stable method for solving the Abel deconvolution problem

A Jacobi-Legendre based method for the Stable Solution of the Abel equation.

Numerical Results

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Introduction

The standard form of the Abel integral equation is given as follows, T

_α

u(x) =

Z

x

0

u(t )

(x − t)

^1−α

dt = f (x), 0 ≤ x ≤ 1.

where 0 < α < 1 is a positive real number, f (·) is the data function and u(·) is the unknown function to be computed.

In this talk, we built stable methods for the solution of the ill-posed problem.

These methods are explicit and they are based on the use of various families of orthogonal polynomials of the Legendre and Jacobi types.

Moreover, they have the advantage to ensure the stability of the

solution under a fairly weak condition on the functional spaces to

which the data function belongs. Also, we provide some numerical

examples that illustrate our proposed methods.

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A Jacobi based method for Stable Solution of Abel equation A Legendre based method for Stable Solution

Numerical Results

Theme 2 : Stable solution of Abel integral equation by means of

orthogonal polynomials

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Numerical Results

Two stable methods for solving the standard Abel integral equation

we study two stable methods for solving of the standard Abel integral equation

T

_α

u(x) = Z

x

0

u (t)

(x − t)

^1−α

dt = f (x), 0 ≤ x ≤ 1. (1) The first method is based on the use of Jacobi-types

orthogonal polynomials that constitute a basis of

L

²

([0, 1], (1 − x)

^α

x

^β

dx ). More precisely, for real numbers α, β > −1, we consider the set of orthonormal polynomials Q

_n^α,β

(x) =

s

n!(2n + α + β + 1)Γ(n + α + β + 1)

Γ(n + α + 1)Γ(n + β + 1) P

_n^α,β

(2x − 1), (2)

where P

n^(α,β)

(x) denotes the n−th degree Jacobi polynomial.

P

n^α,β

(x) =

⁽⁻¹⁾₂nn!ⁿ

(1 − x)

^−α

(1 + x)

^−β_dx^dⁿn

((1 − x)

^n+α

(1 + x)

^n+β

).

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Numerical Results

Two stable methods for solving the standard Abel integral equation

we study two stable methods for solving of the standard Abel integral equation

T

_α

u(x) = Z

x

0

u(t)

(x − t)

^1−α

dt = f (x), 0 ≤ x ≤ 1. (1)

The first method is based on the use of Jacobi-types orthogonal polynomials that constitute a basis of

L

²

([0, 1], (1 − x)

^α

x

^β

dx ). More precisely, for real numbers α, β > −1, we consider the set of orthonormal polynomials Q

_n^α,β

(x) =

s

n!(2n + α + β + 1)Γ(n + α + β + 1)

Γ(n + α + 1)Γ(n + β + 1) P

_n^α,β

(2x − 1), (2)

where P

n^(α,β)

(x) denotes the n−th degree Jacobi polynomial.

P

n^α,β

(x) =

⁽⁻¹⁾₂nn!ⁿ

(1 − x)

^−α

(1 + x)

^−β_dx^dⁿn

((1 − x)

^n+α

(1 + x)

^n+β

).

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Numerical Results

Two stable methods for solving the standard Abel integral equation

we study two stable methods for solving of the standard Abel integral equation

T

_α

u(x) = Z

x

0

u(t)

(x − t)

^1−α

dt = f (x), 0 ≤ x ≤ 1. (1) The first method is based on the use of Jacobi-types

orthogonal polynomials that constitute a basis of

L

²

([0, 1], (1 − x )

^α

x

^β

dx ). More precisely, for real numbers α, β > −1, we consider the set of orthonormal polynomials Q

_n^α,β

(x) =

s

n!(2n + α + β + 1)Γ(n + α + β + 1)

Γ(n + α + 1)Γ(n + β + 1) P

_n^α,β

(2x − 1), (2) where P

n^(α,β)

(x) denotes the n−th degree Jacobi polynomial.

P

n^α,β

(x) =

⁽⁻¹⁾₂nn!ⁿ

(1 − x)

^−α

(1 + x)

^−β_dx^dⁿn

((1 − x)

^n+α

(1 + x)

^n+β

).

(8)

Numerical Results

Two stable methods for solving the standard Abel integral equation

we study two stable methods for solving of the standard Abel integral equation

T

_α

u(x) = Z

x

0

u(t)

(x − t)

^1−α

dt = f (x), 0 ≤ x ≤ 1. (1) The first method is based on the use of Jacobi-types

orthogonal polynomials that constitute a basis of

L

²

([0, 1], (1 − x )

^α

x

^β

dx ). More precisely, for real numbers α, β > −1, we consider the set of orthonormal polynomials Q

_n^α,β

(x) =

s

n!(2n + α + β + 1)Γ(n + α + β + 1)

Γ(n + α + 1)Γ(n + β + 1) P

_n^α,β

(2x − 1), (2) where P

n^(α,β)

(x) denotes the n−th degree Jacobi polynomial.

n

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Numerical Results

The second method is based on the expansion of the solution with respect to the basis of Legendre polynomials P

_n

that are orthonormal over [0, 1].

The following results given in [Ammari-Karoui, 2010][2], show the utility of the choice for Jacobi polynomials :

∀ ω ∈ [0, 1], we have

T

α

(w

^β

Q

_n^µ,β

)(w) = Γ(α) q

Γ(n+µ−α+1)Γ(n+β+1)

Γ(n+β+α+1)Γ(n+µ+1)

w

^β+α

Q

_n^{µ−α,β+α}

(w ). We know that if u, f in L

²

([0, 1], dx ), then the solution of the Abel equation T

_α

u = f satisfies the following identity

R

x

0

u(t)dt =

^sin(απ)_π

T

1−α

f (x), 0 ≤ x ≤ 1.

Let E

_α

denote the normed space given as follows, E

_α

=

( f =

∞

P

n=0

f

_n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx); P

n≥0

|f

_n

| n

^α+³²

< ∞ )

. A norm k · k

_α

is defined on E

α

by, kf k

²_α

= P

n≥0

f

_n²

(n + 2)

^2α

.

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Numerical Results

The second method is based on the expansion of the solution with respect to the basis of Legendre polynomials P

_n

that are orthonormal over [0, 1].

The following results given in [Ammari-Karoui, 2010][2], show the utility of the choice for Jacobi polynomials :

∀ ω ∈ [0, 1], we have

T

α

(w

^β

Q

_n^µ,β

)(w) = Γ(α) q

Γ(n+µ−α+1)Γ(n+β+1)

Γ(n+β+α+1)Γ(n+µ+1)

w

^β+α

Q

_n^{µ−α,β+α}

(w ). We know that if u, f in L

²

([0, 1], dx ), then the solution of the Abel equation T

_α

u = f satisfies the following identity

R

x

0

u(t)dt =

^sin(απ)_π

T

1−α

f (x), 0 ≤ x ≤ 1.

Let E

_α

denote the normed space given as follows, E

_α

=

( f =

∞

P

n=0

f

_n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx); P

n≥0

|f

_n

| n

^α+³²

< ∞ )

. A norm k · k

_α

is defined on E

α

by, kf k

²_α

= P

n≥0

f

_n²

(n + 2)

^2α

.

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Numerical Results

The second method is based on the expansion of the solution with respect to the basis of Legendre polynomials P

_n

that are orthonormal over [0, 1].

The following results given in [Ammari-Karoui, 2010][2], show the utility of the choice for Jacobi polynomials :

∀ ω ∈ [0, 1], we have

T

_α

(w

^β

Q

_n^µ,β

)(w) = Γ(α) q

Γ(n+µ−α+1)Γ(n+β+1)

Γ(n+β+α+1)Γ(n+µ+1)

w

^β+α

Q

_n^{µ−α,β+α}

(w ).

We know that if u, f in L

²

([0, 1], dx ), then the solution of the Abel equation T

_α

u = f satisfies the following identity

R

x

0

u(t)dt =

^sin(απ)_π

T

1−α

f (x), 0 ≤ x ≤ 1.

Let E

_α

denote the normed space given as follows, E

_α

=

( f =

∞

P

n=0

f

_n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx); P

n≥0

|f

_n

| n

^α+³²

< ∞ )

. A norm k · k

_α

is defined on E

α

by, kf k

²_α

= P

n≥0

f

_n²

(n + 2)

^2α

.

(12)

Numerical Results

The second method is based on the expansion of the solution with respect to the basis of Legendre polynomials P

_n

that are orthonormal over [0, 1].

The following results given in [Ammari-Karoui, 2010][2], show the utility of the choice for Jacobi polynomials :

∀ ω ∈ [0, 1], we have

T

_α

(w

^β

Q

_n^µ,β

)(w) = Γ(α) q

Γ(n+µ−α+1)Γ(n+β+1)

Γ(n+β+α+1)Γ(n+µ+1)

w

^β+α

Q

_n^{µ−α,β+α}

(w ).

We know that if u, f in L

²

([0, 1], dx ), then the solution of the Abel equation T

_α

u = f satisfies the following identity

R

x

0

u(t)dt =

^sin(απ)_π

T

1−α

f (x), 0 ≤ x ≤ 1.

Let E

_α

denote the normed space given as follows, E

_α

=

( f =

∞

P

n=0

f

_n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx); P

n≥0

|f

_n

| n

^α+³²

< ∞ )

. A norm k · k

_α

is defined on E

α

by, kf k

²_α

= P

n≥0

f

_n²

(n + 2)

^2α

.

(13)

Numerical Results

The second method is based on the expansion of the solution with respect to the basis of Legendre polynomials P

_n

that are orthonormal over [0, 1].

The following results given in [Ammari-Karoui, 2010][2], show the utility of the choice for Jacobi polynomials :

∀ ω ∈ [0, 1], we have

T

_α

(w

^β

Q

_n^µ,β

)(w) = Γ(α) q

Γ(n+µ−α+1)Γ(n+β+1)

Γ(n+β+α+1)Γ(n+µ+1)

w

^β+α

Q

_n^{µ−α,β+α}

(w ).

We know that if u, f in L

²

([0, 1], dx ), then the solution of the Abel equation T

_α

u = f satisfies the following identity

R

x

0

u(t)dt =

^sin(απ)_π

T

1−α

f (x), 0 ≤ x ≤ 1.

Let E

α

denote the normed space given as follows, E

_α

=

( f =

∞

P

n=0

f

_n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx); P

n≥0

|f

_n

| n

^α+³²

< ∞ )

. A norm k · k

_α

is defined on E

α

by, kf k

²_α

= P

n≥0

f

_n²

(n + 2)

^2α

.

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Numerical Results

The following proposition give the formula of the exact solution of equation (1) by the first method

Proposition

Let 0 < α < 1, and assume that f =

∞

P

n=0

f

_n

Q

_n^1−α,0

∈ E

_α

, then the solution of the equation T

α

u = f is given by the following formula :

u(x) = sin(απ) (1 − α)π

X

n≥0

f

_n

(−1)

ⁿ

p

(2n + 2 − α) (2 − α)

n

h

_n

(x)

= sin(απ) (1 − α)π

X

n≥0

f

n

(−1)

ⁿ

Γ(1 − α) p

(2n + 2 − α)

Γ(n + 2 − α) h

n

(x), (3)

where h

_n

(x) =

_dx^dⁿ⁺¹n+1

((1 − x)

ⁿ

x

^n+1−α

), x ∈ [0, 1].

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Numerical Results

The following proposition give the formula of the exact solution of equation (1) by the first method

Proposition

Let 0 < α < 1, and assume that f =

∞

P

n=0

f

_n

Q

_n^1−α,0

∈ E

_α

, then the solution of the equation T

α

u = f is given by the following formula :

u(x) = sin(απ) (1 − α)π

X

n≥0

f

_n

(−1)

ⁿ

p

(2n + 2 − α) (2 − α)

n

h

_n

(x)

= sin(απ) (1 − α)π

X

n≥0

f

n

(−1)

ⁿ

Γ(1 − α) p

(2n + 2 − α)

Γ(n + 2 − α) h

n

(x), (3)

where h

_n

(x) =

_dx^dⁿ⁺¹n+1

((1 − x)

ⁿ

x

^n+1−α

), x ∈ [0, 1].

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Numerical Results

The following proposition give the formula of the exact solution of equation (1) by the first method

Proposition

Let 0 < α < 1, and assume that f =

∞

P

n=0

f

_n

Q

_n^1−α,0

∈ E

_α

, then the solution of the equation T

α

u = f is given by the following formula :

u(x) = sin(απ) (1 − α)π

X

n≥0

f

_n

(−1)

ⁿ

p

(2n + 2 − α) (2 − α)

n

h

_n

(x)

= sin(απ) (1 − α)π

X

n≥0

f

n

(−1)

ⁿ

Γ(1 − α) p

(2n + 2 − α)

Γ(n + 2 − α) h

n

(x), (3)

where h

_n

(x) =

_dx^dⁿ⁺¹n+1

((1 − x)

ⁿ

x

^n+1−α

), x ∈ [0, 1].

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Numerical Results

Stability of the first method of equation (1)

It is important to mention that the previous method for solving (1) is stable. This is given by the following theorem. Theorem

Let 0 < α < 1 and assume that f ∈ E

_α

, where the space E

_α

is given by the previous formula. Then, the solution u of the equation T

_α

u = f belongs to the weighted space

L

²

([0, 1], d σ) = L

²

([0, 1], (1 − x)x

^α

dx). Moreover, this solution is stable in the sense that if e f =

∞

P

n=0

f e

n

Q

n^1−α,0

∈ E

α

and e u ∈ L

²

([0, 1], d σ) satisfy T

α

e u = e f , then we have

ku − e uk

_σ

≤ 1 Γ(α)

f − e f

_α

, (4)

where k · k

_σ

is the usual norm of L

²

([0, 1], d σ).

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Numerical Results

Stability of the first method of equation (1)

It is important to mention that the previous method for solving (1) is stable. This is given by the following theorem.

Theorem

Let 0 < α < 1 and assume that f ∈ E

_α

, where the space E

_α

is given by the previous formula. Then, the solution u of the equation T

_α

u = f belongs to the weighted space

L

²

([0, 1], d σ) = L

²

([0, 1], (1 − x)x

^α

dx). Moreover, this solution is stable in the sense that if e f =

∞

P

n=0

f e

n

Q

n^1−α,0

∈ E

α

and e u ∈ L

²

([0, 1], d σ) satisfy T

α

e u = e f , then we have

ku − e uk

_σ

≤ 1 Γ(α)

f − e f

_α

, (4)

where k · k

_σ

is the usual norm of L

²

([0, 1], d σ).

(19)

Numerical Results

Stability of the first method of equation (1)

It is important to mention that the previous method for solving (1) is stable. This is given by the following theorem.

Theorem

Let 0 < α < 1 and assume that f ∈ E

_α

, where the space E

_α

is given by the previous formula. Then, the solution u of the equation T

_α

u = f belongs to the weighted space

L

²

([0, 1], d σ) = L

²

([0, 1], (1 − x)x

^α

dx). Moreover, this solution is stable in the sense that if e f =

∞

P

n=0

f e

n

Q

n^1−α,0

∈ E

α

and e u ∈ L

²

([0, 1], d σ) satisfy T

α

e u = e f , then we have

ku − e uk

_σ

≤ 1 Γ(α)

f − e f

_α

, (4)

where k · k

_σ

is the usual norm of L

²

([0, 1], d σ).

(20)

Numerical Results

An error analysis associated with the truncated solution is given by the following proposition.

Proposition

Let N ≥ 1 be a positive integer and let

e u

N

(x) = sin(απ) π

N−1

X

n=0

e f

n

(−1)

ⁿ

Γ(1 − α) p

(2n + 2 − α) Γ(n + 2 − α) h

n

(x ),

be the approximate solution of equation T

_α

e u = e f . Here, the

e f

n

0≤n≤N−1

are the expansion coefficients of the perturbed function e f . Assume that kf − e f k

_α

≤ Γ(α)ε, where f is the noise free function, satisfying T

α

u = f . Then, we have

ku − e u

N

k

_σ

≤ 1 Γ(α)

v u u t

∞

X

n=N

f

_n²

(n + 2)

^2α

+ ε

(21)

Numerical Results

An error analysis associated with the truncated solution is given by the following proposition.

Proposition

Let N ≥ 1 be a positive integer and let

e u

N

(x) = sin(απ) π

N−1

X

n=0

e f

n

(−1)

ⁿ

Γ(1 − α) p

(2n + 2 − α) Γ(n + 2 − α) h

n

(x ),

be the approximate solution of equation T

_α

e u = e f . Here, the

e f

n

0≤n≤N−1

are the expansion coefficients of the perturbed function e f . Assume that kf − e f k

_α

≤ Γ(α)ε, where f is the noise free function, satisfying T

α

u = f . Then, we have

ku − e u

N

k

_σ

≤ 1 Γ(α)

v u u t

∞

X

n=N

f

_n²

(n + 2)

^2α

+ ε

(22)

Numerical Results

An error analysis associated with the truncated solution is given by the following proposition.

Proposition

Let N ≥ 1 be a positive integer and let

e u

N

(x) = sin(απ) π

N−1

X

n=0

e f

n

(−1)

ⁿ

Γ(1 − α) p

(2n + 2 − α) Γ(n + 2 − α) h

n

(x ),

be the approximate solution of equation T

_α

e u = e f . Here, the

e f

n

0≤n≤N−1

are the expansion coefficients of the perturbed function e f . Assume that kf − e f k

_α

≤ Γ(α)ε, where f is the noise free function, satisfying T

α

u = f . Then, we have

ku − e u

N

k

_σ

≤ 1 Γ(α)

v u u t

∞

X

n=N

f

_n²

(n + 2)

^2α

+ ε

(23)

Numerical Results

Remark : In practice, for smooth enough function f , the remainder term

v u u t

∞

X

n=N

f

_n²

(n + 2)

^2α

has a fast decay. In fact under the assumption that there exist a positive real number s > α + 1/2 and a constant k > 0 such that |f

_n

| ≤ k (n + 2)

^−s

, ∀ n ≥ 1, we have P

n≥N

f

_n²

(n + 2)

^2α

≤ k

²

P

n≥N+2

n

^2(α−s)

≤ k

²

R

+∞

N+1

x

^2(α−s)

dx =

(2(α−s)+1)(N+1)^k² ^{2(s−α)−1}

. The appropriate number of modes N

_ε

, to be used is given in this case is given by, N

ε

= inf

n

N ≥ 1,

_2(α−s)+1^k² _(N+1)2(s−α)−1¹

≤ kf − e f k

²_α

o .

The following theorem give the formula of the exact solution

of equation (1) by the second method

(24)

Numerical Results

Remark : In practice, for smooth enough function f , the remainder term

v u u t

∞

X

n=N

f

_n²

(n + 2)

^2α

has a fast decay. In fact under the assumption that there exist a positive real number s > α + 1/2 and a constant k > 0 such that |f

_n

| ≤ k (n + 2)

^−s

, ∀ n ≥ 1, we have P

n≥N

f

_n²

(n + 2)

^2α

≤ k

²

P

n≥N+2

n

^2(α−s)

≤ k

²

R

+∞

N+1

x

^2(α−s)

dx =

(2(α−s)+1)(N+1)^k² ^{2(s−α)−1}

. The appropriate number of modes N

_ε

, to be used is given in this case is given by, N

ε

= inf

n

N ≥ 1,

_2(α−s)+1^k² _(N+1)2(s−α)−1¹

≤ kf − e f k

²_α

o .

The following theorem give the formula of the exact solution

of equation (1) by the second method

(25)

Numerical Results

Theorem

Let 0 < α < 1 and let f = P

s≥0

f

s

P

s

(t) ∈ L

²

([0, 1]). If u(t) = P

m≥0

α

_m

P

_m

(t) ∈ L

²

([0, 1]) is the solution of the Abel integral equation T

α

u = f , then ∀ m ≥ 0, we have α

m

=

P

s≥0

f

_s

(−1)

^s

√2m+1√ 2s+1 m!Γ(α)

β

_m,α

m

P

l=0

(−1)

^{l m}_l

^s−1

Q

j=0

(l − α − j )

_{Γ(l−α+s+2)}^Γ(m+l+1)

.

where β

_m,α

= Γ(α)

q

_Γ(m−α+1)

Γ(m+α+1)

∼ m

^−α

Let σ

₀

(α) > 0, be the positive real number, given as follows,

σ

0

(α) = inf







η > 0, X

m≥0

X

s≥0

< P

_s

, F

_m

>

²

β

_m,α²

(1 + (s(s + 1))

^η

)

²

< +∞







.

(5)

where F

m

(x) = (1 − x )

^−α

Q

m^−α,α

(x).

(26)

Numerical Results

Theorem

Let 0 < α < 1 and let f = P

s≥0

f

s

P

s

(t) ∈ L

²

([0, 1]). If u(t) = P

m≥0

α

_m

P

_m

(t) ∈ L

²

([0, 1]) is the solution of the Abel integral equation T

α

u = f , then ∀ m ≥ 0, we have α

m

=

P

s≥0

f

_s

(−1)

^s

√2m+1√ 2s+1 m!Γ(α)

β

_m,α

m

P

l=0

(−1)

^{l m}_l

^s−1

Q

j=0

(l − α − j )

_{Γ(l−α+s+2)}^Γ(m+l+1)

.

where β

_m,α

= Γ(α)

q

_Γ(m−α+1)

Γ(m+α+1)

∼ m

^−α

Let σ

₀

(α) > 0, be the positive real number, given as follows,

σ

0

(α) = inf







η > 0, X

m≥0

X

s≥0

< P

_s

, F

_m

>

²

β

_m,α²

(1 + (s(s + 1))

^η

)

²

< +∞







.

(5)

where F (x) = (1 − x)

^−α

Q

^−α,α

(x).

(27)

Numerical Results

Stability of the second method for solving equation (1)

For any real number σ > 0, we defined the space D(A

^σ

) =

n u = P

n≥0

b u

n

P

n

, P

n≥0

(1 + (n(n + 1))

^σ

)

²

| b u

n

|

²

< +∞ o , equipped by a norm k · k

_σ

given by

ku k

²_σ

= P

n≥0

(1 + (n(n + 1))

^σ

)

²

| b u

n

|

²

. And A : u 7→

_dx^d

[(1 − x)x

^du_dx

]

Under the above notation, the following proposition shows that our second method for solving the Abel equation (1) is stable.

Proposition

Let 0 < α < 1 and let σ be any positive real number satisfying

σ > σ

₀

(α). Assume that f ∈ D(A

^σ

), then the solution of the

equation T

_α

u = f , given by the previous method is stable.

(28)

Numerical Results

Stability of the second method for solving equation (1)

For any real number σ > 0, we defined the space D (A

^σ

) = n

u = P

n≥0

b u

_n

P

_n

, P

n≥0

(1 + (n(n + 1))

^σ

)

²

| b u

_n

|

²

< +∞ o , equipped by a norm k · k

_σ

given by

ku k

²_σ

= P

n≥0

(1 + (n(n + 1))

^σ

)

²

| b u

n

|

²

. And A : u 7→

_dx^d

[(1 − x)x

^du_dx

]

Under the above notation, the following proposition shows that our second method for solving the Abel equation (1) is stable.

Proposition

Let 0 < α < 1 and let σ be any positive real number satisfying

σ > σ

₀

(α). Assume that f ∈ D(A

^σ

), then the solution of the

equation T

_α

u = f , given by the previous method is stable.

(29)

Numerical Results

Stability of the second method for solving equation (1)

For any real number σ > 0, we defined the space D (A

^σ

) = n

u = P

n≥0

b u

_n

P

_n

, P

n≥0

(1 + (n(n + 1))

^σ

)

²

| b u

_n

|

²

< +∞ o , equipped by a norm k · k

_σ

given by

ku k

²_σ

= P

n≥0

(1 + (n(n + 1))

^σ

)

²

| b u

n

|

²

. And A : u 7→

_dx^d

[(1 − x)x

^du_dx

]

Under the above notation, the following proposition shows that our second method for solving the Abel equation (1) is stable.

Proposition

Let 0 < α < 1 and let σ be any positive real number satisfying

σ > σ

₀

(α). Assume that f ∈ D(A

^σ

), then the solution of the

equation T

_α

u = f , given by the previous method is stable.

(30)

Numerical Results

Stability of the second method for solving equation (1)

For any real number σ > 0, we defined the space D (A

^σ

) = n

u = P

n≥0

b u

_n

P

_n

, P

n≥0

(1 + (n(n + 1))

^σ

)

²

| b u

_n

|

²

< +∞ o , equipped by a norm k · k

_σ

given by

ku k

²_σ

= P

n≥0

(1 + (n(n + 1))

^σ

)

²

| b u

n

|

²

. And A : u 7→

_dx^d

[(1 − x)x

^du_dx

]

Under the above notation, the following proposition shows that our second method for solving the Abel equation (1) is stable.

Proposition

Let 0 < α < 1 and let σ be any positive real number satisfying

σ > σ

₀

(α). Assume that f ∈ D(A

^σ

), then the solution of the

equation T

α

u = f , given by the previous method is stable.

(31)

Numerical Results

Corollary

Let 0 < α < 1, let σ > σ

0

(α) be a positive real number and let N ≥ 0, be a positive integer. Assume that f ∈ D(A

^σ

) and let e f , be the perturbation of the function f , given by e f = f + η, where η ∈ L

²

([0, 1]), is the noise function. If u, e u

N

, denote the solutions of T

α

u = f , T

α

e u

_N

= (π

_N

e f ), then we have ∀0 < < σ − σ

0

(α).

ku − e u

_N

k

_L2([0,1])

≤

√Cαkfk_σ

((N+1)(N+2))^σ−σ⁰⁽^α)−

+ √

C

_α

kπ

_N

ηk

_σ₀_(α)+

.

Remark : Numerical evidences show that P

s≥0

<P_s,F_m>²

(1+(s(s+1))^η)²

= O m

^−4η

. Moreover, since β

²_m,α

∼ m

^−2α

, then we can chose σ

0

(α) =

¹₄

+

^α₂

defined by (5). This means that our Legendre based method for solving (1) is numerically valid on a fairly large subspace E e

α,

of L

²

([0, 1]), given by

E e

α,

= (

f =

∞

P

n=0

f

n

P

n

∈ L

²

([0, 1]); P

n≥0

(1 + (n(n + 1))

^1/4+α/2+

)

²

(f

n

)

²

< ∞

)

(32)

Numerical Results

Corollary

Let 0 < α < 1, let σ > σ

0

(α) be a positive real number and let N ≥ 0, be a positive integer. Assume that f ∈ D(A

^σ

) and let e f , be the perturbation of the function f , given by e f = f + η, where η ∈ L

²

([0, 1]), is the noise function. If u, e u

N

, denote the solutions of T

α

u = f , T

α

e u

_N

= (π

_N

e f ), then we have ∀0 < < σ − σ

0

(α).

ku − e u

_N

k

_L2([0,1])

≤

√Cαkfk_σ

((N+1)(N+2))^σ−σ⁰⁽^α)−

+ √

C

_α

kπ

_N

ηk

_σ₀_(α)+

.

Remark : Numerical evidences show that P

s≥0

<P_s,F_m>²

(1+(s(s+1))^η)²

= O m

^−4η

. Moreover, since β

²_m,α

∼ m

^−2α

, then we can chose σ

0

(α) =

¹₄

+

^α₂

defined by (5). This means that our Legendre based method for solving (1) is numerically valid on a fairly large subspace E e

α,

of L

²

([0, 1]), given by

E e

α,

= (

f =

∞

P

n=0

f

n

P

n

∈ L

²

([0, 1]); P

n≥0

(1 + (n(n + 1))

^1/4+α/2+

)

²

(f

n

)

²

< ∞

)

(33)

Numerical Results

Example 1 : solved by the first method

We first consider α = 1/10 and we apply our proposed method with N = 25 to solve equation (1), with the noisy data function e f

₁

(x) = f

₁

(x) + η(x), where f

₁

(x) = 10(x −

¹₄

)

^0.9

1

_[¹

4,1]

(x),

η(x) = 0.01 sin(100πx), 0 ≤ x ≤ 1.

Figure: (a) graphs of the exact and numerical solution, (b) graph of the

error corresponding to the function e f

1

.

(34)

Numerical Results

Example 2 : solved by the second method

In this example, we consider the value α = 0.5 and the smooth function f

₂

, given by f

₂

(x) = 16

3Γ(0.5) (x

^5/2

− x

^3/2

). In this case, the exact solution of (1) is given by u(x) = 5x

²

− 4x. Then, we contaminate the function f

₂

with a 35 db SNR Gaussian white noise η

₂

. Here, the SNR denotes the signal to noise ratio, given by the formula SNR = 20 log

₁₀

kf k

₂

kηk

₂

, where f is the signal

function and η is the noise. The graph of noisy function e f

2

, is

given by Figure 2(a). Next, we apply the Legendre polynomials

based method with N = 8 and compute an approximate solution

e u

_N

(x) to the exact solution u(x). The graphs of u(x) and e u

_N

are

given by Figure 2(b).

(35)

Numerical Results

Figure: (a) graph of the noised function e f

₂

, (b) graphs of the exact and

numerical solution corresponding to e f

₂

.

(36)

A Jacobi-Legendre based method for the Stable Solution of the Abel equation.

Numerical Results

Theme 3 : Another Stable method for solving the Abel deconvolution problem

T

α

u(x) = Z

x

0

u(t)

(x − t)

^1−α

dt = f (x) =

∞

X

n=0

f

n

Q

_n^α,β

(37)

Numerical Results

We use a Jacobi-Legendre based method for the stable solution of the problem (1). This method is valid under the weak condition that f (·) belongs to the following functional space

F

α

=



 

  f =

∞

P

n=0

f

n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx); kf k = P

n≥0

|f

n

| √

2n + 2 − α < ∞



 

  For a given three real numbers α, β, µ satisfying

0 < α < 1, 2α > β > −1, µ > 0, we consider the weighted Sobolev type functional space F

α,β,µ

, given by

F

α,β,µ

=

f = P

n≥0

f

n

x

^α+β

Q

_n^{µ−α,β+α}

∈ L

²

([0, 1], x

^−α−β

(1 − x)

^µ−α

dx)

; P

n≥0

(f

n

)

²

(1 + n)

^2α

< +∞

Then, we prove that the Abel equation (1) has a stable solution

belonging to L

²

([0, 1], (1 − x)

^µ

x

^−β

dx), whenever f ∈ F

_α,β,µ

.

(38)

Numerical Results

We use a Jacobi-Legendre based method for the stable solution of the problem (1). This method is valid under the weak condition that f (·) belongs to the following functional space

F

α

=



 

  f =

∞

P

n=0

f

n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx); kf k = P

n≥0

|f

n

| √

2n + 2 − α < ∞



 

  For a given three real numbers α, β, µ satisfying

0 < α < 1, 2α > β > −1, µ > 0, we consider the weighted Sobolev type functional space F

α,β,µ

, given by

F

α,β,µ

=

f = P

n≥0

f

n

x

^α+β

Q

_n^{µ−α,β+α}

∈ L

²

([0, 1], x

^−α−β

(1 − x)

^µ−α

dx)

; P

n≥0

(f

n

)

²

(1 + n)

^2α

< +∞

Then, we prove that the Abel equation (1) has a stable solution

belonging to L

²

([0, 1], (1 − x)

^µ

x

^−β

dx), whenever f ∈ F

_α,β,µ

.

(39)

Numerical Results

We use a Jacobi-Legendre based method for the stable solution of the problem (1). This method is valid under the weak condition that f (·) belongs to the following functional space

F

α

=



 

  f =

∞

P

n=0

f

n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx);

kf k = P

n≥0

|f

n

| √

2n + 2 − α < ∞



 

 

For a given three real numbers α, β, µ satisfying

0 < α < 1, 2α > β > −1, µ > 0, we consider the weighted Sobolev type functional space F

α,β,µ

, given by

F

α,β,µ

=

f = P

n≥0

f

n

x

^α+β

Q

_n^{µ−α,β+α}

∈ L

²

([0, 1], x

^−α−β

(1 − x)

^µ−α

dx)

; P

n≥0

(f

n

)

²

(1 + n)

^2α

< +∞

Then, we prove that the Abel equation (1) has a stable solution

belonging to L

²

([0, 1], (1 − x)

^µ

x

^−β

dx), whenever f ∈ F

_α,β,µ

.

(40)

Numerical Results

We use a Jacobi-Legendre based method for the stable solution of the problem (1). This method is valid under the weak condition that f (·) belongs to the following functional space

F

α

=



 

  f =

∞

P

n=0

f

n

Q

_n^1−α,0

∈ L

²

([0, 1], (1 − x)

^1−α

dx);

kf k = P

n≥0

|f

n

| √

2n + 2 − α < ∞



 

  For a given three real numbers α, β, µ satisfying

0 < α < 1, 2α > β > −1, µ > 0, we consider the weighted Sobolev type functional space F

α,β,µ

, given by

F

α,β,µ

=

f = P

n≥0

f

n

x

^α+β

Q

_n^{µ−α,β+α}

∈ L

²

([0, 1], x

^−α−β

(1 − x )

^µ−α

dx)

; P

n≥0

(f

n

)

²

(1 + n)

^2α

< +∞

Then, we prove that the Abel equation (1) has a stable solution

belonging to L

²

([0, 1], (1 − x)

^µ

x

^−β

dx), whenever f ∈ F

_α,β,µ

.

(41)

Numerical Results

The following theorem is needed to compute the exact solution of the Abel equation (1) by a Jacobi-Legendre based method.

Theorem

For any positive integers n, k and any real 0 < α < 1, we have

I

_n,k

(α) = √

2n + 2 − α √

2k + 1 − Z

1

0

x

^1−α

Q

_n^0,1−α

(x)P

_k⁰

(x)dx

= sin(πα) π

(−1)

^n+k

√

2n + 2 − α √ 2k + 1 n!

n

X

l=0

n l

Γ(n + l + 2 − α)Γ(l − α + 1)Γ(k − l + α)

Γ(k + l + 2 − α) (6)

(42)

Numerical Results

The following theorem is needed to compute the exact solution of the Abel equation (1) by a Jacobi-Legendre based method.

Theorem

For any positive integers n, k and any real 0 < α < 1, we have

I

_n,k

(α) = √

2n + 2 − α √

2k + 1 − Z

1

0

x

^1−α

Q

_n^0,1−α

(x)P

_k⁰

(x)dx

= sin(πα) π

(−1)

^n+k

√

2n + 2 − α √ 2k + 1 n!

n

X

l=0

n l

Γ(n + l + 2 − α)Γ(l − α + 1)Γ(k − l + α)

Γ(k + l + 2 − α) (6)

(43)

Numerical Results

Explicit solution of (1)

Theorem

Let 0 < α < 1, and assume that f ∈ F

α

. If u(t) = P

k≥0

β

k

P

k

(t) is the solution of the equation T

_α

u = f , then for all k ≥ 1 β

_k

is given by the following formula

β

_k

= √

2k + 1β

₀

− sin(απ) π

Z

₁

0

T

1−α

f (x)P

_k⁰

(x)dx

= 1

Γ(α)

k−1

X

n=0

f

_n

n!

Γ(n + 2 − α) I

_n,k

(α) (7) +

√ 2k + 1 Γ(α)

∞

X

n=k

f

n

n!

Γ(n + 2 − α)

√

2n + 2 − α, (8)

where I

n,k

(α) is as given by previous theorem.

(44)

Numerical Results

Stability of the proposed solution of the Abel equation

To prove the stability of the method, we need to prove an almost sharp upper bound of the quantity I

_n,k

(α). This is the objective of the following theorem

Theorem

For any positive integers n, k and any real number 0 < α < 1,we have

|I

_n,k

(α)| ≤

√ 2n + 2 − αΓ(n + 2 − α)

n! χ

k

(α), (9)

with χ

_k

(α) =

√2k+1k^α

(k+1−α)

2

^α

− 1 + 2

^{α α}_2k

≤ 2

^1/2+α

1 +

_2−α^α

k

^α−¹²

.

(45)

Numerical Results

Stability of the proposed solution of the Abel equation

To prove the stability of the method, we need to prove an almost sharp upper bound of the quantity I

_n,k

(α). This is the objective of the following theorem

Theorem

For any positive integers n, k and any real number 0 < α < 1,we have

|I

_n,k

(α)| ≤

√ 2n + 2 − αΓ(n + 2 − α)

n! χ

k

(α), (9)

with χ

_k

(α) =

√2k+1k^α

(k+1−α)

2

^α

− 1 + 2

^{α α}_2k

≤ 2

^1/2+α

1 +

_2−α^α

k

^α−¹²

.

(46)

Numerical Results

Theorem

Let 0 < α < 1, be a real number and let U

_α

be the subspace of L

²

([0, 1]), given by

U

_α

=







u ∈ L

²

([0, 1]) = X

k≥0

β

_k

P

_k

; kuk

_α

= sup

k≥0

|β

_k

| (k + 1)

¹²^−α

< +∞







Let f =

∞

P

n=0

f

_n

Q

n^1−α,0

∈ F

_α

, e f =

∞

P

n=0

f e

_n

Q

n^1−α,0

∈ F

_α

and let u = P

k≥0

β

_k

P

_k

(x) ∈ U

_α

, e u = P

k≥0

f β

_k

P

_k

(x) ∈ U

_α

be the solutions of the equation T

α

u = f , andT

α

e u = e f respectively. Then, we have

ku − e uk

_α

≤ C

_α

f − e f

.

with C

_α

=

_Γ(α)¹

max

1

(1−α)

, 2

^2+α

(1 +

_2−α^α

)

(47)

Numerical Results

Remark :

In practice, for a given integer N ≥ 1, we only compute e u

N

(x), a truncated version of e u(x), given by the following formula,

e u

_N

(x) =

N−1

X

k=0

β e

_k

P

_k

(x), 0 ≤ x ≤ 1, (10)

where the β e

_k

are as given by Theorem 2.

Note that if kf − e f k = , and sup

n≥N+1

|β

_n

|(n + 1/2)

^1/2−α

−→

N7→∞

0 then by using the previous theorem, one obtains the following bound of the approximation error,

ku− e u

N

(x)k

_α

≤ ku−u

_N

k

_α

+ku

_N

− e u

N

k

_α

≤ sup

n≥N+1

|β

_n

|(n+1/2)

^1/2−α

+C

α

.

(11)

(48)

Numerical Results

Remark :

In practice, for a given integer N ≥ 1, we only compute e u

N

(x), a truncated version of e u(x), given by the following formula,

e u

_N

(x) =

N−1

X

k=0

β e

_k

P

_k

(x), 0 ≤ x ≤ 1, (10)

where the β e

_k

are as given by Theorem 2.

Note that if kf − e f k = , and sup

n≥N+1

|β

_n

|(n + 1/2)

^1/2−α

−→

N7→∞

0 then by using the previous theorem, one obtains the following bound of the approximation error,

ku− e u

N

(x)k

_α

≤ ku−u

_N

k

_α

+ku

_N

− e u

N

k

_α

≤ sup

n≥N+1

|β

_n

|(n+1/2)

^1/2−α

+C

α

.

(11)

(49)

Numerical Results

Second method

A second but a more classical stability result concerning the

solution of the Abel integral equation (1) based on the use of

Jacobi polynomials is given by the following theorem.

(50)

Numerical Results

Theorem

Let α, β, µ, be any real numbers, satisfying 0 < α < 1, µ > 0, and 2α > β > −1. If f (·) ∈ F

_α,β,µ

, then the function

u(x) = X

n≥0

f

_n

η

_n,β,µ

(α)x

^β

Q

_n^µ,β

(x), (12)

η

_n,β,µ

(α) = 1 Γ(α)

s

Γ(n + β + α + 1)Γ(n + µ + 1) Γ(n + µ − α + 1)Γ(n + β + 1) (13) belongs to L

²

([0, 1], (1 − x)

^µ

x

^−β

dx) and it is a solution of (1).

Moreover, we have kuk

²

≤ (1 + µ)

^α

(1 + β )

^α

Γ

²

(α)

X

n≥0

|f

_n

|

²

(1+n)

^2α

= (1 + µ)

^α

(1 + β)

^α

Γ

²

(α) kf k

²

.

(14)

(51)

Numerical Results

Remark :

We should note that the solution of (1) given by the second method via (12) is stable and easier to compute than the one given by the Jacobi-Legendre based method. Nonetheless, this second method has some practical drawbacks. In fact, if β ≥ 0, then one has to compute the expansion coefficients of the auxiliary function g (x) = f (x)

x

^α+β

. In this case, the computation of g (x) in the neighborhood of the origin may result on some numerical

instabilities. If β < 0, then the computation of the solution u(x) via formula (1) suffers from a loss of accuracy in the neighborhood of the origin.

By the work of R. Gorenflo and M. Yamamato [4], we can obtain similar stability results associated with the solution of the generalized problem of Abel type by Jacobi type polynomials.

Ku(x) = Z

x

0

k (x, t )

(x − t )

^1−α

u(t)dt = f (x), 0 ≤ x ≤ 1.

Here, k(·) is a smooth function with k (0) 6= 0.

(52)

Numerical Results

Remark :

We should note that the solution of (1) given by the second method via (12) is stable and easier to compute than the one given by the Jacobi-Legendre based method. Nonetheless, this second method has some practical drawbacks. In fact, if β ≥ 0, then one has to compute the expansion coefficients of the auxiliary function g (x) = f (x)

x

^α+β

. In this case, the computation of g (x) in the neighborhood of the origin may result on some numerical

instabilities. If β < 0, then the computation of the solution u(x) via formula (1) suffers from a loss of accuracy in the neighborhood of the origin.

By the work of R. Gorenflo and M. Yamamato [4], we can obtain similar stability results associated with the solution of the generalized problem of Abel type by Jacobi type polynomials.