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On the Numerical Solution of Integro-Differential Equations of Prandtl’s Type

Maria Rosaria Capobianco

CNR - National Research Council of Italy Institute for Computational Applications “Mauro Picone”, Naples, Italy.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 1

OUTLINE OF THE COURSE

• Introduction on Integro-Differential Equations of Prandtl’s Type
• Mapping Properties of Hypersingular Operators
• Collocation and Quadrature Methods for Linear Equations
• Fast Algorithms for Linear Equations
• Collocation Method and Iterative Schemes for Nonlinear Equations
• Fast Algorithms for Nonlinear Equations

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 2

g(x)v(x) − 1 π 1

−1

v′(t) t − xdt + 1 π 1

−1

h(x, t)v(t)dt = f(x), (1) v(−1) = v(1) = 0. (2) For a function v ∈ Lp(−1, 1) possessing a generalized derivative v′ ∈ Lp(−1, 1),we have d dx 1

−1

v(t) t − xdt = 1

−1

v′(t) t − xdt − v(−1) 1 + x + v(1) 1 − x, x ∈ (−1, 1),

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 3

• Eq. (??) together with (??) can be written in the form

g(x)v(x)−1 π 1

−1

v(t) (t − x)2dt+1 π 1

−1

h(x, t)v(t)dt = f(x), −1 < x < 1 (3) where the hypersingular integral operator has to be understood in the sense of 1

−1

v(t) (t − x)2dt = d dx 1

−1

v(t) t − xdt (4) Most of the physical problems we can model with such equations, suggest that the solution of (??)-(??) or (??)-(??) has an endpoint behavior of the form √1 − x2. Thus, it is convenient to represent v as the product v(x) = ϕ(x)u(x), ϕ(x) =

• 1 − x2

(5)

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 4

◮ MAPPING PROPERTIES Multiplication Operator Γ(x) = g(x)ϕ(x), (MΓ)u(x) = Γ(x)u(x) (6) Cauchy Singular Integral Operator For real numbers a and b with a − ib = eiπα, 0 < α < 1, β = 1 − α, define the Jacobi weight function vα,β(x) = (1 − x)α(1 + x)β and the singular integral operator of Cauchy type (Au)(x) = avα,β(x)u(x) + b π 1

−1

u(t) t − xvα,β(t)dt (7)

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 5

If a = 0 and b = −1 (i.e. α = β = 1

2 )

(Su)(x) = −1 π 1

−1

u(t) t − xϕ(t)dt (8) Hypersingular Integral Operator (DAu)(x) = a d dx[vα,β(x)u(x)] + b π 1

−1

u(t) (t − x)2 vα,β(t) dt (9) V = DS, D = d dx, (α = β = 1 2) (10)

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 6

Kernel Integral Operator (Hu)(x) = 1 π 1

−1

h(x, t)v(t)dt, (11) We assume that the function h is continuous on [−1, 1]2. At first, we consider the hypersingular integral equation , written in operator form: (MΓ + V + H)u = f (12)

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 7

Let vγ,δ(x) = (1 − x)γ(1 + x)δ, γ, δ > −1 be a Jacobi weight and L2

γ,δ, γ, δ > −1 denote the weighted space

• f square integrable functions on the interval [−1, 1] endowed with

the scalar product u, vγ,δ = 1 π 1

−1

u(x)v(x)vγ,δ(x)dx, and the norm uγ,δ =

• u, uγ,δ.

Let pγ,δ

n

refer as the normalized Jacobi polynomial (with positive leading coefficient) of degree n with respect to the Jacobi weight vγ,δ.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 8

For real numbers s ≥ 0 define the weighted Sobolev space L2,s

γ,δ

by L2,s

γ,δ =

• u ∈ L2

γ,δ : ∞

• n=0

(1 + n)2s u, pγ,δ

n γ,δ

• 2 < ∞
• ,

with the norm uγ,δ,s = ∞

• n=0

(1 + n)2s u, pγ,δ

n γ,δ

• 2

1/2 .

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 9

In the following we summarize some results concerning the properties

• f weighted Sobolev spaces, of interpolation operators with respect

to the zeros of the orthogonal polynomials pγ,δ

n , the multiplication

• perator MΓ defined by (??), the hypersingular integral operator

V defined by (??) and the kernel operator H defined by (??). By L(X, Y) we will denote the Banach space of all bounded linear

• perators between the Banach spaces X and Y .

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 10

Lemma 1. [Berthold,Hoppe and Silbermann] For 0 ≤ s < t the space L2,t

γ,δ is compactly imbedded in L2,s γ,δ.

Lemma 2. [Junghanns] If the operator B belongs to L(L2,s1

α1,β1, L2,s2 α2,β2)

and L(L2,t1

α1,β1, L2,t2 α2,β2) then B ∈ L(L2,s(τ) α1,β1 , L2,t(τ) α2,β2), where s(τ) = (1 − τ)s1

and t(τ) = (1 − τ)s2 + τ t2 , 0 ≤ τ ≤ 1. Lemma 3. [Berthold,Hoppe and Silbermann] Let r ≥ 0 be an integer. Then u ∈ L2,r

γ,δ

if and only if u(k)ϕk belongs to L2

γ,δ for all k = 0, . . . , r. Moreover, the norms ||u||γ,δ,r and

||u||γ,δ,r,ϕ = r

k=0 ||u(k)ϕk||γ,δ are equivalent.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 11

Let xγ,δ

nk with −1 < xγ,δ nn < . . . < xγ,δ n1 be the zeros of pγ,δ n

and denote by Lγ,δ

n

the Lagrange interpolation operator Lγ,δ

n f = n

• k=1

f(xγ,δ

nk )lγ,δ nk ,

lγ,δ

nk (x) = n

• j=1,j=k

x − xγ,δ

nj

xγ,δ

nk − xγ,δ nj

. Lemma 4. [C.,Mastroianni] For s > 1/2 we have (a) limn→∞ ||f − Lγ,δ

n f||γ,δ,s = 0 for all f ∈ L2,s γ,δ,

(b) ||f − Lγ,δ

n f||γ,δ,t ≤ const nt−s||f||γ,δ,s if 0 ≤ t ≤ s.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 12

By Cr

ϕ,

r ≥ 0 an integer, we denote the space

• f

all r times differentiable functions u : (−1, 1) → C satisfying the conditions u(k)ϕk ∈ C[−1, 1] for k = 0, 1, . . . , r. Let ||u||Cr

ϕ = r

k=0 ||u(k)ϕk||∞.

Lemma 5. [Junghanns] Let r ≥ 0 be an integer and Γ ∈ Cr

ϕ.

Then the multiplication operator MΓ belongs to L(L2,r

γ,δ, L2,r γ,δ)

and ||MΓ||L2,r

γ,δ→L2,r γ,δ ≤ const||Γ||Cr ϕ.

Lemma 6. Taking into account Lemma 1, under the assumptions of Lemma 5, if MΓ ∈ L(L2

γ,δ, L2 γ,δ), the condition

Γ ∈ Cr

ϕ implies MΓ ∈ L(L2,r γ,δ, L2,r γ,δ) for 0 ≤ s ≤ r.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 13

We use the notations L2,s

ϕ

= L2,s

1/2,1/2, ., .ϕ = ., .1/2,1/2, and pϕ n = p1/2,1/2 n

, and L2,s,0

γ,δ

=

• f ∈ L2,s

γ,δ : f, pγ,δ

γ,δ = 0

• .

Lemma 7. [Berthold,Hoppe and Silbermann] For all s ≥ 0, the Cauchy singular integral operator A belongs to L(L2,s

α,β, L2,s −α,−β).

Moreover, A : L2,s

α,β → L2,s,0 −α,−β is a bijection, and the inverse

• perator is given by

A−1 = A, ( Af)(t) := av−α,−β − b π 1

−1

f(x) x − tv−α,−β(x) dx. (13) Lemma 8. [Proosdorf, Silbermann] For the Cauchy singular integral

• perator A defined in (??)( we recall that α, β > 0) we have the

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 14

relation Apα,β

n

= p−α,−β

n+1

, n = 0, 1, 2, . . . Lemma 9. For all s ≥ 0 and γ, δ > −1, the operator D of generalized differentiation is a continuous isomorphism from L2,s+1,0

γ,δ

• nto L2,s

1+γ,1+δ. Moreover, For each s ≥ 0, the finite part

integral operator DA is a continuous isomorphism between the spaces L2,s+1

α,β

and L2,s

β,α. Finally, for u ∈ L2,s+1 α,β ,

DAu =

∞

• n=0

(n + 1)u, pα,β

n

α,βpβ,α

n

. (14) In our case a = 0, b = −1 i.e. α = β = 1/2, it follows that

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 15

V ∈ L(L2,s+1

ϕ

, L2,s

ϕ ) and

V u = DSu =

∞

• n=0

(n + 1)u, pϕ

nϕpϕ n.

(15)

• Remark. We remember that to prove the previous Lemmas we need

also these two relations for the orthonormal polynomials: vγ,δ(x)pγ,δ

n (x) = −[n(n + γ + δ + 1)]−1/2 d

dx

• v1+γ,1+δ(x)p1+γ,1+δ

n−1

(x)

• , n

d dxpγ,δ

n (x) =

• n(n + γ + δ + 1)p1+γ,1+δ

n−1

(x) , n = 1, 2, . . . and that α + β = 1 .

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 16

(Hu)(x) = 1 π 1

−1

h(x, t)u(t)vα,β(t) dt We assume that the function h is continuous on [−1, 1]2. Lemma 10. [Berthold, Hoppe, Silbermann] If h(., t) ∈ L2,s

γ,δ uniformly

w.r.t. t ∈ [−1, 1] then H ∈ L(L2

α,β, L2,s γ,δ).

(Hu)(x) = a x

−1

h(x, t)u(t)vα,β(t) dt+ − b π 1

−1

h(x, t) ln |x − t|u(t)vα,β(t) dt ; (Hu)(x) = 1 π 1

−1

h(x, t)|x − t|−ηu(t)vα,β(t) dt .

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 17

◮Collocation method We investigate the simple hypersingular integral equation Bu := (MΓ + V + H)u = f. (16) The collocation method consists in looking for an approximate solution un ∈ Pn−1 of equation by solving the equation Lϕ

n(MΓ + V + H)un = Lϕ nf .

In view of relation (14) this equation is equivalent to Bnun := [V + Lϕ

n(MΓ + H)]un = Lϕ nf.

(17)

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 18

We can observe, that in view of Lemma 9 on the operator V , each solution un ∈ L2,s+1

ϕ

• f equation (16) is a polynomial (Invariance

Property), therefore we can consider equation (16) in the pair of spaces (L2,s+1

ϕ

, L2,s

ϕ ).

In order to prove the convergence of the collocation method we need some assumptions. 1) For f ≡ 0 equation has only the trivial solution u ≡ 0 in L2,1

ϕ .

2) Γ ∈ Cr

ϕ for some integer r ≥ 0.

3) h(., t) ∈ L2,

s ϕ

uniformly w.r.t. t ∈ [−1, 1]. The first step is to show the invertibility of the operator B.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 19

Recalling Lemma 5 and 6 on the mapping properties of the multiplication operator MΓ, Lemma 10 on the mapping property of the kernel operator H and Lemma 1 about the imbedding property

• f the Sobolev spaces, we can conclude that the operators MΓ and

H are compact in the pair of spaces (L2,s+1

ϕ

, L2,s

ϕ ), as well as in the

pair of spaces (L2,1

ϕ , L2 ϕ).

Furthermore, by Lemma 9 on the operator V and Lemma 1 about the imbedding property of the Sobolev spaces, we can conclude that the

• perator B : L2,t+1

ϕ

→ L2,t

ϕ

is invertible for 0 ≤ t ≤ s and equation possesses a unique solution u∗ ∈ L2,s+1

ϕ

.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 20

Now we are able to prove the convergence of the collocation method. Theorem 1. Assume that

• s > 1/2;
• f ∈ L2,s

ϕ ;

• For f ≡ 0 equation has only the trivial solution u ≡ 0 in L2,1

ϕ ;

• Γ ∈ Cr

ϕ for some integer r, 0 ≤ s ≤ r;

• h(., t) ∈ L2,s

ϕ

uniformly w.r.t. t ∈ [−1, 1].

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 21

Then, for all sufficiently large n, the approximate equation (16) is uniquely solvable, and the solution u∗

n converges to the

unique solution u∗ of hypersingular integral equation (15) in the norm of L2,s+1

ϕ

. Moreover, for 0 ≤ t ≤ s, ||u∗

n − u∗||ϕ,t+1 ≤ const nt−s||u∗||ϕ,s+1.

Proof. By Lemma 6 we can deduce the compactness of the multiplication operator MΓ : L2,t+1

ϕ

→ L2,t

ϕ

for 0 ≤ t ≤ s. Since H : L2,s+1

ϕ

→ L2,s

ϕ

is compact, it follows lim

n→∞ ||Bn − B||L2,s+1

ϕ

→L2,s

ϕ = lim

n→∞ ||(I − Lϕ n)(MΓ + H)||L2,s+1

ϕ

→L2,s

ϕ = 0

taking into account assertion (a) of Lemma 4.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 22

By assertion (b) of Lemma 4, recalling that r ≥ 1, we can deduce ||(MΓ − Lϕ

nMΓ)u||ϕ ≤ const n−1||u||ϕ,1

for u ∈ L2,1

ϕ ,

and again by assertion (b) of Lemma 4 and Lemma 10, ||(H − Lϕ

nH)u||ϕ ≤ const n−s||u||ϕ,s

for u ∈ L2,s

ϕ ,

and therefore ||Bn − B||L2,1

ϕ →L2 ϕ ≤ constn−

s,

where s = min{s, 1}. This inequality, together with limn→∞ ||Bn − B||L2,s+1

ϕ

→L2,s

ϕ = 0,

and Lemma 1 imply the existence and uniform boundedness of

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 23

B−1

n

∈ L(L2,t

ϕ , L2,t+1 ϕ

) for 0 ≤ t ≤ s and for all sufficiently large n. Consequently, since u∗

n − u∗ = B−1 n

• Lϕ

nf − f +

• I − Lϕ

n

• (MΓ + H)u∗

, we have, with the help of assertion (b) of Lemma 4 ||u∗

n − u∗||ϕ,t+1 ≤ const nt−s (||f||ϕ,s + ||(MΓ + H)u∗||ϕ,s) ,

and this prove the Theorem. ✷

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 24

◮ Quadrature method We consider the gaussian quadrature rule with respect to the Jacobi weight vγ,δ, i.e. 1 π 1

−1

u(t)vγ,δ(t) dt ≈ Qγ,δ

n (u) := n

• k=1

λγ,δ

nk u(xγ,δ nk )

with λγ,δ

nk = 1

π 1

−1

lγ,δ

nk (t)vγ,δ(t) dt.

If we choose vγ,δ = ϕ we can approximate the operator H by (Hnu)(x) =

n

• k=1

λϕ

nkh(x, xϕ nk)u(xϕ nk).

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 25

The quadrature or discrete collocation method consists in solving the equation V + Lϕ

n(MΓ + Hn)un = Lϕ nf.

The solution of this equation again belongs to Pn−1 . Since, for such un , we have (Hun)(x) = Qϕ

n

• unLϕ

n[h(x, .)]

• =

= 1 π 1

−1

un(t)Lϕ

nt[h(x, t)]ϕ(t) dt =: (

Hnun)(x) , (18) the approximate equation is equivalent to

• Bnun := V + Lϕ

n(MΓ +

Hn)un = Lϕ

nf .

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 26

Lemma 11. Assume h(x, .) ∈ L2,s

ϕ

for some s > 1/2 uniformly w.r.t. x ∈ [−1, 1] . Then, for 0 ≤ t ≤ s and u ∈ L2

ϕ ,

||Lγ,δ

m (

Hn − H)u||γ,δ,t ≤ constmtn−s||u||ϕ . With this Lemma, we can prove the same result as that in Theorem 1. We can generalize all these results in the more general hypersingular integral equation MΓ + DA + H = f where DA is defined in (14) and the kernel operator can have logarithmic or algebraic singularity.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 27

◮Weighted Uniform Convergence To prove similar results in suitable weighted uniform norm, we need to investigate a regularized version of the hypersingular integral equation (MΓ + V + H)u = f. Our first aim is to describe the inverse operator of V : L2,s+1

ϕ

→ L2,s

ϕ

and to study the mapping properties of such operator in appropriate pairs of weighted spaces of continuous functions. We recall Lemma 9. For each s ≥ 0, the hypersingular integral operator V is a continuous isomorphism between L2,s+1

ϕ

and L2,s

ϕ

and V u = DSu =

∞

• n=0

(n + 1)u, pϕ

nϕpϕ n.

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 28

In view of this Lemma the operator V −1 : L2,s

ϕ

→ L2,s+1

ϕ

is uniquely determined by V −1pϕ

n =

1 n + 1pϕ

n,

n = 0, 1, 2, ....... and by continuity since the set of polynomials is dense in L2,s

γ,δ.

Moreover, we can show that Lemma 12. [C.,Criscuolo,Junghanns,Luther] The inverse operator V −1 : L2,s

ϕ

→ L2,s+1

ϕ

• f the hypersingular integral operator V

can be written in the form: V −1 = −S−1W S (19)

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 29

with (W u)(x) = 1 π 1

−1

ϕ−1(t) ln |x − t|u(t)dt, (S−1u)(x) = 1 π 1

−1

u(t) t − xϕ−1(t)dt We can generalize this result recalling Lemma 9 in the general case and that α + β = 1. Lemma 13. [C.,Criscuolo,Junghanns,Luther] The inverse operator (DA)−1 : L2,s

β,α → L2,s+1 α,β

• f the hypersingular integral operator

DA can be written in the form: (DA)−1 = A−1W B (20)

M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 30

where the continuous operators B : L2,s

β,α → L2,s −β,−α,

W : L2,s

−β,−α → L2,s+1 β−1,α−1 = L2,s+1 −α,−β,

and A−1 : L2,s+1

−α,−β → L2,s+1 α,β

are defined by (Bu)(x) = avβ,α(x)u(x) − b π 1

−1

u(t) t − xvβ,α(t)dt, (W u)(x) = a x

−1

v−β,−α(t)u(t)dt − b π 1

−1

u(t)v−β,−α(t) ln |x − t|u(t)dt, and A−1 is the inverse of the Cauchy singular integral operator defined in (15).

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Equation (V + MΓ + H)u = f, can be write in the regularized equivalent form (I + V −1(MΓ + H))u = V −1f. Therefore, the Collocation Method can be represented with this

• perator equation

(I + V −1Ln(MΓ + H))un = V −1Lnf, and also the Quadrature Method can be represented with this

• perator equation

(I + V −1Ln(MΓ + Hn))un = V −1Lnf.

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The equations here described are studied in suitable pairs of weighted Besov spaces.

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