On the Numerical Solution of Integro-Differential Equations of - PowerPoint PPT Presentation

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

� 1 � 1 v ′ ( t ) g ( x ) v ( x ) − 1 t − xdt + 1 h ( x, t ) v ( t ) dt = f ( x ) , (1) π π − 1 − 1 v ( − 1) = v (1) = 0 . (2) For a function v ∈ L p ( − 1 , 1) possessing a generalized derivative v ′ ∈ L p ( − 1 , 1) ,we have � 1 � 1 v ′ ( t ) d v ( t ) t − xdt − v ( − 1) 1 + x + v (1) t − xdt = 1 − x, x ∈ ( − 1 , 1) , dx − 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 � 1 � 1 g ( x ) v ( x ) − 1 ( t − x ) 2 dt +1 v ( t ) h ( x, t ) v ( t ) dt = f ( x ) , − 1 < x < 1 π π − 1 − 1 (3) where the hypersingular integral operator has to be understood in the sense of � 1 � 1 ( t − x ) 2 dt = d v ( t ) v ( t ) t − xdt (4) dx − 1 − 1 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 − x 2 . Thus, it is convenient to represent v as the product � 1 − x 2 v ( x ) = ϕ ( x ) u ( x ) , ϕ ( x ) = (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 ) , ( M Γ ) u ( x ) = Γ( x ) u ( x ) (6) Cauchy Singular Integral Operator a − ib = e iπα , For real numbers a and b with 0 < α < 1 , β = 1 − α , define the Jacobi weight function v α,β ( x ) = (1 − x ) α (1 + x ) β and the singular integral operator of Cauchy type � 1 ( Au )( x ) = av α,β ( x ) u ( x ) + b u ( t ) t − xv α,β ( t ) dt (7) π − 1 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 ) � 1 ( Su )( x ) = − 1 u ( t ) t − xϕ ( t ) dt (8) π − 1 Hypersingular Integral Operator � 1 ( DAu )( x ) = a d dx [ v α,β ( x ) u ( x )] + b u ( t ) ( t − x ) 2 v α,β ( t ) dt (9) π − 1 D = d ( α = β = 1 V = DS, dx, 2) (10) M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 6

Kernel Integral Operator � 1 ( Hu )( x ) = 1 h ( x, t ) v ( t ) dt, (11) π − 1 We assume that the function h is continuous on [ − 1 , 1] 2 . At first, we consider the hypersingular integral equation , written in operator form: ( M Γ + V + H ) u = f (12) M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 7

Let v γ,δ ( x ) = (1 − x ) γ (1 + x ) δ , γ, δ > − 1 be a Jacobi weight and L 2 γ,δ , γ, δ > − 1 denote the weighted space of square integrable functions on the interval [ − 1 , 1] endowed with the scalar product � 1 � u, v � γ,δ = 1 u ( x ) v ( x ) v γ,δ ( x ) dx, π − 1 and the norm � � u � γ,δ = � u, u � γ,δ . Let p γ,δ refer as the normalized Jacobi polynomial (with positive n leading coefficient) of degree n with respect to the Jacobi weight v γ,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 L 2 ,s γ,δ by � � ∞ � (1 + n ) 2 s � � � 2 < ∞ L 2 ,s u ∈ L 2 � � u, p γ,δ γ,δ = γ,δ : n � γ,δ , n =0 with the norm � ∞ � 1 / 2 � (1 + n ) 2 s � � � 2 � � u, p γ,δ � u � γ,δ,s = n � γ,δ . n =0 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 of weighted Sobolev spaces, of interpolation operators with respect to the zeros of the orthogonal polynomials p γ,δ n , the multiplication operator M Γ 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 operators 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 L 2 ,t γ,δ is compactly imbedded in L 2 ,s γ,δ . If the operator B belongs to L (L 2 , s 1 α 1 ,β 1 , L 2 , s 2 Lemma 2. [Junghanns] α 2 ,β 2 ) α 2 ,β 2 ) then B ∈ L ( L 2 ,s ( τ ) α 1 ,β 1 , L 2 ,t ( τ ) and L ( L 2 ,t 1 α 1 ,β 1 , L 2 ,t 2 α 2 ,β 2 ) , where s ( τ ) = (1 − τ ) s 1 and t ( τ ) = (1 − τ ) s 2 + τ t 2 , 0 ≤ τ ≤ 1 . Lemma 3. [Berthold,Hoppe and Silbermann] Let r ≥ 0 be an if and only if u ( k ) ϕ k belongs to Then u ∈ L 2 ,r integer. γ,δ L 2 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 ) ϕ k || γ,δ are equivalent. M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 11

Let x γ,δ nn < . . . < x γ,δ nk with − 1 < x γ,δ n 1 be the zeros of p γ,δ and n denote by L γ,δ the Lagrange interpolation operator n x − x γ,δ n n � � nj f ( x γ,δ nk ) l γ,δ l γ,δ L γ,δ n f = nk , nk ( x ) = . x γ,δ nk − x γ,δ nj k =1 j =1 ,j � = k Lemma 4. [C.,Mastroianni] For s > 1 / 2 we have n f || γ,δ,s = 0 for all f ∈ L 2 , s (a) lim n →∞ || f − L γ,δ γ,δ , (b) || f − L γ,δ n f || γ,δ,t ≤ const n t − 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

C r By ϕ , r ≥ 0 an integer, we denote the space of all r times differentiable functions u : ( − 1 , 1) → C satisfying u ( k ) ϕ k ∈ C[ − 1 , 1] the conditions for k = 0 , 1 , . . . , r. Let ϕ = � r k =0 || u ( k ) ϕ k || ∞ . || u || C r Let r ≥ 0 be an integer and Γ ∈ C r Lemma 5. [Junghanns] ϕ . Then the multiplication operator M Γ belongs to L ( L 2 ,r γ,δ , L 2 ,r γ,δ ) and || M Γ || L 2 ,r γ,δ ≤ const || Γ || C r ϕ . γ,δ → L 2 ,r Lemma 6. Taking into account Lemma 1, under the assumptions of Lemma 5, if M Γ ∈ L ( L 2 γ,δ , L 2 γ,δ ) , the condition ϕ implies M Γ ∈ L ( L 2 ,r γ,δ , L 2 ,r Γ ∈ C 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 = L 2 ,s L 2 ,s 1 / 2 , 1 / 2 , � ., . � ϕ = � ., . � 1 / 2 , 1 / 2 , and p ϕ n = p 1 / 2 , 1 / 2 , and ϕ n � � L 2 ,s, 0 f ∈ L 2 ,s γ,δ : � f, p γ,δ = � γ,δ = 0 . γ,δ 0 Lemma 7. [Berthold,Hoppe and Silbermann] For all s ≥ 0 , the Cauchy singular integral operator A belongs to L ( L 2 ,s α,β , L 2 ,s − α, − β ) . Moreover, A : L 2 ,s α,β → L 2 ,s, 0 − α, − β is a bijection, and the inverse operator is given by � 1 Af )( t ) := av − α, − β − b f ( x ) A − 1 = � ( � x − tv − α, − β ( x ) dx. A, π − 1 (13) Lemma 8. [Proosdorf, Silbermann] For the Cauchy singular integral operator 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 = p − α, − β Ap α,β , n = 0 , 1 , 2 , . . . n +1 n Lemma 9. For all s ≥ 0 and γ, δ > − 1 , the operator D of generalized differentiation is a continuous isomorphism from L 2 ,s +1 , 0 onto L 2 ,s 1+ γ, 1+ δ . Moreover, For each s ≥ 0 , the finite part γ,δ integral operator DA is a continuous isomorphism between the spaces L 2 ,s +1 and L 2 ,s β,α . Finally, for u ∈ L 2 ,s +1 α,β , α,β ∞ � ( n + 1) � u, p α,β � α,β p β,α DAu = . (14) n n n =0 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 ( L 2 ,s +1 , L 2 ,s ϕ ) and ϕ ∞ � ( n + 1) � u, p ϕ n � ϕ p ϕ V u = DSu = n . (15) n =0 Remark. We remember that to prove the previous Lemmas we need also these two relations for the orthonormal polynomials: � � n ( x ) = − [ n ( n + γ + δ + 1)] − 1 / 2 d v 1+ γ, 1+ δ ( x ) p 1+ γ, 1+ δ v γ,δ ( x ) p γ,δ ( x ) , n n − 1 dx � d n ( n + γ + δ + 1) p 1+ γ, 1+ δ dxp γ,δ n ( x ) = ( x ) , n = 1 , 2 , . . . n − 1 and that α + β = 1 . M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 16