Mixed Collocation for Fractional Differential Equations cois Dubois - PowerPoint PPT Presentation

Groupe de travail Num´ erique Orsay, 03 d´ ecembre 2003. Mixed Collocation for Fractional Differential Equations cois Dubois ∗ † e ‡ and Fran¸ St´ ephanie Mengu´ ∗ Conservatoire Nat. des Arts et M´ etiers, Saint-Cyr-L’Ecole, France , E.U. † CNRS, laboratoire ASCI, Orsay, France, E.U. ‡ Laboratoire Syst` emes de Communication, Universit´ e de Marne La Vall´ ee, Marne La Vall´ ee, France, E.U.

International Conference on Numerical Algorithms, Dedicated to Claude Br´ ezinski. Marrakesh, Marocco, October 1-5, 2001. Mixed Collocation for Fractional Differential Equations cois Dubois ∗ † e ‡ and Fran¸ St´ ephanie Mengu´ ∗ Conservatoire Nat. des Arts et M´ etiers, Saint-Cyr-L’Ecole, France , E.U. † CNRS, laboratoire ASCI, Orsay, France, E.U. ‡ Laboratoire Syst` emes de Communication, Universit´ e de Marne La Vall´ ee, Marne La Vall´ ee, France, E.U.

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. Summary 1) Introduction 2) Mixed collocation numerical scheme 3) First numerical tests 4) Nonlinear model with a singularity 5) Conclusion 3

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. Introduction The letter β, 0 < β < 1 , is a real number, • Γ( • ) is the classical Euler function. Fractional differential operator D β ( • ) : • � t 1 d u d θ ( D β u ) ( t ) ≡ (1) ( t − θ ) 1 − β . Γ (1 − β ) d θ 0 Fractional ordinary differential equation of order β : • D β ( u − u 0 ) � = Φ ( u ( t ) , t ) , t > 0 (2) = 0 , t ≤ 0 . u − u 0 4

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. Discretization • Discretization step h > 0 . Discrete space P h 1 : • continuous functions that are affine in each mesh element ] jh, ( j + 1) h [ . Discrete space Q h 0 : constant functions in each element. • Fractional integrator I β of order β : • � t 1 I β ( v ( • ) , t ) ≡ 0 ( t − θ ) β − 1 v ( θ ) d θ . (3) Γ ( β ) 5

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. Mixed collocation numerical scheme (i) Integrate the equation (2) with the fractional integrator I β (3) : • u ( t ) − u 0 = I β (Φ ( u ( • ) , t )) , (4) t ≥ 0 . Low order ( P 1 Q 0 ) mixed collocation method : choose • a discrete state u h ( • ) satisfying u h ∈ P h 1 a flux f h ≃ Φ ( u ( • ) , t ) according to the condition f h ∈ Q h 0 . • Write the equation (4) at the grid points jh ( j ∈ N ) : u h ( jh ) − u 0 = I β � f h ( • ) , jh � (5) , j ∈ N . 6

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. Mixed collocation numerical scheme (ii) Mean value of the approached flux f h ( • ) : • equal to the mean value of the exact flux in each element : � ( j +1) h � ( j +1) h f h ( θ ) d θ ≡ u h ( θ ) , θ � � (6) Φ d θ . jh jh ”Projection step” on the discrete space Q h 0 : • � 1 f h � u h j (1 − θ ) + θu h � (7) = 0 Φ j +1 , jh + θh d θ , j ∈ N . j + 1 2 7

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. Mixed collocation numerical scheme (iii) • ”State-flux constraint” for the scheme P 1 Q 0 : the relations (5)(7) take the form j − 1 h β h β u h Γ ( β + 1) f h α j − k f h � (8) j +1 − = u 0 + , j ∈ N , j + 1 k + 1 Γ ( β + 1) 2 2 k =0 α k ≡ ( k + 1) β − k β , with k ∈ N . • Newton method for the numerical solution of equations (7) (8) ; ” Semidif ” software, see http : //www.laas.fr/gt-opd/ (free of charge !). 8

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (i) • Elementary tests with β = 0 . 5 and Dynamics Φ ( u , t ) ≡ g ( t ) with g ( • ) chosen as : √ √ π , √ π t , g 1 ( t ) = 1 2 g 3 ( t ) = 3  g 2 ( t ) = √ π t , 2 4  √ (9) √ π t 2 . 8 g 5 ( t ) = 15 g 4 ( t ) = 3 √ π t t , 16  • Then the solution of equation (2) is simply � √ � j , (10) u j ( t ) ≡ t j = 1 , · · · , 5 . as shown on the following figures. 9

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (ii) 1.2 1/2 Solution exacte u 1 (t) = t Schéma de GL à deux points Schéma de GL à trois points Schéma de Msallam 1 Schéma par éléments finis Schéma mixte P 1 Q 0 Schéma mixte P 1 Q 1 0.8 D 1/2 u 1 = π 1/2 /2 0.6 0.4 Solutions exacte et approchées avec 8 points 0.2 0 0 0.2 0.4 0.6 0.8 1 √ Numerical solution of D 1 / 2 u = g 1 ( t ) ; u ( t ) = Figure 1. t. 10

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (iii) 1.2 Solution exacte u 2 (t)= t Schéma de GL à deux points Schéma de GL à trois points Schéma de Msallam 1 Schéma par éléments finis Schéma mixte P 1 Q 0 Schéma mixte P 1 Q 1 0.8 D 1/2 u 2 = 2 (t/ π ) 1/2 0.6 0.4 0.2 Solutions exacte et approchées avec 8 points 0 0 0.2 0.4 0.6 0.8 1 Numerical solution of D 1 / 2 u = g 2 ( t ) ; u ( t ) = t. Figure 2. 11

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (iv) 1.2 3/2 Solution exacte u 3 (t)= t Schéma de GL à deux points Schéma de GL à trois points Schéma de Msallam 1 Schéma par éléments finis Schéma mixte P 1 Q 0 Schéma mixte P 1 Q 1 0.8 D 1/2 u 3 = 3 π 1/2 t / 4 0.6 0.4 0.2 Solutions exacte et approchées avec 8 points 0 0 0.2 0.4 0.6 0.8 1 √ Numerical solution of D 1 / 2 u = g 3 ( t ) ; u ( t ) = t Figure 3. t. 12

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (v) 1.2 2 Solution exacte u 4 (t)= t Schéma de GL à deux points D 1/2 u 4 = 8 t 3/2 / 3 π 1/2 Schéma de GL à trois points Schéma de Msallam 1 Schéma par éléments finis Schéma mixte P 1 Q 0 Schéma mixte P 1 Q 1 0.8 0.6 Solutions exacte et approchées avec 8 points 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Numerical solution of D 1 / 2 u = g 4 ( t ) ; u ( t ) = t 2 . Figure 4. 13

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (vi) 1.2 5/2 Solution exacte u 5 (t)= t Schéma de GL à deux points D 1/2 u 5 = 15 π 1/2 t 2 / 16 Schéma de GL à trois points Schéma de Msallam 1 Schéma par éléments finis Schéma mixte P 1 Q 0 Schéma mixte P 1 Q 1 0.8 0.6 Solutions exacte et approchées avec 8 points 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Numerical solution of D 1 / 2 u = g 5 ( t ) ; u ( t ) = t 2 √ Figure 5. t. 14

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (vii) • Orders of convergente with mesh steps h , 1 h = 2 n , 3 ≤ n ≤ 13 . 2 relatively to the norm L 2 and e n ∞ for the norm L ∞ : Errors e n • � � j � | u (0) − u 0 | 2 2 n − 1 + | u (1) − u 2 n | 2 √ 2 � � � � � e n � � � (11) 2 ≡ h + � u − u j . � � � 2 n 2 2 � j =1 e n ∞ ≡ max {| u ( jh ) − u j | , j = 0 , · · · , 2 n } (12) 15

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (viii) • Orders of convergence for the previous test case : Mixed scheme P 1 Q 0 L 2 L ∞ g 1 ( t ) ∞ ∞ g 2 ( t ) 1.0000 1.3982 1.4850 1.4677 g 3 ( t ) g 4 ( t ) 1.4722 1.4627 1.4613 1.4564 g 5 ( t ) • Satisfying results ? 16

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (ix) • Tests with β = 0 . 5 and nonlinear dynamics Φ ( u , t ) ≡ f ( u ) : √ π , √ u , √ π u 2  f 1 ( u ) = 1 2 f 3 ( u ) = 3 3 , f 2 ( u ) = √ π  2 4  (13) √ π u 3 4 8 f 5 ( t ) = 15 4 , 5 . f 4 ( u ) = 3 √ π u   16 Then the solution of equation (2) is simply • � √ � j , (14) u j ( t ) ≡ t j = 1 , · · · , 5 . as in the five previous test cases. 17

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (x) 1.2 1/2 Solution exacte u 1 (t) = t Schéma de GL à deux points Schéma de GL à trois points Schéma de Msallam 1 Schéma par éléments finis Schéma mixte P 1 Q 0 Schéma mixte P 1 Q 1 0.8 D 1/2 u 1 = π 1/2 /2 0.6 0.4 Solutions exacte et approchées avec 8 points 0.2 0 0 0.2 0.4 0.6 0.8 1 √ Numerical solution of D 1 / 2 u = f 1 ( u ) ; u ( t ) = Figure 6. t. 18

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (xi) 1.2 Solution exacte u 2 (t)= t Schéma de GL à deux points Schéma de GL à trois points Schéma de Msallam 1 Schéma par éléments finis Schéma mixte P 1 Q 0 Schéma mixte P 1 Q 1 0.8 D 1/2 u 2 = 2 (u 2 / π ) 1/2 0.6 0.4 0.2 Solutions exacte et approchées avec 8 points 0 0 0.2 0.4 0.6 0.8 1 Numerical solution of D 1 / 2 u = f 2 ( u ) ; u ( t ) = t. Figure 7. 19

Int. Conf. on Numerical Algorithms, Marrakesh, October 1-5, 2001. First numerical tests (xii) 1.4 3/2 Solution exacte u 3 (t)= t Schéma de GL à deux points Schéma de GL à trois points Schéma de Msallam 1.2 Schéma par éléments finis Schéma mixte P 1 Q 0 Schéma mixte P 1 Q 1 1 0.8 D 1/2 u 3 = 3 π 1/2 u 3 2/3 / 4 0.6 0.4 0.2 Solutions exacte et approchées avec 8 points 0 0 0.2 0.4 0.6 0.8 1 √ Numerical solution of D 1 / 2 u = f 3 ( u ) ; u ( t ) = t Figure 8. t. 20