High Order Numerical Methods for the Riesz Derivatives and the Space Riesz Fractional Differential Equation

Transcription

1 International Symposium on Fractional PDEs: Theory, Numerics and Applications June 3-5, 013, Salve Regina University High Order Numerical Methods for the Riesz Derivatives and the Space Riesz Fractional Differential Equation Hengfei Ding *, Changpin Li *, YangQuan Chen ** * Shanghai University ** University of California, Merced 1

2 Outline Fractional calculus: partial but important Motivation Numerical methods for Riesz fractional derivatives Numerical methods for space Riesz fractional differential equation Conclusions Acknowledgements

3 Fractional calculus: partial but important Calculus= integration + differentiation Fractional calculus= fractional integration + fractional differentiation Fractional integration (or fractional integral) Mainly one: Riemann-Liouville (RL) integral Fractional derivatives More than 6: not mutually equivalent. RL and Caputo derivatives are mostly used. 3

4 Fractional calculus: partial but important Definitions of RL and Caputo derivatives: d () (), 1. dt m ( m ) + RL D0, t xt = D m 0, t xt m < < m Z d () (), 1. dt m ( m ) + CD0, txt = D0, t xt m < < m Z m t m 1 k ( k ) + C D0, t xt = RL D0, t[ xt x (0)], m 1 < < m Z. k = 0 k! The involved fractional integral is in the sense of RL. D 1 Γ( ) t 0, = = t x t Y x t 0 ( t τ ) 1 x( τ ) dτ, R +. 4

5 Fractional calculus: partial but important Once mentioned it, fractional calculus is taken for granted to be the mathematical generalization of calculus. But this is not true!!! For - th ( m 1< < lim ( m 1) + C D 0, t x( t) = x ( m 1) m Z + ( t) x ) Caputo ( m 1) (0), derivative lim m C, D 0, t fix t x( t) = Classical derivative is not the special case of the Caputo derivative. x ( m) ( t). 5

6 Fractional calculus: partial but important On the other hand, see an example below: Investigate a function in [0,1 + ε ], 1 t, 0 t 1, xt () = t 1, 1 < t 1 + ε, ε > 0. RL D xt in( 0,1 + ε ], (0,1). 0, t x '( t) in [0,1) (1,1 + ε ]. The existence intervals of two kind of derivatives for the same funtion is not the same, neither relation of inclusion. RL derivative can not be regarded as the mathematical generalization of the classical derivative. 6

7 Fractional calculus: partial but important Therefore, fractional calculus, closely related to classical calculus, is not the direct generalization of classical calculus in the sense of rigorous mathematics. For details, see the following review article: [1] Changpin Li, Zhengang Zhao, Introduction to fractional integrability and differentiability, The European Physical Journal-Special Topics, 193(1), 5-6,

8 Fractional calculus: partial but important Where is fractional calculus? It is here. Example xt = t,0 < 1/, is not a solution to dx = f( xt, ), arbitrarily given, dt x(0) = x0, arbitrarily given. But it is a solution to RL D xt = f( xt, ), for some f( xt, ), 0, t 1 RL D0, t xt t= 0= x0, for s x 0 ome. 8

9 Fractional calculus: partial but important Another Example q 1, t= (0,1], ( pq, ) = 1, xt () = p 0, others in [0,1], is not a solution to dx = f( xt, ), arbitrarily given, dt x(0) = x0, arbitrarily given. But it is a solution to RL D0, t xt = 0, (0,1), () 1 RL D0, t xt x= 0= 0. Actually, x = 0 a.e. solves eqn (*). Attention: fractional is very possibly a powerful tool for nonsmooth functions. 9

10 Motivation For Riemann-Liouvile derivative, high order methods were very possibly proposed by Lubich (SIAM J. Math. Anal., 1986) RL n s, n ϖ ax n j j ϖn, j j j= 0 j= 0 D n = + + p f( x ) h f( x ) h f( x ) O( h ). For Caputo derivative, high order schemes were constructed by Li, Chen, Ye (J. Comput. Phys., 011) In this talk, we focus on high order algorithms for Riesz fractional derivatives and space Riesz fractional differential equation. 10

11 Motivation The Riesz fractional derivative is defined on (a, b) as follows, in which, uxt (,) x 1 π sec, n 1 + Ψ = < < n Z, n x 1. u ξ, t D, uxt (,) = dξ, Γ n + 1 n x x ξ 11 ( ) RL Da, x RL Dx, b u( xt) u( ξ, t) ( ξ ) RL a x n a = Ψ + n b 1 RL x, b = n n 1 ( n ) Γ + x x x D uxt (,) dξ.,,

12 Motivation The space Riesz fractional differential equation reads as (, ) u( xt, ) u xt = K + f( xt, ),1< <, a< x< b,0< t T t x RL Da, x; tuxt (, ) x= a = φ ( t), 0 t T, (1) RL Dx, b; tuxt (, ) x= b = ϕ( t), 0 t T, ux (,0) = ψ ( x), a x b. The above boundary value conditions are of Dirichlet type. Neumann or Rubin boundary value conditions can be similarly proposed. Unsuitable initial or boundary value conditions: ill-posed. 1

13 Numerical methods for Riesz fractional derivatives Already existed (typical) numerical schemes: Let h be the step size with x = a + mh, m = 0,1,, M, and t = nτ, n= 0,1,, N, where h= b a / M, τ = T / N. n 1) The first order scheme m m M m ux ( m,) t Ψ = ϖk u( xm k, t) ϖk u( xm k, t) O( h), + x h + + k= 0 k= 0 13 k ( 1) Γ ( 1+ ) ( 1 ) ( 1 ) k Γ + k Γ + k ( ) k in which ϖ k = 1 =.

14 Numerical methods for Riesz fractional derivatives ) The shifted first order scheme ux (,) t = u x, t + u( x, t) + O( h), x m+ 1 M m+ 1 m Ψ ϖ k ( m k+ 1 ) ϖk m+ k 1 h k= 0 k= 0 k ( 1) Γ ( 1+ ) ( 1 ) ( 1 ) k Γ + k Γ + k ( ) k in which ϖ k = 1 =. 14

15 Numerical methods for Riesz fractional derivatives 3) The L numerical scheme If 1< <, then one has m 1 k = 0 ( ) ( 1 )( ) u( xm, t) ( M m) ( ) ( ) u( x1, t) u( x0, t) u xm, t Ψ 1 u x0, t = + 1 x Γ( 3 ) h m m + dk ux ( m k+ 1,) t ( uxm k,) t+ u xm k 1, t ( ) u( x, t) u( x, t) M M M m 1 + dk ux ( m+ k 1,) t ( uxm+ k,) t+ u x k = 0 m ( t) O( h) m+ k+ 1 in which dk = ( k+ 1) k, k = 0,1,,max( m 1, M m 1)., +, 15

16 Numerical methods for Riesz fractional derivatives 4) The second order numerical scheme based on the spline interpolation method (, t) u x x m in which z ( ) Ψ ( ) ( = z ) mk, u xk, t + O h, Γ ( ) mk, ( 4 ) 16 M h k = 0 is defined below, zmk,, k < m 1, zmm, 1 + z mm, 1, k = m 1, zmk, = zmm, + z mm,, k = m, zmm, z mm, + 1, k = m+ 1, ( ) z mk,, k > m+ 1.

17 Numerical methods for Riesz fractional derivatives cm 1, k cmk, + cm+ 1, k, k m 1, cmk, cm 1, k, k m, + + = zmk, = cm+ 1, k, k = m+ 1, 0, k > m+ 1, in which 3 + = j 1 j ( j 3 ), k 0, cjk, = j k+ 1 j k + j k 1, 1 k j 1, 1, k = j; 17

18 Numerical methods for Riesz fractional derivatives 0, k < m 1, c m 1, m 1, k m 1, = z mk, = c mm, + c m 1, m, k = m, c m 1, k c mk, + c m + 1, k, m+ 1 k M, in which 1, k = j c jk, = ( k j+ 1) ( k j) + ( k j 1 ), j+ 1 k M 1 3 ( 3 M + j)( M j) + ( M j 1 ), k = M with j= m 1, mm,

19 Numerical methods for Riesz fractional derivatives 5) The second order scheme based on the fractional centered difference method u x x m, t 1 ( =, ) hu xm t + O h. h The so called fractional centered difference operator is defined k ( 1) Γ ( + 1) hu x, t = u( x kh, t). k = Γ k+ 1 Τ + k+ 1 by 19

20 Numerical methods for Riesz fractional derivatives 6) The second order scheme based on the weighted and shifted Grünwald-Letnikov (GL) formula m+ 1 m+ m, t Ψ = ν 1 ϖk u xm k+, t + ν ϖk u xm k+, t u x x h ( ) 0 k= 0 k= 0 M m+ M m ν 1 ϖk u xm+ k, t + ν 1 ϖk u xm+ k, t k= 0 k= 0 + O h ( 1 ) ( 1 ), in which ν =, ν =,, are two integers.

21 Numerical methods for Riesz fractional derivatives 7) The third order scheme based on the weighted and shifted GL formula m+ 1 m+ ( m, t) Ψ = κ1 ϖk u( xm k, t) κ ϖk u( xm k, t ) u x x h m 3 m= 0 k= 0 m+ M m+ k u( xm k+, t) k u( xm+ k, t) κ ϖ + κ ϖ 3 1 k= 0 m= 0 M m+ M m+ 3 k u( xm+ k, t) k u( xm+ k, t) κ ϖ + κ ϖ 3 = 0 m= 0 + O h, ( ) in which, κ1 =, 1 + κ ,, = κ 3 = ,, are three integers. 1 3

22 Numerical methods for Riesz fractional derivatives In our work, we construct new three kinds of second order schemes and a fourth order numerical scheme. We first derive second order schemes. (, ) Lemma 1. Assume that u xt, with respect to xsufficiently smooth. For arbitrarily different numbers p, q have u x t s = ( x x ) u( x, t) + ( x x ) u( x, t) s q p p s q x ( ) q s p s p x + O x x x x q. and s, we

23 Numerical methods for Riesz fractional derivatives Lemma. For any positive number, we have k = 0 ϖ ( ) k =0, ( ) k 1 Γ 1+ where ϖ k = 1 =. k Γ ( 1+ k) Γ ( 1+ k) k Let µ h u ( x, t) + f ( x jh, t), =. ( u ( x jh t) ) 3

24 Numerical methods for Riesz fractional derivatives Define ( u xt ) u xt, = µ,, where AC h h C h ( ) u xt, = u x k ht,. C h ϖ k k = 0 Then one has 1 u xt, u x k ht,. = ϖ ( ) AC h k k = 0 4

25 Numerical methods for Riesz fractional derivatives (, ) RL, x k ( h) k = 0 + RL, x Theorem 1. Let u xt, and the Fourier transform of the D u xt, with respect to x both be in L R, then AC 1 hu xt ( = ) RL D, xu xt, O h, i.e., + h 1 ( ) ( ) ( D u xt, = ϖ u x k ht, + O h ). 5

26 Numerical methods for Riesz fractional derivatives By choices of x, x and x, one s p q kinds of second order sch ems: has new three m u( xm, tn) Ψ ( ) = 1 ϖ k u xm k, x ( h) k = 0 ( t) m M m + ϖk u xm ( k 1), t + 1 ϖk u xm+ k, t k= 0 k= 0 M m ( ) + ϖ + k = 0 k u xm+ ( k 1), t Oh, (I) 6

27 Numerical methods for Riesz fractional derivatives m 1 u( xm, tn) Ψ ( ) = 1 ϖ k u xm k, + x ( h) k = 0 ( t) m M m 1 1 ϖk u( xm ( k+ 1), t) ϖk u xm+ k, t k= 0 k= 0 M m 1 ( ) ϖ k u( xm+ ( k+ 1) t) + Oh k = 0,, (II), n Ψ 1 = + ( h) 4 ( t ) u x x m m 1 k = 0 M m 1 k = 0 ( ) k u( xm ( k+ 1) t) ( ) m 1 k = ( ) ϖ, ( ) 4 7 M m 1 k = 0 ϖ ϖ k ( ) k u( xm ( k 1), t) ( ) k u( xm+ ( k 1) t) u x ϖ, ( II I) + ( m+ ( k + 1), t) Oh.

28 Numerical methods for Riesz fractional derivatives New kinds of third order numerical schemes. Skipped Now directly derive a fourth order scheme. 8

29 This image cannot currently be displayed. Numerical methods for Riesz fractional derivatives 1 7 Theorem. Let u xt, lie in C R whose partial derivatives up to order seven with respect to x belong to ( ) ( θ) L R. Set Lu xt, = g u x k+ ht,, θ = 1, 0,1, θ k = ( 1 k ) ( 1) in which g = Γ + k, Γ k+ 1 Γ + k 1 then one has k u xt, 1 ( 4 = L ) 1u xt, Lu 1 xt, (1 ) Lu 0 xt, O h. x h

30 Numerical methods for Riesz fractional derivatives Based on Therom, m 1 ( m, t) ( ) 4h a fourth order scheme can be established as k= M+ m+ 1 m 1 k= M+ m+ 1 ( ) (, ) + ( 1) (, ) ( 1) m 1 1 ( ) ( g, ) k u xm kt O h, + 1 h k= M+ m+ 1 where g u x x k = + 4h ( 1) Γ ( + 1) g u x t k m k g u x t k m k k =. Γ k+ 1 Γ + k 1 30

31 Numerical method for space Riesz fractional differential equation 31 Recall Equ (1). (, ) u( xt, ) u xt = K + f( xt, ),1< <, a< x< b,0< t T t x RL Da, x; tuxt (, ) x= a = φ ( t), 0 t T, (1) RL Dx, b; tuxt (, ) x= b = ϕ( t), 0 t T, ux (,0) = ψ ( x), 0 x L.

32 Numerical method for space Riesz fractional differential equation n Let u be the approximation solution of u x, t, one m m n has the following finite difference scheme for Equ (1). m 1 m 1 m 1 n n n n m µ 1 k m k 1 µ 1 k m k+ 1+ µ k m k k= M+ m+ 1 k= M+ m+ 1 k= M+ m+ 1 u g u g u g u = τ τ u g u g u g u f f m 1 m 1 m 1 n 1 n 1 n 1 n 1 n n 1 m + µ 1 k m k 1+ µ 1 k m k+ 1 µ k m k+ m+ m k= M+ m+ 1 k= M+ m+ 1 k= M+ m+ 1 τ K where µ 1 =, µ = 48h τk 1. h + 1, () 3

33 Numerical method for space Riesz fractional differential equation 33 n n n n T n n n n = 1 1 = 1 M 1 ( ) M Set U u, u,, u, F ( f, f,, f ). Then system () can by written in a compact form: n n 1 τ n τ n 1 I + H U = I H U + F + F, (3) where I is an identity matrix with order M -1, + H = µ G- µ G - µ G.The expressions of GG,, G are omitted here.

34 Numerical method for space Riesz fractional differential equation 34 Theorem 3. ( Stability) Difference scheme (3) (or ()) is unconditionally stable. Theorem 4. Convergence u x, t u C h. k ( 4) m k m τ +

35 Example 1 Numerical Examples Consider the function ux = x 1 x, x 0,1. Its exact Riesz fractional derivative is u x x [ ] π 1 = sec x + ( 1 x) Γ( 3 ) 6 4 x 3 + Γ 3 1 } 4 4 x 1 x. ( 5 ) Γ 35 x

36 Numerical Examples We use the second order numerical fomulas (I),(II)and (III) to compute the given function, the absolute errors and convergence orders at x=0.5 are shown in Tables

37 Numerical Examples 37

38 Numerical Examples 38

39 Numerical Examples 39

40 Numerical Examples 40

41 Numerical Examples 41

42 4 Example Numerical Examples Consider the following Riesz fractional differential equation u xt, uxt (,) = K + f( xt,),0< x< 1,0< t 1 t x where ( 6 ) π ( 1 x) ( x) f( t) = ( + 1) t x (1 x) + t sec x + 1 ( 5 ) Γ x + + Γ Γ uxt (, ) = t ( 7 ) 6 ( 1 x) x ( 1 x) x ( 1 x) Γ( 8 ) Γ( 9 ) together with the initial condition ux (,0) = 0 and homogenerous boundary value conditions. The exact solution is x 4 (1 x). x + 6

43 Numerical Examples The following table shows the maximum error, time and space convergence orders which confirms with our theoretical analysis. 43

44 Numerical Examples 44

45 Numerical Examples Figs.6.1 and 6. show the comparison of the analytical and numerical solutions with = 1.1 at t = 0., = 1.9 at t =

46 Numerical Examples 46

47 47 Numerical Examples Figs display the numerical and analytical solution surface with = 1.1 and = 1.8

48 Numerical Examples 48

49 Numerical Examples 49

50 Numerical Examples 50

51 Comments 1) Proper and exact applications of fractional calculus ) Long-term integrations at each step, induced by historical dependencies and/or long-range interactions, require more computational time and storage capacity. Therefore the effective and economical numerical methods are needed. 3) Fractional calculus + Stochastic process Stochastics and nonlocality may better characterize our complex world. So numerical fractional stochastic differential equations have been placed on the agenda. And more 51

52 Conclusions Conclusions? Not yet, but Go Fractional with some lists. 5

53 Conclusions Numerical calculations: [1] Changpin Li, An Chen, Junjie Ye, J Comput Phys, 30(9), , 011. [] Changpin Li, Zhengang Zhao, Yangquan Chen, Comput Math Appl 6(3), , 011. [3] Changpin Li, Fanhai Zeng, Finite difference methods for fractional differential equations, Int J Bifurcation Chaos (4), (8 pages), 01. Review Article [4] Changpin Li, Fanhai Zeng, Fawang Liu, Fractional Calculus and Applied Analysis 15 (3), , 01. [5] Changpin Li, Fanhai Zeng, Numer Funct Anal Optimiz 34(), , 013. [6] Hengfei Ding, Changpin Li, J Comput Phys 4(9), ,

54 Conclusions Fractional dynamics: [1] Changpin Li, Ziqing Gong, Deliang Qian, Yangquan Chen, Chaos 0(1), 01317, 010. [] Changpin Li, Yutian Ma, Nonlinear Dynamics 71(4), , 013. [3] Fengrong Zhang, Guanrong Chen, Changpin Li, Jürgen Kurths, Chaos synchronization in fractional differential systems, Phil Trans R Soc A 371 (1990), (6 pages), 013. Review article 54

55 Acknowledgements Q & A! Thank Professor George Em Karniadakis for cordial invitation. Thank you all for coming. 55