Several Problems in Fractional Ordinary Differential Equations

1. Several Problems in Fractional Ordinary Differential Equations Changpin Li Reach me @ Dept Math of Shanghai Univ Email: lcp@shu.edu.cn July 6, 2010

2. Outline I. Fractional integral and fractional derivatives II. Applications III. Comparison between fractional Ordinary differential equation (FODE) and typical differential equation (ODE) IV. Our work V. Acknowledgements

3. I. Fractional integral and fractional derivatives Classical calculus = Integration + Differentiation Fractional calculus = Fractional integration + Fractional differentiation Fractional integral: only one Fractional derivatives: more than six, not equivalent to each other Riemann-Liouville and Caputo derivatives are mostly used. I.1 Fractional integral I.2 Riemann-Liouville derivative I.3 Caputo derivative

4. I.1 Fractional integral The convolution kernel of order Definition 1 The fractional integral (or, the Riemann-Liouville integral) Property 1 (1) The Laplace transform: (2) The convolution property: (3) Consistency:

5. I.2 Riemann-Liouville derivative Definition 2 The Riemann-Liouville derivative with Property 2 (1) (2) c is a constant.

6. I.3 Caputo derivative Definition 3The Caputo derivative with Property 3 (1) (2) (3) c is a constant.

7. The relations between the classical derivative and the fractional derivatives Terminal properties for RL and Caputo derivatives

8. From the results in the preceding page and this page, it seems that the RL derivative is the generalization of the classical derivative.

9. Seemingly speaking, the classical derivative is the special case of the fractional derivatives. Actually this is not so. x’(0+) exists but x’(1) does not exist, it is quite the reverse for the RL derivative. One the other hand, its Caputo derivative exists although its classical derivative does not at t=1. So we can not regard it as the generalization of the typical derivative.

10. II. Applications This part comes from I. Podlubny’s PPT.

15. III. Comparison between FODE and ODE To be continued

16. To be continued.

18. IV. Our Work Our work is divided into two parts: IV.1 Stability analysis and dynamics of the fractional ordinary differential equations IV.2 Numerical method and computation of the fractional differential equations

19. IV.1 Stability analysis and dynamics of fractional ordinary differential equations Here the derivative is in Caputo’s sense.

20. where [1] Deng, Li and Lv, Nonlinear Dyn (2007) 48: 409-416.

23. Qian, Li, et al., Mathe Comput Modelling, in press.

24. Although an autonomous system can not define a dynamical system in the sense of semi-group because of the memory property determined by the fractional derivative, we can still use the Lyapunov exponents to discuss its dynamical evolution. Li, Gong, Qian and Chen, Chaos 20,013127, 2010.

25. IV.2 Numerical method and computation of fractional differential equations Adams method: At first, we introduce the numerical analyses of fractional ordinary differential equation.

27. The known result:

28. The further results:

29. Li and Tao, Comput Math Appl 58 (2009) 1573-1588.

30. The continued work can be found from following works: Sun, Chen, Li and Chen, Finite difference schemes for variable-order time fractional diffusion equation, submitted. CP Li and An Chen, Numerical computation in fractional integration and fractional differentiation, accepted by FDA10.

31. V. Acknowledgements Q&A! Thank Drs Chen and Li for cordial invitation! Thank you all for coming!