On tempered and substantial fractional calculus

1. On tempered and substantial fractional calculus YangQuan Chen Collaborators: Jianxiong Cao, Changpin Li School of Engineering, University of California, Merced Department of Mathematics, Shanghai University E: yangquan. chen@ucmerced.edu

2. Outline • Preliminaries of fractional calculus • Tempered and substantial fractional operators • Numerical solution of a tempered diffusion equation • Conclusion

3. Preliminaries • Calculus=integration+differentiation • Fractional Calculus= fractional integration+ fractional differentiation • Fractional integral: Mainly one: Riemann-Liouville integral • Fractional derivatives More than 6: not mutually equivalent RL and Caputo derivatives are mostly used

4. Preliminaries • Left Riemann-Liouville integral • Right Riemann-Liouville Integral • Left Riemann-Liouville fractional derivative • Right Riemann-liouville Fractional derivative

5. Left Caputo fractional derivative • Right Caputo fractional derivative • Riesz fractional derivative • Riesz-Caputo fractional derivative where

6. Slide-6/1024 Tempered and substantial operator Let be piecewise continuous on ,and integrable on any subinterval of , • The left Riemann-Liouville tempered fractional integral is • The right Riemann-Liouville tempered fractional integral

7. (Phys. Rev. E 76, 041105, 2007 ) Suppose that be times continuously differentiable on , and times derivatives be integrable on any subinterval of , then the left and right Riemann-Liouville tempered derivative are defined and

8. Remark I: From the above definitions, we can see if then they reduce to left and right Riemann-Liouville Fractional derivatives. Remark II: The variants of the left and right Riemann-Liouville tempered fractional derivatives (Boris Baeumer, JCAM, 2010)

9. (Phys. Rev. Lett. 96, 230601, 2006) Substantial fractional operators Let be piecewise continuous on , and integrable on any subinterval of , then the substantial fractional integral is defined by And the substantial derivative is defined as where

10. Theorem: If the parameter is a positive constant, the tempered and substantial operators are equivalent. See our paper for the details of proof. Recently, tempered fractional calculus is introduced in Mark M. Meerschaert et al. JCP(2014).

11. Numerical computation of a tempered diffusion equation We consider the following tempered diffusion equation where

12. The exact solution of above problem is We use finite difference method and shifted Grunwald method to numerically solve it, we choose different value Of parameter , and compare the results with its exact solution

16. From the figures, we can see that the numerical solution are in good accordance with exact solutions. • It is easily shown that the peak of the solutions of tempered diffusion equation becomes more and more smooth as exponential factor λ increases.

17. Conclusion • Tempered and substantial fractional calculus are the generalizations of fractional calculus. • The two fractional operators are equivalent, although they are introdued from different physical backgrounds. • Tempered fractional operators are the best tool for truncated exponential power law description

18. Thanks for your attention