MESA LABMESA LABMESA LABMESA LAB

On tempered and substantial fractional On tempered and substantial fractional calculuscalculus

YangQuan Chen

Collaborators: Jianxiong Cao, Changpin LiSchool of Engineering,

University of California, MercedDepartment of Mathematics, Shanghai University

E: yangquan. chen@ucmerced.edu

MESA LABMESA LABMESA LABMESA LAB

Outline

• Preliminaries of fractional calculus

• Tempered and substantial fractional operators

• Numerical solution of a tempered diffusion

equation

• Conclusion

MESA LABMESA LABMESA LABMESA LAB

Preliminaries

• Calculus=integration+differentiation• Fractional Calculus= fractional integration+

fractional differentiation

• Fractional integral: Mainly one: Riemann-Liouville integral • Fractional derivatives More than 6: not mutually equivalentRL and Caputo derivatives are mostly used

MESA LABMESA LABMESA LABMESA LAB

Preliminaries

• Left Riemann-Liouville integral

• Right Riemann-Liouville Integral

• Left Riemann-Liouville fractional derivative

• Right Riemann-liouville Fractional derivative

MESA LABMESA LABMESA LABMESA LAB

• Left Caputo fractional derivative

• Right Caputo fractional derivative

• Riesz fractional derivative

• Riesz-Caputo fractional derivative

where

MESA LABMESA LABMESA LABMESA LAB

Tempered and substantial operator

Let be piecewise continuous on ,and integrable on any subinterval of ,

1)The left Riemann-Liouville tempered fractional integral is

2)The right Riemann-Liouville tempered fractional integral

Slide-6/1024

MESA LABMESA LABMESA LABMESA LABSlide-7/1024

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

(Phys. Rev. E 76, 041105, 2007 )

and

MESA LABMESA LABMESA LABMESA LABSlide-8/1024

Remark I: From the above definitions, we can see if then they reduce to left and right Riemann-LiouvilleFractional derivatives. Remark II: The variants of the left and right Riemann-Liouville

tempered fractional derivatives (Boris Baeumer, JCAM, 2010)

MESA LABMESA LABMESA LABMESA LABSlide-9/1024

(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

MESA LABMESA LABMESA LABMESA LABSlide-10/1024

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 inMark M. Meerschaert et al. JCP(2014).

MESA LABMESA LABMESA LABMESA LABNumerical computation of a tempered diffusion

equation

We consider the following tempered diffusion equation

where

Slide-11/1024

MESA LABMESA LABMESA LABMESA LABSlide-12/1024

The exact solution of above problem is

We use finite difference method and shifted Grunwald method to numerically solve it, we choose different valueOf parameter , and compare the results with its exact solution

MESA LABMESA LABMESA LABMESA LABSlide-13/1024

MESA LABMESA LABMESA LABMESA LABSlide-14/1024

MESA LABMESA LABMESA LABMESA LABSlide-15/1024

MESA LABMESA LABMESA LABMESA LABSlide-16/1024

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.

MESA LABMESA LABMESA LABMESA LAB

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

MESA LABMESA LABMESA LABMESA LAB

Thanks for your attention