Fractional Calculus – The Murky Bits

Aditya Jaishankar August 13th 2010 The Many Definitions

• The Reimann-Liouville definition – Differentiation after integration:

• The Caputo definition - Integration after differentiation:

• Differences arise during physical interpretation – Initial conditions are straightforward in the Caputo definition – Constants are not constants! • The differences are easy to see in the Laplace space.

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 2 The Many Definitions

• The Laplace Transform:

• Two important properties of the Laplace Transform:

– Special case is the Riemann – Liouville integral

Igor Pdolubny, Fractional Differential Equations. “Mathematics in Science and Engineering V198”, Academic Press 1999

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 3 The Many Definitions

• Laplace Transform of the R-L derivative:

• Laplace Transform of the Caputo derivative:

• order term appears only in the multiplier in Caputo derivatives.

This Laplace transform of the Reimann-Liouville fractional derivative is well known. However, its practical applicability is limited by the absence of the physical interpretation of the limit values of fractional derivatives at the lower terminal t=0. At the time of writing, such an interpretation is not known. - Igor Podlubny

Igor Pdolubny, Fractional Differential Equations. “Mathematics in Science and Engineering V198”, Academic Press 1999

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 4 The Many Definitions

• Fractional derivative of a constant is not zero using the R-L definition, while it is always zero using Caputo definition – Makes Caputo much more amenable to physical problems – One needs multiple values of different derivatives at t=0 for Caputo definition

•Might be unphysical but still solvable. Heymans and Podlubny use a combination of integration of the constitutive equation along with the zero time limit to extract fractional initial conditions

Nicole Heymans and Igor Podlubny, Rheol Acta (2006) 45: 765–771 DOI 10.1007/s00397-005-0043-5

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 5 Unphysical Yet Solvable • Fractional Maxwell model – Spring and spring-pot in series

• Stress relaxation – Step strain applied at t=0 K

• To find the boundary condition, we integrate the constitutive equation, and let

Nicole Heymans and Igor Podlubny, Rheol Acta (2006) 45: 765–771 DOI 10.1007/s00397-005-0043-5

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 6 Unphysical Yet Solvable

• Likewise for strain impulse response applied at t=0

• The Generalized Maxwell Model:

Pan Yang, Yee Cheong Lam, Ke-Qin Zhu, J. Non-Newtonian Fluid Mech. 165 (2010) 88–97

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 7 Generalized Maxwell Model

Step loading response

• Also, zero shear viscosity is given by

– Case 1:

– Case 2:

– Case 3: This Diverges!

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 8 Generalized Maxwell Model • Another approach, followed by Friedrich and Braun, is to use the modified Cole-Cole relaxance equation

• Only ensures the existence of a Newtonian viscosity at low frequencies

Chr. Friedrich and H. Braun, Rheol Acta 31:309-322 (1992) Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 9 Fastest Relaxation

• Coefficient of first normal stress difference doesn’t exist – hence redefine the relaxance function.

• Fastest initial relaxation at ? – Any relaxation function can be written as

C Friedrich, Acta Polymer 46, 385-390 (1995) Mario N. Berberan-Santos, J. Math. Chem. Vol. 38, No. 4, Nov. 2005 Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 10 Overview

Generalized Fractional Maxwell Model Generalized Fractional Kelvin-Voigt Model

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 11 Conclusion

• There are different definitions for fractional derivatives – Choose depending on application • Order of fractional derivatives must be chosen carefully – integrals can diverge and give unphysical results • Fastest relaxation at ! What happens if system relaxes faster? • Models can be made as complex as necessary – agrees with experiments?

Thank you. Questions?

Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits 12