Fractional Calculus The Murky Bits

Transcription

1 Aditya Jaishankar August 13 th 2010

2 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. 2

3 The Laplace Transform: The Many Definitions 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

4 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

5 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: DOI /s

6 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: DOI /s

7 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)

8 Generalized Maxwell Model Also, zero shear viscosity is given by Step loading response Case 1: Case 2: Case 3: This Diverges! 8

9 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: (1992) 9

10 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, (1995) Mario N. Berberan-Santos, J. Math. Chem. Vol. 38, No. 4, Nov

11 Overview Generalized Fractional Maxwell Model Generalized Fractional Kelvin-Voigt Model 11

12 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? 12