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Fractional Calculus – The Murky Bits
Aditya JaishankarAugust 13th 2010
Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits
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.
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Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits
The Many Definitions
• The Laplace Transform:
• Two important properties of the Laplace Transform:
– Special case is the Riemann – Liouville integral
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Igor Pdolubny, Fractional Differential Equations. “Mathematics in Science and Engineering V198”, Academic Press 1999
Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits
The Many Definitions
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• 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
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
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Nicole Heymans and Igor Podlubny, Rheol Acta (2006) 45: 765–771 DOI 10.1007/s00397-005-0043-5
•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
Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits
Unphysical Yet Solvable• Fractional Maxwell model – Spring and spring-pot in series
• Stress relaxation – Step strain applied at t=0
• To find the boundary condition, we integrate the constitutive equation, and let
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K
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
Unphysical Yet Solvable
• Likewise for strain impulse response applied at t=0
• The Generalized Maxwell Model:
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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
Generalized Maxwell Model
• Also, zero shear viscosity is given by
– Case 1:
– Case 2:
– Case 3:
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This Diverges!
Step loading response
Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits
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
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Chr. Friedrich and H. Braun, Rheol Acta 31:309-322 (1992)
Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits
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
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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
Overview
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Generalized Fractional Maxwell Model Generalized Fractional Kelvin-Voigt Model
Aditya JaishankarAditya Jaishankar Fractional Calculus – The Murky Bits
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?
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Thank you. Questions?