Many physical processes appear to exhibit fractional order behavior thatmay vary with time or space. The continuum of order in the fractionalcalculus allows the order of the fractional operator to be considered asa variable. This paper develops the concept of variable and distributedorder fractional operators. Definitions based on the Riemann–Liouvilledefinition are introduced and the behavior of the new operators isstudied. Several time domain definitions that assign different argumentsto the order q in the Riemann–Liouville definition are introduced. Foreach of these definitions various characteristics are determined. Theseinclude: time invariance of the operator, operator initialization,physical realization, linearity, operational transforms, and memorycharacteristics of the defining kernels.
A measure (m 2) for memory retentiveness of the order history isintroduced. A generalized linear argument for the order q allows theconcept of `tailored' variable order fractional operators whose m 2 memory may be chosen for a particular application. Memory retentiveness (m 2) andorder dynamic behavior are investigated and applications are shown.
The concept of distributed order operators where the order of thetime based operator depends on an additional independent (spatial)variable is also forwarded. Several definitions and their Laplacetransforms are developed, analysis methods with these operators aredemonstrated, and examples shown. Finally operators of multivariable anddistributed order are defined and their various applications areoutlined.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Lorenzo, C. F. and Hartley, T. T., 'Initialization, conceptualization, and application in the generalized fractional calculus', NASA/TP-1998–208415, 1998.
Hartley, T. T. and Lorenzo, C. F., 'Fractional system identification: An approach using continuous order distributions', NASA/TM-1999–209640, 1999.
Bagley, R. L. and Torvik, P. J., 'On the existence of the order domain and the solution of distributed order equations, Part I', International Journal of Applied Mathematics 2(7), 2000, 865–882, Part II, 2(8), 2000, 965–987.
Bland, D. R., The Theory of Linear Viscoelasticity, Pergamon Press, New York, 1960.
Bagley, R. L., 'The thermorheologically complex material', International Journal of Engineering Science 29(7), 1991, 797–806.
Smit, W. and deVries, H., 'Rheological models containing fractional derivatives', Rheologica Acta 9, 1970, 525–534.
Glöckle, W. G. and Nonnenmacher, T. F., 'A fractional calculus approach to self-similar protein dynamics', Biophysical Journal 68, 1995, 46–53.
Klass, D. L. and Martinek, T. W., 'Electroviscous fluids. I. Rheological properties', Journal of Applied Physics 38(1), 1967, 67–74.
Shiga, T., 'Deformation and viscoelastic behavior of polymer gel in electric fields', Proceedings of Japan Academy, Series B, Physical and Biological Sciences 74, 1998, 6–11.
Davis, L. C., 'Model of magnetorheological elastomers', Journal of Applied Physics 85(6), 1999, 3342–3351.
Lorenzo, C. F. and Hartley, T. T., 'Initialized fractional calculus', International Journal of Applied Mathematics 3(3), 2000, 249–265.
DeRusso, P. M., Roy, J. R., and Close, C. M., State Variables for Engineers, Wiley, New York, 1965.
Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
Erdelyi, A. (ed.), Tables of Integral Transforms, McGraw-Hill, New York, 1954.
Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, 1993.
About this article
Cite this article
Lorenzo, C.F., Hartley, T.T. Variable Order and Distributed Order Fractional Operators. Nonlinear Dynamics 29, 57–98 (2002). https://doi.org/10.1023/A:1016586905654
- variable order fractional operator
- distributed order fractional operator
- order distribution
- Laplace transform
- tailored fractional order operator
- memory measure