Abstract
This paper demonstrates the power of the functional-calculus definition oflinear fractional (pseudo-)differential operators via generalised Fouriertransforms.
Firstly, we describe in detail how to get global causal solutions of linearfractional differential equations via this calculus. The solutions arerepresented as convolutions of the input functions with the related impulseresponses. The suggested method via residue calculus separates an impulseresponse automatically into an exponentially damped (possibly oscillatory)part and a `slow' relaxation. If an impulse response is stable it becomesautomatically causal, otherwise one has to add a homogeneous solution to getcausality.
Secondly, we present examples and, moreover, verify the approach alongexperiments on viscolelastic rods. The quality of the method as an effectivefew-parameter model is impressively demonstrated: the chosen referenceexample PTFE (Teflon) shows that in contrast to standard classical modelsour model describes the behaviour in a wide frequency range within theaccuracy of the measurement. Even dispersion effects are included.
Thirdly, we conclude the paper with a survey of the required theory. Therethe attention is directed to the extension from the L 2-approachon the space of distributions \({\mathcal{D}'}\)′.
Similar content being viewed by others
References
Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.
Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives, Gordon and Breach, London, 1993.
Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
Kempfle, S. and Schäfer, I., 'Fractional differential equations and initial conditions', Fractional Calculus & Applied Analysis 3(4), 2000, 387–400.
Kempfle, S. and Schäfer, I., 'Functional calculus method versus Riemann-Liouville approach', in Transform Methods & Special Functions, AUBG'99 (Proceedings of 3rd International Workshop), P. Rusev, I. Dimovski, and V. Kiryakowa (eds.), IMI-BAS, Sofia, 1999, pp. 210–226.
Beyer, H. and Kempfle, S., 'Definition of physically consistent damping laws with fractional derivatives' Zentralblatt für angewandte Mathematik und Mechanik 75(8), 1995, 623–635.
Kempfle, S. and Beyer, H., 'Global and causal solutions of fractional differential equations', in Transform Methods & Special Functions, Varna'96 (Proceedings of the 2nd International Workshop), P. Rusev, I. Dimovski, and V. Kiryakowa (eds.), IMI-BAS, Sofia, 1998, pp. 210–226.
Kempfle, S., 'Causality criteria for solutions of linear fractional differential equations', Fractional Calculus & Applied Analysis 1(4), 1998, 351–364.
Kempfle, S., Schäfer, I., and Beyer, H., 'Fractional differential equations and viscoelastic damping', in Proceedings of the European Control Conference 2001, Porto, J. L.Martins de Carvalho, F. A. C. C. Fontes, and M. d. R. de Pinho (ed.), 2001, pp. 1744–1751.
Nolte, B., Kempfle, S., and Schäfer, I., 'Applications of fractional differential operators to the damped structure borne sound in viscoelastic solids,' in Proceedings of ICTCA 2001, Peking, 2001, to appear.
Schäfer, I., 'Beschreibung der Dämpfung in Stäben mittels fraktionaler Zeitableitungen', Zentralblatt für angewandte Mathematik und Mechanik 80, 2000, 1–5.
Conway, J. B., Functions of One Complex Variable, Springer-Verlag, New York, 1978.
Marsden, J. E. and Hoffman, M. J., Basic Complex Analysis, Freeman, New York, 1999.
Gorenflo, R. and Mainardi, F., 'Fractional calculus: Integral and differential equations of fractional order', in: Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer-Verlag, New York, 1997, pp. 223–276.
Gorenflo, R. and Rutman, R., 'On ultraslow and intermediate processes', in Transform Methods & Special Functions, Sofia' 94 (Proceedings of 1st International Workshop) P. Rusev, I. Dimovski, and V. Kiryakowa (eds.), SCTP, Singapore, 1995, pp. 61–81.
Podlubny, I., 'Solution of linear fractional differential equations with constant coefficients', in Transform Methods & Special Functions, Sofia'94 (Proceedings of 1st International Workshop) P. Rusev, I. Dimovski, and V. Kiryakowa (eds.), World Scientific, Singarore, 1995, pp. 227–237.
Schäfer, I. and Seifert, H.-J., 'Description of the impulse response in rods by fractional derivatives', Zentralblatt für angewandte Mathematik und Mechanik 81, 2001, 423–427.
Rudin, W., Functional Analysis, McGraw-Hill, New York, 1978.
Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971.
Grubb, G., 'Pseudodifferential boundary problems and applications', Deutsche Mathematiker Vereinigung, Jahresbericht 99(3), 1997, 110–121.
Wong, M. W., An Introduction to Pseudo-Differential Operators, World Scientific, Singapore, 1991.
Yoshida, K., Functional Analysis, Springer-Verlag, Berlin, 1978.
Walter, W., Einführung in die Theorie der Distributionen, 3. Aufl. Bibliographisches Institut Mannheim, 1994.
Gel'fand, I. M. and Shilov, G. E., Generalized Functions, Vol. 1, Academic Press, New York, 1964.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kempfle, S., Schäfer, I. & Beyer, H. Fractional Calculus via Functional Calculus: Theory and Applications. Nonlinear Dynamics 29, 99–127 (2002). https://doi.org/10.1023/A:1016595107471
Issue Date:
DOI: https://doi.org/10.1023/A:1016595107471