Fractional Calculus via Functional Calculus: Theory and Applications

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 L₂-approachon the space of distributions .