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Definition of physically consistent damping laws with fractional derivatives. (English) Zbl 0865.70014

Summary: The generalized damping equation \(E:(D^2+aD^q+b)x(t)=f(t)\); \(q\in (0,2)\) is treated. It is shown that for \(q\neq 1\) and \(x,f\in \mathbb{L}_C^2(\mathbb{R})\) there are arbitrarily many proper definitions of \(E\) corresponding to the choice of branches of \((i\omega)^q\) in the definition of the characteristic functions \(p(\omega)=(i\omega)^2+a(i\omega)^q+b\). The only restriction is that \(p(\omega)\) is measurable. General conditions and results concerning uniqueness and causality of the solutions of \(E\) are developed. Physically reasonable ones are: \(E\) has unique solutions if \(p(\omega)\) is continuous and has no real zeros. If, furthermore, \(p\) is restricted to the principal branch, the solutions become causal if and only if \(a,b>0\). For demonstration purposes, a general analytic solution of the causal impulse response is given and discussed.

MSC:

70J99 Linear vibration theory
70J40 Parametric resonances in linear vibration theory
47A60 Functional calculus for linear operators
26A33 Fractional derivatives and integrals
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