Definition of physically consistent damping laws with fractional derivatives

The generalized damping equation E: (D 2 + aD q + b) x(t) = f(t); q ∈ (0,2) is treated. It is shown that for q = 1 and x, f ∈ L C 2 (R) there are arbitrarily many proper definitions of E corresponding to the choice of branches of (iω) q in the definition of the characteristic functions p(ω) = (iω) 2 + a(iω) q + b. The only restriction is that p(ω) 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(ω) is continuous and has no real zeros. If, furthermore, p is restricted to the principal branch, the solutions then 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.