DEFINITION OF PHYSICALLY CONSISTENT DAMPING LAWS WITH FRACTIONAL DERIVATIVES

The generalized damping equation E: (D-2 + aD(q) + b) x(t) = f(t); q is an element of (0, 2) is treated It is shown that for q not equal 1 and x, f is an element of L(C)(2)(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 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.