Analytical solution of fractional oscillation equation with two Caputo fractional derivatives

Analytical solution of initial value problem for the fractional oscillation equation with two Caputo fractional derivatives x''(t)+ aD.. t x(t)+ cx'(t)+ bD.. t x(t)+ kx(t) = q(t), where the coefficients and orders satisfy a, b, c, k > 0, 1 <.. = 2 and 0 <.. = 1, is investigated by using the Laplace transform and complex inverse integral method on the principal Riemann surface. It is proved by using the argument principle that the characteristic equation has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface under the assumption that.. and.. are not both integers. Then three fundamental solutions, the unit impulse response, the unit initial displacement response, and the unit initial rate response, are derived analytically. Each of these solutions is expressed into a superposition of a classical damped oscillation decaying exponentially and a real Laplace integration decaying in a negative power law. Finally, the asymptotic behaviors of these analytical solutions for sufficiently large t are determined as monotonous decays in a power of negative exponent.