Exact Solutions of Fractional Order Oscillation Equation with Two Fractional Derivative Terms

Abstract The fractional oscillation equation with two derivative terms in the sense of Caputo, where orders $$\alpha$$ α and $$\beta$$ β satisfy $$1<\alpha \le 2$$ 1 < ≤ 2 $$0<\beta 1$$ 0 , is investigated, unit step response initial value responses are obtained forms by using different methods inverse Laplace transform. first method yields series solutions nonnegative powers t which converge fast for small . second our emphasis, complex path integral formula transform used. In order to determine singularities integrand we seek roots characteristic equation, a transcendental four parameters, coefficients noninteger power exponents. existence conditions properties on principal Riemann surface given. Based results derive these as sum classical exponentially damped oscillation, vanishes an indicated case, infinite type, converges large steady component response. Asymptotic behaviors derived algebraic decays negative laws characterized system exhibits magical transition from monotonic decay law.