Exact Solutions of Fractional Order Oscillation Equation with Two Fractional Derivative Terms
Jun‐Sheng Duan, Jun-Yan Zhang, Xiang Qiu
Abstract
Abstract The fractional oscillation equation with two fractional derivative terms in the sense of Caputo, where the orders $$\alpha$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> satisfy $$1<\alpha \le 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>α</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and $$0<\beta \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>β</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , is investigated, and the unit step response and two initial value responses are obtained in two forms by using different methods of inverse Laplace transform. The first method yields series solutions with the nonnegative powers of t , which converge fast for small t . The second method is our emphasis, where the complex path integral formula of the inverse Laplace transform is used. In order to determine singularities of integrand we first seek for the roots of the characteristic equation, which is a transcendental equation with four parameters, two coefficients and two noninteger power exponents. The existence conditions and properties of the roots on the principal Riemann surface are given. Based on the results on the characteristic equation, we derive these responses as a sum of a classical exponentially damped oscillation, which vanishes in an indicated case, and an infinite integral of the Laplace type, which converges fast for large t , with a steady component in the unit step response. Asymptotic behaviors of solutions for large t are derived as algebraic decays in negative power laws characterized by the orders $$\alpha$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> . The fractional system exhibits a magical transition from oscillation to monotonic decay in negative power law.