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General solution of the Bagley-Torvik equation with fractional-order derivative. (English) Zbl 1221.34020

Summary: This paper investigates the general solution of the Bagley–Torvik equation with \(1/2\)-order derivative or \(3/2\)-order derivative. This fractional-order differential equation is changed into a Sequential Fractional-order Differential Equation (SFDE) with constant coefficients. Then the general solution of the SFDE is expressed as the linear combination of fundamental solutions that are in terms of \(\alpha \)-exponential functions, a kind of functions that have a similar role as the classical exponential function. Because the number of fundamental solutions of the SFDE is greater than 2, the general solution of the SFDE depends on more than two free (independent) constants. This paper shows that the general solution of the Bagley–Torvik equation involves actually two free constants only, and it can be determined fully by the initial displacement and initial velocity.

MSC:

34A08 Fractional ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
45J05 Integro-ordinary differential equations
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