Cauchy problems for some classes of linear fractional differential equations

Abstract

Cauchy problems for a class of linear differential equations with constant coefficients and Riemann-Liouville derivatives of real orders, are analyzed and solved in cases when some of the real orders are irrational numbers and when all real orders appearing in the derivatives are rational numbers. Our analysis is motivated by a forced linear oscillator with fractional damping. We pay special attention to the case when the leading term is an integer order derivative. A new form of solution, in terms of Wright's function for the case of equations of rational order, is presented. An example is treated in detail.MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 34K37, 35R11, 60G22

Exact Stability and Instability Regions for Two-Dimensional Linear Autonomous Multi-Order Systems of Fractional-Order Differential Equations

“…This can also be verified by the method proposed in [11]. Indeed, for (q 1 , q 2 ) = ( 1 4 , 1 2 ), system (3.6) is equivalent to the following system of three fractional differential equations of the same order q = 1 4 : [25], as arg(λ 2 ) = arg(λ 3 ) = 0, it follows that system (3.7) is unstable.…”

“…Case 1. The special case (q 1 , q 2 ) = ( 1 2 , 1 4 ) has been considered in [11], and it has been shown, by transforming the corresponding system to a system of three fractional differential equations with the same order 1 4 , that in this particular case, system (3.6) is globally asymptotically stable.…”

“…Multi-term fractional-order differential equations [1] and their stability properties are closely related to multi-order systems of fractional differential equations. Very recently, the stability of two-term fractional-order differential and difference equations has been analyzed in [6,7,18].…”

Modeling the rheological properties of four commercially available composite core build-up materials

“…The first property is obvious, while the second one holds large class of functions. 20,21 For rheological test the Fourier transform is necessary Fft=f^ω=true∫−∞∞fte−itωdt of Riemann-Liouville fractional derivative and it reads 22,20,21…”

Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives

“…It is important to note that multi-term fractional-order differential equations [18] and their qualitative properties are sharply linked to multi-order systems of fractional differential equations. We refer to [11] for a thorough presentation of the relationship between these two concepts.…”