Abstract
This paper proposes a series-representations for the solution of initial value problems of linear inhomogeneous fractional differential equation with continuous variable coefficients. It is proved that the solution of the problem is determined by adding the solution of the inhomogeneous differential equations with the homogeneous initial conditions to the linear combination of the canonical fundamental system of solution for corresponding homogeneous fractional differential equation and the inhomogeneous initial values. The effectiveness of the theoretical analysis is illustrated with two examples.
Highlights
The fractional modelings have aroused much attention in the fields of both engineering and mathematics due to their significant applications in diverse scientific areas such as electromagnetism [1], behaviors of physical phenomena [2], signal processing [3], and control engineering [4]
In [22], a solution of general linear inhomogeneous fractional differential equations with constant coefficients has been obtained by using the Adomian decomposition method and one proved that this solution is equal to the solution represented by Green’s function
A theory on the system of linear inhomogeneous fractional differential equation has been studied, and the solution was represented in terms of the Green function for the case
Summary
The fractional modelings have aroused much attention in the fields of both engineering and mathematics due to their significant applications in diverse scientific areas such as electromagnetism [1], behaviors of physical phenomena [2], signal processing [3], and control engineering [4]. In [22], a solution of general linear inhomogeneous fractional differential equations with constant coefficients has been obtained by using the Adomian decomposition method and one proved that this solution is equal to the solution represented by Green’s function. A theory on the system of linear inhomogeneous fractional differential equation has been studied, and the solution was represented in terms of the Green function for the case. In [24], a power series solution method for some linear fractional differential equations with continuous variable coefficients has been presented. Our work proposes series-representations for the solution of linear inhomogeneous fractional differential equation with continuous variable coefficients and inhomogeneous initial conditions.
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