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On representation formulas for solutions of linear differential equations with Caputo fractional derivatives. (English) Zbl 1488.34036

Summary: In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix is carried out. In particular, the Hölder continuity of this matrix with respect to both variables is proved, and its dual definition is given. Based on this, two representation formulas for the solution of the Cauchy problem are proposed and justified.

MSC:

34A08 Fractional ordinary differential equations
34A30 Linear ordinary differential equations and systems
34A05 Explicit solutions, first integrals of ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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