On linear fractional differential equations with variable coefficients

• We study and solve linear ordinary differential equations, with fractional order derivatives of either Riemann–Liouville or Caputo types, and with variable coefficients which are either integrable or continuous functions. • The solutions are given explicitly by a convergent infinite series involving compositions of fractional integrals, and its uniqueness is proved in suitable function spaces. • For the case of constant coefficients, the solutions can be expressed by the multivariate Mittag–Leffler function. We study and solve linear ordinary differential equations, with fractional order derivatives of either Riemann–Liouville or Caputo types, and with variable coefficients which are either integrable or continuous functions. In each case, the solution is given explicitly by a convergent infinite series involving compositions of fractional integrals, and its uniqueness is proved in suitable function spaces using the Banach fixed point theorem. As a special case, we consider the case of constant coefficients, whose solutions can be expressed by using the multivariate Mittag–Leffler function. Some illustrative examples with potential applications are provided.