Solution of Nonhomogeneous Linear System of Caputo Fractional Differential Equations with Initial Conditions

The solution of a nonhomogeneous linear Caputo fractional differential equation of order nq,(n−1)<nq<n with Caputo fractional initial conditions can be expressed using suitable Mittag–Leffler functions. In order to extend this result to such a nonhomogeneous linear Caputo fractional differential equation of order nq,(n−1)<nq<n, that also includes lower order fractional derivative terms, we can reduce such a problem to an n-system of Caputo fractional differential equations of order q,0<q<1, with corresponding initial conditions. In this work, we use an approximation method to solve the resulting system of Caputo fractional differential equations of order q with initial conditions, using the fundamental matrix solutions involving the matrix Mittag–Leffler functions. Furthermore, we compute the fundamental matrix solution using the standard eigenvalue method. This fundamental matrix solution then allows us to express the component-wise solutions of the system using initial conditions, similar to the scalar case. As a consequence, we obtain solutions to linear nonhomogeneous Caputo fractional differential equations of order nq,(n−1)<nq<n, with Caputo fractional initial conditions having lower-order Caputo derivative terms. We illustrate the method with several examples for two and three system, considering cases where the eigenvalues are real and distinct, real and repeated, or complex conjugates.