The paper presents the numerical solution of multi-order fractional differential equations of the general (possibly nonlinear) form
with and , for all and . Its linear case is . The initial conditions have the form , . The derivatives are understood in the Caputo sense.
A generalization of an approach employed in the solution of ordinary differential equations of order two or higher converting such equation to a system of equations of order one is used. It uses the fact that any real number can be approximated arbitrarily closely by a rational number. Thereby, the assumption on the commensuracy for fractional order equations can be ensured by an appropriate order approximation. A simple generalization of the theorem on the equivalence of a nonlinear system and the linear systems theory is presented first. Then, the nonlinear problem includes two Gronwall-type results for a two-term equation, the general existence-uniqueness as well as the structural stability results. A convergent and stable Adams-type numerical method is proposed including a specific numerical example.