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Multi-order fractional differential equations and their numerical solution. (English) Zbl 1060.65070

The paper presents the numerical solution of multi-order fractional differential equations of the general (possibly nonlinear) form

y (α) (t)=f(t,y(t),y (β 1 ) (t),y (β 2 ) (t),,y (β n ) (t))

with α>β n >β n-1 >···>β 1 and α-β n 1, β j -β j-1 1 for all j and 0<β 1 1. Its linear case is y (α) (t)=λ o y(t)+ j=1 n λ j y (β j ) (t)+f(t). The initial conditions have the form y k (t)=y o (k) , k=0,1,,α-1. 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.

MSC:
65L05Initial value problems for ODE (numerical methods)
65L20Stability and convergence of numerical methods for ODE
34A34Nonlinear ODE and systems, general
26A33Fractional derivatives and integrals (real functions)
65L06Multistep, Runge-Kutta, and extrapolation methods