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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

${y}^{\left(\alpha \right)}\left(t\right)=f\left(t,y\left(t\right),{y}^{\left({\beta }_{1}\right)}\left(t\right),{y}^{\left({\beta }_{2}\right)}\left(t\right),\cdots ,{y}^{\left({\beta }_{n}\right)}\left(t\right)\right)$

with $\alpha >{\beta }_{n}>{\beta }_{n-1}>···>{\beta }_{1}$ and $\alpha -{\beta }_{n}\le 1$, ${\beta }_{j}-{\beta }_{j-1}\le 1$ for all $j$ and $0<{\beta }_{1}\le 1$. Its linear case is ${y}^{\left(\alpha \right)}\left(t\right)={\lambda }_{o}y\left(t\right)+{\sum }_{j=1}^{n}{\lambda }_{j}{y}^{\left({\beta }_{j}\right)}\left(t\right)+f\left(t\right)$. The initial conditions have the form ${y}^{k}\left(t\right)={y}_{o}^{\left(k\right)}$, $k=0,1,\cdots ,⌈\alpha ⌉-1$. 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:
 65L05 Initial value problems for ODE (numerical methods) 65L20 Stability and convergence of numerical methods for ODE 34A34 Nonlinear ODE and systems, general 26A33 Fractional derivatives and integrals (real functions) 65L06 Multistep, Runge-Kutta, and extrapolation methods