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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 $$y^{(\alpha )}(t)=f(t, y(t), y^{(\beta_1)}(t), y^{(\beta_2)}(t),\dots, y^{(\beta_n)}(t))$$ with $\alpha>\beta_n>\beta_{n-1}>...>\beta_1$ and $\alpha -\beta_n\leq1$, $\beta_j -\beta_{j-1}\leq 1$ for all $j$ and $0<\beta_1\leq 1$. Its linear case is $y^{(\alpha )}(t)=\lambda_oy(t)+ \sum_{j=1}^n\lambda_jy^{(\beta_j)}(t)+f(t)$. The initial conditions have the form $y^{k}(t)=y_o^{(k)}$, $k=0,1,\dots,\lceil\alpha\rceil -1$. 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
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Full Text: DOI
References:
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