Multi-order fractional differential equations and their numerical solution
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.
multi-term fractional differential equation
scientific article; zbMATH DE number 2105538
scientific article