A difference method for initial value problems for ordinary fractional differential equations (Q2761890)

The author of this interesting paper investigates the solvability of ordinary differential equations \(_cD_t^pu(t)+\lambda u(t)=f(t)\), \(t>0\), under the initial data \(u(0),u^{(1)}(0),\dots ,u^{(n-1)}(0)\), with \(c\leq 0\), \(p\) is nonintegral, and \(n\) is an integer that satisfies \(n-1<p<n\). It is assumed that there exists a proper function \(u_{in}(t)\) defined for \(t<0\) that satisfies the initial values at \(t=0\), \(u(t)=u_{in}(t)\), \(t\leq 0\). This function, referred to as the initial function, represents the response of species in an experiment to which the difference equation is applied to external forcing during \(t<0\) in order to set the initial condition of the experiment. For instance \(u_{in}(t)=\sum_{k=0}^{n-1}u^{k}(0)t^k/k! \). It is shown that the Riemann-Liouville (RL) derivative is equivalent to the Caputo (C) derivative. The numerical solutions by the difference method obtained are in good agreement with the analytic solutions by the Laplace transformation method.