For the impulsive fractional-order system (IFrOS) of order ϵ∈(1,2), there have appeared some conflicting equivalent integral equations in existing studies. However, we find two fractional-order properties of piecewise function and use them to verify that these given equivalent integral equations have some defects to not be the equivalent integral equation of the IFrOS. For the IFrOS, its limit property shows the linear additivity of the impulsive effects. For the IFrOS, we use the limit analysis and the linear additivity of the impulsive effects to find its correct equivalent integral equation, which is a combination of some piecewise functions with two arbitrary constants; that is, the solution of the IFrOS is a general solution. Finally, a numerical example is given to show the equivalent integral equation and the non-uniqueness of the solution of the IFrOS.

It is found that there are two piecewise functions satisfy the conditions in the impulsive fractional partial differential system (IFrPDS), which deduce that the three different integral solutions of the IFrPDS given in the cited papers are inappropriate. Next, by applying two limit properties of the IFrPDS and the properties of piecewise function, the new formula of solution of the IFrPDS is discovered that is the integral equation with an arbitrary continuously differentiable function of t on [0 ,c] to reveal the non-uniqueness of the IFrPDE’s solution. Finally, an example is provided to expound the computation of the solution of the IFrPDS.