A document by Xianmin Zhang. Click on the document to view its contents.

It is found that there are two piecewise functions to satisfy the conditions in the impulsive fractional partial differential system (IFrPDS), which deduce that the three different equivalent integral equations 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 equivalent integral equation 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 equivalent integral equation of the IFrPDS.

The fractional derivatives are not equal for different expressions of the same piecewise function, which caused that the equivalent integral equations of impulsive fractional order system (IFrOS) proposed in existing papers are incorrect. Thus we reconsider two generalized IFrOSs that both have the corresponding impulsive Caputo fractional order system and the corresponding impulsive Riemann-Liouville fractional order system as their special cases, and discover that their equivalent integral equations are two integral equations with some arbitrary constants, which reveal the non-uniqueness of solution of the two generalized IFrOSs. Finally, two numerical examples are offered for explaining the non-uniqueness of solution to the two generalized IFrOSs.