A Dilation Invariance Method and the Stability of Inhomogeneous Wave Equations
Round 1
Reviewer 1 Report
The Authors in this manuscript give analysis of the generalized Hyers-Ulam stability of a wave equation with source term. The material presented could be of interest to the readership sand is worth to be published in Mathematics.
Optionally, the Authors may comment on the application of the presented method to the general time fractional wave equations with source term.
Author Response
The first reviewer optionally suggests the application of the presented method to the general time fractional wave equations with a source term. This suggestion is very nice and valuable. It is very sorry that I have little knowledge about the time fractional differential equations. I would like to think over this subject in a next research. I thank you very much for the first reviewer for this valuable suggestion.
Reviewer 2 Report
The paper deals with the proof of the generalized Hyers- Ulam stability of the linear wave equation with a source term. The authors consider special solutions invariant with respect to a scaling group both on one space
dimension and $n$ spatial variables.
The paper is clearly written and the results widely discussed. However, some corrections need to be done.
At the beginning of Section 2 the authors observe that if $u(x,t)$ is a solution then the dilated solution $u(ax, at)$ is a solution of the inhomogeneous wave equation if the source is such that
$a^2 f(ax,at)=f(x,t)$, that is $f(x,t)$ is a homogeneous function of degree $-2$. This is true also when $n$ space variables are involved.
Nevertheless, when they consider their special solution $u(x,t)=t v(|x|/t)$, the dilation invariance is different since both independent variables and the unknown $u$ are scaled homogeneously. In fact, in this case, the source term has to be a homogeneous function of degree $-1$. The same occurs when the case of $n$ space variables are considered. The authors are requested to fix this item.
The paper could be accepted for publication provided that the authors take into account the above comment.
Author Response
The reviewer 2 suggested a valuable idea concerning the dilation invariance. I revised my original manuscript in accordance with the suggestion (please see the parts colored with red pen on the page 3 of the revised manuscript). I thank the reviewer for his/her nice suggestion.
