All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The jurists clarified that a wife whose husband died is obliged to stay throughout her waiting period, which is four months and ten days, in her house where she was when he died. Therefore, it is not permissible for her to go out of it except for a necessity or a dire need.
Indeed, the prohibition of travelling in her case is even more confirmed and stressed since the jurists are of the view that it is forbidden for her (a widow in her waiting period) to travel even for Hajj.
As regards your statement about your mother “There are no proper arrangements to look after her” is a general statement. What we can say here is that if she is not really harmed by staying away from you, or that there are means to repel the harm off her, then it is not permissible for her to travel to you.
On the other hand, you should remind your brother and your sister of the right of their mother on them and that they are obliged to be kind and dutiful to her. You may also provide a trustworthy maid to serve her if she is in need of this.
For more benefit on the cases when it is permissible for her to travel at times of necessity and when travelling becomes a must in order to repel the harm, please refer to Fataawa 136302 and 88684.
Allaah Knows best.