Should the mother shorten her prayer or complete it during the first four days of visiting her son, if her frequent visits to him are similar to her residing in her own home, and her husband knows about and approves of this?

The majority of scholars hold that the intention to stay for four days severs the ruling of travel. Therefore, as long as your mother stays with you for that duration, she is considered a resident; she completes her prayers and is obligated to fast Ramadan (definition).