What should a grandmother do who vowed to fast and did so, but doubts the validity of her fast, while her daughter-in-law is certain that she fasted?

The basic principle is that whoever doubts having performed an act of worship must perform it, for one's obligation is not discharged except with certainty. If your grandmother is in a normal state and does not suffer from obsessive thoughts, then she must make up the fasts. However, if her daughter-in-law is trustworthy and certain of her fasting, then she is not required to make them up. Similarly, if the grandmother is very forgetful or suffers from obsessive thoughts, she should not pay attention to the doubt, as the cure for obsessive thoughts is to disregard them.