How does a patient with a chronic illness, who often falls into a coma, make up for missed prayers and fasting, and is it permissible for his children to pray on his behalf?

The sick person must pray according to their ability, as long as their mind is sound. If they are able, they should pray standing; otherwise, sitting, or on their side. What they are unable to do is waived. If they lose consciousness, prayer is not obligatory for them, and there is no make-up prayer for someone who was unconscious for the entire time. As for fasting, if they are unable to fast due to an illness from which recovery is not expected, they must feed a poor person for each day. If it is an illness from which recovery is hoped, they must make up the fast after recovery. If the illness leads directly to their death, then nothing is required of them.