Swearing by Him to adjure another person means swearing that the other person shall certainly do such-and-such. If that person causes the oath to be broken by failing to fulfill it, then, according to the great majority of jurists, the expiation is due from the person who swore, not the person adjured. For example, if someone swears that his slave, child, or friend shall do something, and that person does not do it, the expiation is due from the one whose oath has been broken.

As for saying:

“I ask you by Allah to do such-and-such,”

this is a request, not an oath. The hadith states:

“Give to whoever asks you by Allah.”

No expiation is due from the person making such a request if it is not granted. All created beings ask Allah, believers and unbelievers alike, and Allah may answer the supplications of unbelievers. Unbelievers ask Allah for provision, and He provides for them and gives them water. When distress strikes them at sea, all those they invoke vanish from them except Him. Yet when He delivers them safely to land, they turn away; man is ever ungrateful. Those who swear an oath calling upon Allah and have Him fulfill it, however, are particular individuals.

Examples of petitioning Allah include saying:

“I ask You by the fact that all praise belongs to You. You are Allah, the Bestower, the Originator of the heavens and the earth, O Possessor of Majesty and Honor.”

“I ask You by the fact that You are Allah, the One, the Eternal Refuge, who neither begets nor is born, and to whom none is equal.”

“I ask You by every name that belongs to You, by which You have named Yourself, which You have revealed in Your Book, which You have taught to any of Your creation, or which You have kept to Yourself in the knowledge of the unseen that is with You.”

This is petitioning Allah Most High by His names and attributes, not swearing an oath calling upon Him. His acts are entailed by His names and attributes: His forgiveness and mercy are entailed by His names, the All-Forgiving and the Most Merciful, and His pardon by His name, the Pardoner. Hence, when “‘Ā’ishah said to the Prophet