What is the expiation for a vow, for someone who vowed to memorize a specific part of the Qur'an within a set time and did not fulfil it?

You should thank Allah for your dedication to memorizing the Quran and exert your utmost effort in it, even if it's memorizing two or three verses daily. You are not obliged to vow to memorize a part of it within a limited time, as that might be difficult for you or cause boredom. Also, a vow does not bring anything about; rather, it is used to extract from the miserly. However, since you have made a vow and are unable to fulfill it, you must offer an expiation for an oath, which is: feeding ten poor people, or clothing them, or freeing a slave. If you are unable to do that, then fasting for three days.