James 5:15 says that the prayer of faith will save the sick. Is there a specific prayer of faith?
The verse is not referring to any specific prayer, but rather to the prayer of faith (ευχη της πιστεως) as opposed to prayer that is undertaken without faith. For this reason James emphasizes a verse later:
The entreaty of a righteous man hath much strength
He further provides Elijah as an example (v.17-18):
Elijah was a man subject to like passions as we are, and he prayed earnestly that it might not rain: and it rained not on the earth by the space of three years and six months. And he prayed again, and the heaven gave rain, and the earth brought forth her fruit.
The prayer of faith is not the "Prayer of Faith". This means, that there is not a specific prayer that a person can recite in order to heal someone else. God listens to His children. And He answers according to His will and to the measure of faith of the person asking. This is the prayer of faith that James is talking about.
A prayer is not something to be repeated without understanding, as Jesus said in His template of a prayer, now known as the Lord's Prayer; which oddly enough is repeated today throughout the world without understanding and just as vain repetition.
It can be found in Matthew 6:
7 And when you pray, do not use vain repetitions as the heathen do. For they think that they will be heard for their many words.
8 “Therefore do not be like them. For your Father knows the things you have need of before you ask Him. 9 In this manner, therefore, pray:
Our Father in heaven, Hallowed be Your name. 10 Your kingdom come. Your will be done On earth as it is in heaven. 11 Give us this day our daily bread. 12 And forgive us our debts, As we forgive our debtors. 13 And do not lead us into temptation, But deliver us from the evil one. For Yours is the kingdom and the power and the glory forever. Amen.[c]
Prayer is meant to bring you in communion with God. It is meant to grow your faith and relationship with the Father and Christ. It is not to let God know something, as He already knows.
Steps to Christ p. 93
Prayer is the opening of the heart to God as to a friend. Not that it is necessary in order to make known to God what we are, but in order to enable us to receive Him. Prayer does not bring God down to us, but brings us up to Him.
When Jesus was upon the earth, He taught His disciples how to pray. He directed them to present their daily needs before God, and to cast all their care upon Him. And the assurance He gave them that their petitions should be heard, is assurance also to us.
For further information of prayer, I encourage you to read this small e-book: