How does fasting affect violent situations not under my control? I read an article yesterday about Pope Francis announcing a day of fasting on Sept. 7th for the situation in Syria. According to Catholic/Orthodox/Anglican doctrine, exactly how does my fasting affect violent situations in the middle East?
Our own wickedness is the cause of the wickedness around the world. Personal fasting is a sign of personal repentance that, it is hoped, will lead God to mitigate similar evils in the world.

Pope Francis said this:

“We will gather in prayer and in a spirit of penance, invoking God’s great gift of peace upon the beloved nation of Syria and upon each situation of conflict and violence around the world. Humanity needs to see these gestures of peace and to hear words of hope and peace.”

The full text of his September 1st Angelus Address can be found here.
In making this proclamataion, he joins thousands of others in explicitly seeing the fallen condition of the world not merely a result of some generic sin, but our sin.  In calling us to fast, he is calling us to repent of our part in making this world a fallen place.
During the American Civil War, Abraham Lincoln called us to a similar fast.  He asked the nation to &quot;fast and humble&quot; itself as an outward manifestation of our repentance.  The full text can be seen here.  Within that, he wrote a justification as follows:

And, insomuch as we know that, by His divine law, nations like individuals are subjected to punishments and chastisements in this world, may we not justly fear that the awful calamity of civil war, which now desolates the land, may be but a punishment, inflicted upon us, for our presumptuous sins, to the needful end of our national reformation as a whole People? We have been the recipients of the choicest bounties of Heaven. We have been preserved, these many years, in peace and prosperity. We have grown in numbers, wealth and power, as no other nation has ever grown. But we have forgotten God. We have forgotten the gracious hand which preserved us in peace, and multiplied and enriched and strengthened us; and we have vainly imagined, in the deceitfulness of our hearts, that all these blessings were produced by some superior wisdom and virtue of our own. Intoxicated with unbroken success, we have become too self-sufficient to feel the necessity of redeeming and preserving grace, too proud to pray to the God that made us!
It behooves us then, to humble ourselves before the offended Power, to confess our national sins, and to pray for clemency and forgiveness.

In a nutshell - God is punishing us by bringing these events to come to pass.  Maybe if we show our contrition for what we have done, God will relent.
The idea is both simple and has a long basis in history.
Indeed, Paul, in Romans acknowledges the link, in Romans 1:

28 Furthermore, just as they did not think it worthwhile to retain the knowledge of God, so God gave them over to a depraved mind, so that they do what ought not to be done. 29 They have become filled with every kind of wickedness, evil, greed and depravity. They are full of envy, murder, strife, deceit and malice.

The people of Nineveh, when confronted by Jonah (in Chapter 3), made almost the exact same leap:

6 When Jonah’s warning reached the king of Nineveh, he rose from his throne, took off his royal robes, covered himself with sackcloth and sat down in the dust. 7 This is the proclamation he issued in Nineveh:
“By the decree of the king and his nobles:
Do not let people or animals, herds or flocks, taste anything; do not let them eat or drink. 8 But let people and animals be covered with sackcloth. Let everyone call urgently on God. Let them give up their evil ways and their violence. 9 Who knows? God may yet relent and with compassion turn from his fierce anger so that we will not perish.”
10 When God saw what they did and how they turned from their evil ways, he relented and did not bring on them the destruction he had threatened.

The logic is simple:

Violence is a result of our wickedness
We are sorry for this
Perhaps if God sees our sorrow, he will relent, and change the situation.