Does anyone know how to validate a quad which you have designed it from the scratch.(Starting from all the math equations and writing the algorithms)
If yes can someone please help me?
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I'm not aware of any real-time software approach to this matter.
However, a multi-rotor can be suspended in the air (see the Youtube video here: http://diydrones.com/forum/topics/arducopter-tuning-guide?page=9&am...) - or mounted on a camera tripod with a loose swivel, to make tests and measurements.
I've seen videos of test-halls (U.S. and German universities), where they have a sort of cage with soft nettings around. I bet that keeps the cost down for a lot of crashes.
If you'd like to physically measure the static thrust of a given motor/propeller/voltage arrangement, using an L-shaped device (a torsion balance) and a kitchen scale can take you a long way.
I fear that I don't understand your question quite right; you might have to elaborate a bit.
If you mean to calculate if a creation will take off/hover/fly from its weight, (number of) motors, propellers, ESCs, and battery, the answer is eCalc.
It lets you fill in your values from scratch, or select among already known equipment. I love it to death, but I don't know (all) the equations and calculations behind it.
There's even a nice help page, which you should probably start reading.