How to solve fixed-fixed beam with finite difference method?

What equations to use on this system to form a matrix $A$ with dimensions $[n,n]$ and load vector $q$ with dimension $[n]$ ? I am trying to get vertical displacement $w$.

$$w = A^{-1}\times q$$

Boundary conditions are as follows: $$w(o) = 0$$ $$w(L) = 0$$ $$\phi(o) = 0$$ $$\phi(L) = 0$$ It is becouse in any point of beam I can't make equation: $$d^2y/dx^2*E*I=M=0$$ so I can't get the exact values of displacement. The problem is that everywhere I look for solution it is done on a beam with continous load over entire beam or with at least one joint and I have only half of the beam covered with continous load and no joints.

From $d^4y/dx^4*E*I=q=0$ again I have too many unknown values.

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Hmm... Hi Franko. Welcome to Physics.SE. Could you phrase your question (just add it in quotes) here? –  Waffle's Crazy Peanut Feb 6 '13 at 13:39
What equations to use on this system to form a matrix A dimensions [n,n] and load vector q [n] ? I am trying to get vertical displacent w. w = A^(-1)*q –  Franko Feb 6 '13 at 13:47
Hi again Franko. Now, we've got a problem. This site deals with conceptual Physics Q&A. We don't encourage homework questions that doesn't involve any sort of work done by the author (which is you) and asks other users to solve the problem. If you think you could clarify your question, add what you've done along with your question. We're ready to help you. If you aren't clear, Please have a look at our homework policy for more info. After improving the post, flag it for moderator attention. –  Waffle's Crazy Peanut Feb 6 '13 at 13:54