Answer1::

The Solution as: Maplefile

In this homework the frist qustion is to solve the Laplacian in 2-d by sepration
of variables with given boundary data(conditions) which is absloutaly constrained 
on my solution, here are 3-plots which shows the solutions itself,the plot for
finding the eigenvalues numericaly, the third is the contoure plot for the solution..

Solving laplac's equation:
[U_xx+U_yy=0
U_x(0,y)=0
U_y(x,0)=0
U_y(x,1)+U(x,1)=0
U(1,y)=g(y)]
and g(y)=10,12

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   This plot represents the solution and look at the boundary of the plot
  it sholud match the given boundary condations , there's more plots showing this
  case will see it later.

   This plot represents the zeros for given function -xsin(x)+cos(x) where 
  x- represent the eigenvalues.

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  In this plot is trying to find the boundary function g(y) to get solution in the 
  interval[0,1/2}^2 approximatly =10.

  for Q2:
  The Solution of the Convection diffusion problem:
  -e(U_xx+U_yy)+U_y=0
  U(0,y)=U_y(x,1)=U(1,y)=0
  U(x,0)=g(x)
  ]
  

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  for last:Damped eq
  [U_tt+2aU_t-U_xx=0.
  U(0,t)=U(Pi,t)=0
  U(x,0)=f(x),U_t(x,0)=0
  a=1/4]
  

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  This 3D solution via seperation of variables and given function f(x)=x*(x-Pi)*(x-1)
  

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  This 3D solution via seperation of variables and given function f(x)=Pi/2*|x-Pi/2|
  

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  and down is the Numerical approximation of the solution over  f(x):
  

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