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PDE solutions using Fourier Transforms

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Exercise 1

Solve, using Fourier Transforms frac{partial^{2}u}{partial x^{2}}+frac{partial^{2}u}{partial y^{2}}=0
for 0<y<H, -infty<x<infty subject to the initial/boundry
conditionsbegin{gathered}u(x,0)=0
frac{partial u}{partial y}(x,H)+hu(x,H)=f(x)end{gathered}

Exercise 2

Solve, using Fourier Transforms frac{partial^{2}u}{partial x^{2}}+frac{partial^{2}u}{partial y^{2}}=0
for x<0, -infty<y<infty subject to u(0,y)=g(y).

Exercise 3

Solvefrac{partial u}{partial t}+v_{0}cdotnabla u=knabla^{2}u
subject to the initial condition u(x,y,0)=f(x,y)Show how the
influence function is altered bt the convection term v_{0}cdotnabla u
.

Exercise 4

Solve, via Fourier Transforms:frac{partial u}{partial t}=k_{1}frac{partial^{2}u}{partial x^{2}}+k_{2}frac{partial^{2}u}{partial y^{2}}
with the initial condition u(x,y,0)=f(x,y)

Exercise 5

Solve, via Fourier Transformsfrac{partial u}{partial t}=kleft(frac{partial^{2}u}{partial x^{2}}+frac{partial^{2}u}{partial y^{2}}right)
with x>0 and y>0 and the initial condition u(x,y,0)=f(x,y) and
the bound conditionsu(0,y,t)=0qquadfrac{partial u}{partial y}(x,0,t)=0

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Green's Functions, Parabolic Equations and Heat Equation

I am having great difficulty understanding how you derive Green's functions, particularly how the boundary conditions are incorporated. I've also not studied Fourier series before and it appears that these are also used particularly in developing solutions for parabolic PDEs. My text does not have any specific worked examples (only the theory) and so I can't see how it all works.

The question I have is to find in 0<x'<a, t'>0 the Green's function G(x,t,x',t') satisfying P=v_xx + v_t and the additional conditions G_x'(x,t,a,t') = G(x,t,0,t') = 0.

I've also attached two files (one in Word format, one PDF) where the equations/formulae are in a more readable format, along with the actual answer to this problem.

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