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PDE solutions using Fourier Transforms

See attached PDF for the compiled LaTex.
I would prefer any solution include the Latex Source, however, if you are unable to use LaTex, you may use another format.

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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This solution is comprised of a detailed explanation to answer PDE solutions using Fourier Transforms.

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