# 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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