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MATHS 361:  Partial Differential Equation

Tutorial 7:  End of Laplace, start of Fourier

1. Use Laplace transforms to solve the partial differential equation:

2ut + ux = u,

u(x,0) = ex ,u(0,t) = t.

(This question is repeated from last weeks tutorial, as not all students would have had the Laplace tools necessary at the time of their tutorial)

2. What if we instead had 2ut − ux = 0? Would this equation work, or would it cause problems? Why?

3. Use the definition of Fourier transforms to show that all even functions have a purely real Fourier transform.

4. Consider the‘sgn’function.  sgn(t) = 1 if t > 0 and sgn(t) = −1 if t < 0.  Can you calculate the Fourier transform of sgn(t) directly? What limit could you take in order to calculate F(sgn(t))?

5. Use Fourier transforms to solve the partial differential equation:

utt = 4uxx ,u(x,0) = e−|x|,ut(x, 0) = 0

You may assume u → 0 at infinity for all t