Problem B:
a) (3 points) Verify that the function u(t,x) = sin(sin(t+x)) is a solution of the transport equation
u_{t}(t,x) = u_{x}(t,x).b) (7 points) The partial differential equation f _{t}+f f_{x} = f_{xx} is called Burgers equation
and describes waves. Verify that
Remarks: You can solve this problem without technology or, use technology if you like. Here is how you can verify with Mathematica that a function satisfies a partial differential equation. In the following example we verify that a function satisfies the heat equation. This function appears in the PDE handout. f[t_,x_]:=(1/Sqrt[t])*Exp[-x^2/(4t)]; Simplify[ D[f[t,x],t] == D[f[t,x],{x,2}]]If you should not have Mathematica installed, you can also plug in and differentiate functions into Wolfram Alpha. For example, D[(x/t)*Sqrt[1/t]*Exp[-x^2/(4 t)]/(1+ Sqrt[1/t] Exp[-x^2/(4t)]),x]computes the derivative of the above function with respect to x. Entering (x/t)*Sqrt[1/t]*Exp[-x^2/(4 t)]/(1+ Sqrt[1/t] Exp[-x^2/(4t)])plots the function. Note that Mathematica can be a bit picky if you enter things in a different way. f[t_,x_]:=(x/t)*Sqrt[1/t]*Exp[-x^2/(4 t)]/(1+ Sqrt[1/t] Exp[-x^2/(4t)]);should define the function under consideration. |