Introduction to Partial Differential Equations
Exercise Sheet 2
Ross Ogilvie 9th September,
2024
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We consider now the inhomogeneous transport equation
with initial value given by a function , namely . It had an explicit solution
on the domain , with initial condition . Note, the function will not be constant along the characteristic, but its value along the characteristic will be determined by its initial value. (6 points)
Duhamel’s principle is a method to find a solution to an inhomogeneous PDE if one can solve the homogeneous PDE for any initial condition. In this exercise we give the general idea and show how it applies to the transport equation. Consider an inhomogeneous PDE on of the following form
where is a linear differential operator on with constant coefficients. The idea is to instead consider the following family of homogeneous equations
Suppose that we can find such solutions . Prove that
is a solution to the inhomogeneous problem. (Do not worry about convergence problems.)
Use this method to solve the inhomogeneous transport. (2 + 4 points)