4. Strong and Weak Solutions¶
4.1. Conservative and Non-conservative Form of the Burgers’ Equation¶
4.1.1. Non-Conservative Form¶
\[{\partial u \over \partial t} + u {\partial u \over \partial x} = 0\]
4.1.2. Conservative Form¶
\[{\partial u \over \partial t} + {\partial F \over \partial x} = 0\]
where the Flux is \(F\) is (for Burgers’ Equation):
\[F = {u^2 \over 2}\]
May also write:
\[{\partial u \over \partial t} + A {\partial u \over \partial x} = 0\]
where the Jacobian \(A\) is (for Burgers’ Equation):
\[A = A(u) = u\]
For more than 1 dimension, A is a matrix
\[A = {\partial F_i \over \partial u_j}\]
4.2. Hyperbolic¶
The Equation is hyperbolic \(\Rightarrow\) all of the eigenvalues of A are real.
4.3. Strong Solution¶
u is continuous, but bounded discontinuties in derivatives may occur
4.4. Weak Solution¶
u is continuous almost everywhere, i.e. it has a jump (e.g. a shock wave)