Consider the coupled transport (i.e. advection) system

$$ \begin{align} \partial_t u + b\partial_x \phi &= 0,\\ \partial_t \phi + b\partial_x u &= 0, \end{align} $$

where $u(x,t),\phi(x,t) \in C^2\left([0,1]\times \mathbb{R}^+\right)$ are unknown, and $b\in \mathbb{R}$ is given and constant.

By combining the equations, one can recover two classical, decoupled, wave equations:

$$\partial_{tt} u - b^2 \partial_{xx} u = 0$$

and similar for $\phi$. The latter accepts one initial condition for the unknown $u/\phi$, and another one for its time derivative $\partial_t u / \partial_t \phi$. The reason for this property - to my understanding - is the structure of d'Alembert's solution, see e.g. https://mathworld.wolfram.com/dAlembertsSolution.html for a derivation.

How can I use this knowledge to find the "correct" initial conditions for the coupled transport system? How many conditions are required and which?

Intuitively, I would think that I need one initial condition for $u$ and one for $\phi$, as the time derivatives are first order. Is this correct?

The coupled transport system and the two wave equations are analytically equivalent, as far as I understand. Can I recover the very initial conditions I have to choose for the coupled transport system, such that the problems are equivalent (based on the chosen initial conditions for the wave equation)?


1 Answer 1


You are correct: for the system of first order PDEs \begin{align} \partial_t u + b\partial_x \phi &= 0, \tag{1a} \\ \partial_t \phi + b\partial_x u &= 0, \tag{1b} \end{align} you need one initial condition for $u$ and one for $\phi$, specifically \begin{align} u(x,0) &= u_0(x), \tag{2a} \\ \phi(x,0) &= \phi_0(x). \tag{2b} \end{align} On the other hand, for the second order PDE $$ \partial_{tt} u - b^2 \partial_{xx} u = 0, \tag{3} $$ you need two initial conditions, $u(x,0)$ and $\partial_t u(x,0)$. The first is provided by $(2\text{a})$, and the second can be derived from $(1\text{a})$ and $(2\text{b})$: $$ \partial_t u(x,0)=-b\partial_x\phi(x,0)=-b\phi_0'(x). \tag{4} $$

  • 1
    $\begingroup$ Thank you, very simple! $\endgroup$ Nov 7, 2023 at 16:15

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