Is it useful to convert a higher order PDE into a 1st order system?

Question

I just learned how a higher order PDE can be converted into a system of PDEs.I am just wondering whether this is a standard way to solve some higher order PDEs which are immune to other methods or is it more efficient?

Answer 1

In addition to @GiuseppeNegro's answer, it has several advantages to convert a high-order PDE into a first-order system, e.g. a system of the form $\boldsymbol{u}_t + \boldsymbol{f}(\boldsymbol{u})_x = \boldsymbol{r}(\boldsymbol{u})$.

1. For some particular linear hyperbolic systems, we may be able to decouple the equations, so that the method of characteristics can be used to solve initial- and boundary-value problems.
2. For some particular hyperbolic systems, we may be able to derive an analytical solution to the Riemann problem (a particular initial-value problem).
3. We may be able to exploit theoretical results available for this kind of system (existence, uniqueness, smoothness).
4. We can make use of well-suited numerical methods, such as finite-volume methods of Godunov type.

However, not every higher-order PDE can be written in the previous form.

Answer 2

It's not clear to me how this would work for PDE's. For ODE's:

Yes, any higher-order ODE is equivalent to a system of first-order ODE's. For example $$y''=f(y,y',t)$$is equivalent to the system $$y_1'=y_2, y_2'=f(y_1,y_2,t).$$I don't know that this is important for solving ODE's, but it's certainly useful in proving things about them. Because that system of two ODE"s is just $$Y'=F(Y,t)$$for a suitable $F$, where $Y=(y_1,y_2)$. Proofs of things for ODE's often work just as well for "vector-valued" ODE's, which then have corollaries for higher-order (scalar-valued) ODE's. Hmm, I suppose the same holds for numerical methods for approximate solutions.

Answer 3

This is connected to the Hamiltonian formalism, and is also done in PDEs sometimes. I am far from being an expert, so let me just present the case of the wave equation $u_{tt} = \Delta u$. Introduce the two-component vector (position-momentum): $$ \boldsymbol{u}:=\begin{bmatrix} u \\ u_t\end{bmatrix}.$$ Then the wave equation is rewritten as $$ \partial_t \boldsymbol{u} = J H \boldsymbol{u},\qquad J=\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \qquad H=\begin{bmatrix} -\Delta & 0 \\ 0 & 1 \end{bmatrix}.$$

Is this useful? Well, to begin with, it lets the abstract theory of semigroups come into play. (That theory is the generalization to the infinite-dimensional case of the equation $\dot{x}= A x$, where $A$ is replaced with a linear operator on some Banach space). It also reveals a conservation law; if $\boldsymbol{u}$ solves the wave equation then $$ \begin{split} \partial_t \langle H\boldsymbol{u}(t), \boldsymbol{u}(t)\rangle &= \langle HJH\boldsymbol{u}| \boldsymbol{u}\rangle + \langle H\boldsymbol{u}|JH\boldsymbol{u}\rangle \\ &= -\langle H\boldsymbol{u}|JH\boldsymbol{u}\rangle + \langle H\boldsymbol{u}|JH\boldsymbol{u}\rangle =0, \end{split} $$ where $$ \langle (f_1, g_1) |(f_2, g_2)\rangle:= \int f_1f_2 + \int g_1g_2,$$ and where we used the fact that $H$ is a symmetric operator, while $J$ is an anti-symmetric one. We have thus shown that $$ \langle H \boldsymbol{u}|\boldsymbol{u}\rangle=\int -\Delta u\, u + \int u_t^2 $$ is constant along the wave flow, which is a fundamental fact known as conservation of energy. (The first summand equals $\int |\nabla u|^2$, which is how it usually appears).

This is all very basic, but already shows that, sometimes, this kind of manipulation can be useful.