Transforming a system of PDEs into a first order system of ODEs

[tylerc1991]

Homework Statement

Say we have a system of $N$ PDEs, each with even order. That is, say the $k^{th}$ equation has order $2 m_k$. If $m_i = m_j$ for all $i$ and $j$, then we can transform the system of PDEs into a first order system of ODEs by introducing new variables.

However, if $m_i \neq m_j$ for some $i$ and $j$, then I do not see how to transform the system of PDEs into a first order system of ODEs. Below is a simple example of where I am getting stuck

The Attempt at a Solution

Say $N = 2$, $m_1 = 2$, and $m_2 = 1$. We can write this system as

$x_{1, t} = - x_1^{(4)} + \sum_{j = 1}^2 \sum_{i = 0}^{3} a_{j, i}^1 x_1^{(i)}, \\ x_{2, t} = x_k^{(2)} + \sum_{j = 1}^2 \sum_{i = 0}^{1} a_{j, i}^2 x_2^{(i)}.$

Say we make the change of variables

$u_{1, 1, x} = u_{1, 2} \\ u_{1, 2, x} = u_{1, 3} \\ u_{1, 3, x} = u_{1, 4} \\ u_{1, 4, x} = -\lambda u_{1, 1} + \sum_{j = 1}^2 \sum_{i = 0}^{3} a_{j, i}^1 u_{j, i+1} \\ u_{2, 1, x} = u_{2, 2} \\ u_{2, 2, x} = \lambda u_{2, 1} - \sum_{j = 1}^2 \sum_{i = 0}^{1} a_{j, i}^2 u_{j, i+1} \\ u_{2, 3, x} = u_{2, 4} \\ u_{2, 4, x} = ??$

As far as I can see, $u_{2, 4, x}$ is not defined because it involves the fifth derivative of $x_1$. Does this mean that systems of PDEs can only be transformed into first order systems of ODEs if the orders of each of the equations agree? If anyone could help it would be greatly appreciated.

 

[mighty2000]

Thank you for your question. It is indeed true that systems of PDEs can only be transformed into first order systems of ODEs if the orders of each of the equations agree. This is because the transformation you have proposed relies on being able to express higher order derivatives in terms of lower order derivatives. When the orders of the equations do not agree, this is not possible.

In your example, the transformation you have proposed is not valid because it relies on being able to express the fourth derivative of x_1 in terms of lower order derivatives. However, this is not possible because the fourth derivative of x_1 does not appear in the original system of PDEs.

I hope this clarifies your doubts. If you have any further questions, please do not hesitate to ask.