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

  1. Jun 30, 2012 #1
    1. The problem statement, all variables and given/known data

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

    However, if [itex]m_i \neq m_j[/itex] for some [itex]i[/itex] and [itex]j[/itex], 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

    3. The attempt at a solution

    Say [itex]N = 2[/itex], [itex]m_1 = 2[/itex], and [itex]m_2 = 1[/itex]. We can write this system as

    [itex]
    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)}.
    [/itex]

    Say we make the change of variables

    [itex]
    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} = ???????
    [/itex]

    As far as I can see, [itex]u_{2, 4, x}[/itex] is not defined because it involves the fifth derivative of [itex]x_1[/itex]. 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.
     
  2. jcsd
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