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Partial differential equations

  1. Mar 15, 2009 #1


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    I'm not so well-versed in the topic of partial differential equations, but the following question has arisen.

    Suppose that for some unknown function u(s, t) of two variables, we have a set of differential equations
    [tex]\left\{ \begin{matrix} u_s(s, t) & {} = f(s, t, u(s, t)) \\ u_t(s, t) & {} = g(s, t, u(s, t)) \end{matrix} \right. [/tex]
    where us denotes the partial derivative of u(s, t) w.r.t. s.

    My question is, what the conditions on f would have to be in order to have a good solution for u(s, t). For example, we can integrate the first one to get u(s, T) for fixed t = T, and similarly the second one will give u(S, t) for fixed s = S, but of course u(S, T) must be well-defined (i.e. single-valued).

    I am particularly interested in the case where f and g do not depend on s and t explicitly (i.e. only through u(s, t)) and the case where they do not depend on u(s, t) explicitly.

    Thanks for sharing your thoughts.
  2. jcsd
  3. Apr 2, 2009 #2
    For a solution to exist you certainly want that your two PDEs are compatible, i.e.

    [tex]u_{ts} = u_{st}[/tex] which gives [tex]f_t + g f_u = g_s + g_u f[/tex]

    The two cases your interested in

    (i) f and g independent of u [text]f_t = g_s[/tex]
    (ii) f and g independent of t and s so [tex]g f_u = f g_u[/tex] so [tex]f = c \;g[/tex] for some constant c.
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