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Math Help - Linear 1st Order PDE Query

  1. #1
    Senior Member bugatti79's Avatar
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    Linear 1st Order PDE Query

    Folks,

    Looking at past exam papers. There is one section as follows which I have no idea on.

    Prove that a characteristic curve is parallel to a solution surface and hence explain why a characteristic curve that intersects any intial data must be a solution curve.



    Thanks
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  2. #2
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    Basically a characteristic is a curve along which the directional derivative is known and given by the coefficients. To get these we solve an ODE, and if we are given initial data (Cauchy problem) we have initial conditions to impose, so everything is determined. On the other hand if you want the general solution this initial conditions are missing so you get a family of curves all satisfying the same differential equation, and if this curve intersects a solution surface, by fixing this point and using the existence and uniqueness theorem for ODE, we get that there is an 'interval' around this point in the curve which is contained in the surface; we can then conclude that the whole characteristic is contained in the surface as long as both exist.
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  3. #3
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Jose27 View Post
    Basically a characteristic is a curve along which the directional derivative is known and given by the coefficients. To get these we solve an ODE, and if we are given initial data (Cauchy problem) we have initial conditions to impose, so everything is determined. On the other hand if you want the general solution this initial conditions are missing so you get a family of curves all satisfying the same differential equation, and if this curve intersects a solution surface, by fixing this point and using the existence and uniqueness theorem for ODE, we get that there is an 'interval' around this point in the curve which is contained in the surface; we can then conclude that the whole characteristic is contained in the surface as long as both exist.
    Thanks for the description. I wonder was the question loooking for an algebraic method etc?

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  4. #4
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    Let's refrase the claim to something that may be more familiar (and hopefully that you have seen proved, or can prove): If we are given a Cauchy problem along a characteristic curve (ie. the transversality condition is violated) then this problem has either no solution or an infinite number of solutions.
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