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Math Help - Turning a P.D.E to an O.D.E

  1. #1
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    Turning a P.D.E to an O.D.E

    I have a question, if we define a (for example) parabolic P.D.E in a spatiotemporal surface (1D in space), and also its initial and boundary vlues, would it be possible to solve the equation not in the entire domain, but in a space-time path.

    u-xx+c*u-t=0
    u(x=0)=0, u(x=l)=d, u(t=0)=0

    we limit the domain in
    a*x+b*t=0

    u-xx+c*u-(-a/b*x)=0
    boundary condition??

    therefore we can eliminate t from the main equation, and simply deal with the O.D.E. and with changing the a and b and also interpolation, find the a good approximation of the exact answer of the main P.D.E in the entire domain.
    I do not know is this approach is correct or not, but it if is, the main problem would be modeling the boundary and initial condition in a the path-domain.
    We project the answer in a 2-D somain (space and time) on a 1-D path.
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  2. #2
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    Quote Originally Posted by farshad View Post
    I have a question, if we define a (for example) parabolic P.D.E in a spatiotemporal surface (1D in space), and also its initial and boundary vlues, would it be possible to solve the equation not in the entire domain, but in a space-time path.

    u-xx+c*u-t=0
    u(x=0)=0, u(x=l)=d, u(t=0)=0

    we limit the domain in
    a*x+b*t=0

    u-xx+c*u-(-a/b*x)=0
    boundary condition??

    therefore we can eliminate t from the main equation, and simply deal with the O.D.E. and with changing the a and b and also interpolation, find the a good approximation of the exact answer of the main P.D.E in the entire domain.
    I do not know is this approach is correct or not, but it if is, the main problem would be modeling the boundary and initial condition in a the path-domain.
    We project the answer in a 2-D somain (space and time) on a 1-D path.
    I have no idea what's being said here, but if it was me solving the PDE I'd assume a seperable solution u(x, t) = X(x) T(t). Things re-arrange to:

    \frac{1}{X} \frac{d^2 X}{d x^2} = - \frac{c}{T} \frac{d T}{dt}

    and it's blue sky I would have thought.
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  3. #3
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    Quote Originally Posted by farshad View Post
    I have a question, if we define a (for example) parabolic P.D.E in a spatiotemporal surface (1D in space), and also its initial and boundary values, would it be possible to solve the equation not in the entire domain, but in a space-time path.

    u-xx+c*u-t=0
    u(x=0)=0, u(x=l)=d, u(t=0)=0

    we limit the domain in
    a*x+b*t=0

    u-xx+c*u-(-a/b*x)=0
    boundary condition??

    therefore we can eliminate t from the main equation, and simply deal with the O.D.E. and with changing the a and b and also interpolation, find the a good approximation of the exact answer of the main P.D.E in the entire domain.
    I do not know is this approach is correct or not, but it if is, the main problem would be modeling the boundary and initial condition in a the path-domain.
    We project the answer in a 2-D somain (space and time) on a 1-D path.
    Separation of variables is a well known and simple approach to solve th P.D.Es, and in finite spatial domains, it works for parabolic P.D.Es. My question was different from classic analytical solutions. You may have complicated boundary condition or domain, or a set of coupled diffrential equations. For example for semi infinite spatial domain, separation of variables does not work and we need Laplace transform. I was thinking about some semi-analytic solution for P.D.Es. you may consider the solution of the P.D.E not for the entire domain, but for 1D path, and in this particular domain, we would have an extra equation between two separate variables (x,t) which allows us to eliminate one. and turn the P.D.E into an ODE.
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