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Thread: Partial Differential Equation -- method of characteristics (ode solve)

  1. #1
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    Partial Differential Equation -- method of characteristics (ode solve)

    I've been asked to solve

    $\displaystyle u_t + cos(\pi x) u_x = 0$ using the method of characteristics. This amounts to solving the following odes:

    $\displaystyle dt/ds = 1, t(0) = 0$ which implies $\displaystyle t = s$ and

    $\displaystyle dx/ds = cos(\pi x), x(0) = x_0$.

    This ode yields (if I'm right):

    $\displaystyle |sec(\pi x) + tan(\pi x)| = Ce^{\pi s}$.

    However, I cannot solve this explicitly for x, and therefore cannot solve the pde by the method of characteristics? Can anyone shed insight? Have I done something wrong? Is there a missing trig identity somewhere?

    Thanks
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Partial Differential Equation -- method of characteristics (ode solve)

    Quote Originally Posted by davismj View Post
    I've been asked to solve

    $\displaystyle u_t + cos(\pi x) u_x = 0$ using the method of characteristics. This amounts to solving the following odes:

    $\displaystyle dt/ds = 1, t(0) = 0$ which implies $\displaystyle t = s$ and

    $\displaystyle dx/ds = cos(\pi x), x(0) = x_0$.

    This ode yields (if I'm right):

    $\displaystyle |sec(\pi x) + tan(\pi x)| = Ce^{\pi s}$.

    However, I cannot solve this explicitly for x, and therefore cannot solve the pde by the method of characteristics? Can anyone shed insight? Have I done something wrong? Is there a missing trig identity somewhere?

    Thanks
    Your PDE...

    $\displaystyle u_{t}+ \cos (\pi x)\ u_{x}=0$

    ... is 'nonhomogeneous' and equivalent to then system...

    $\displaystyle d t=\frac{d x}{\cos \pi x}$

    $\displaystyle d u=0$ (2)

    ...the solution of which is...

    $\displaystyle c_{1}= v(x,t)= t-\frac{1}{\pi}\ \ln\ |\tan (\frac{\pi}{2} x +\frac{\pi}{4})|$

    $\displaystyle c_{2}= u$ (3)

    ... so that the solution of (1) is...

    $\displaystyle u= \gamma\{v(x,t)\}$ (4)

    ... where $\displaystyle \gamma(*)$ is any continous function with continous derivative...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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