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Numerical PDE's II - Circular Domain

  1. Mar 29, 2008 #1
    1. The problem statement, all variables and given/known data
    Approximate the solution of:
    [tex]

    \frac{\partial^{2}u }{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}} = 0, 1<r<3, 0<\theta<\pi,

    u(1,\theta) = 1-\cos(\theta),u(2,\theta)=u(r,0)=0 u(r,\pi)=4-2r

    [/tex]

    2. Relevant equations
    [tex]

    \delta r = \frac{r_{outer} - r_{inner}}{N}
    \delta \theta = \frac{\theta_{end} - \theta_{start}}{M}

    [/tex]
    n,m are positve integers

    [tex]
    \delta r_i = r_{inner} + i\delta r
    [/tex]
    [tex]
    \delta theta_j = \theta_{start} + j\delta \theta
    [/tex]
    3. The attempt at a solution
    [tex]
    \frac{u_{i+1,j} - 2U_{i,j} + U_{i-1,j}}{(\delta r)^2} + \frac{1}{r_i}\frac{u_{i+1,j}-u_{i-1,j}}{2\delta r} +\frac{u_{i,j+1} - 2U_{i,j} + U_{i,j-1}}{(\delta \theta)^2} = f_{i,j}
    [/tex]

    Multiply this equation by [tex] (\delta r)^2
    [/tex]
    [tex]
    u_{i+1,j} - 2u_{i,j} + u_{i-1,j} + \frac{1}{r_i}\frac{u_{i+1,j}-u_{i-1,j}}{2}\delta r + \frac{1}{r_i}\frac{\delta r}{\delta \theta}(u_{i+1,j}-2u{i,j}+u{i,j-1} = f_{i,j}(\delta r)^2

    [/tex]

    What do I do next>
     
    Last edited: Mar 29, 2008
  2. jcsd
  3. Mar 29, 2008 #2

    Integral

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    Staff Emeritus
    Science Advisor
    Gold Member

    And the question is?
     
  4. Mar 30, 2008 #3
    Nevermind, I got it.

    But does anyone know if such a problem with those given BC's can be applied elsewhere?
    When would you model something with a circular annulus?
     
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