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Math Help - Heat equation

  1. #1
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    Heat equation

    Hello,

    it is
    \frac{\partial u}{\partial t} = k*\frac{\partial^2 u}{\partial x^2} (as known as heat equation)

    Show that

    \frac{\partial^2 u}{\partial x^2} \approx \frac{u(t,x_{i+1})-2u(t,x_i) + u(t,x_{i-1})}{h^2}

    Anyone knows how to show this?

    Best Regards,
    Rapha
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Rapha View Post
    Show that

    \frac{\partial^2 u}{\partial x^2} \approx \frac{u(t,x_{i+1})-2u(t,x_i) + u(t,x_{i-1})}{h^2}

    Anyone knows how to show this?
    For any twice differentiable function f, you have the asymptotic expansion: f(x+h)=f(x)+f'(x)h+\frac{f''(x)}{2}h^2+o(h^2), and (substituting -h for h): f(x-h)=f(x)-f'(x)h+\frac{f''(x)}{2}h^2+o(h^2). Sum these expansions, and you get:
    f(x+h)+f(x-h)=2f(x)+f''(x)h^2+o(h^2),
    and finally:
    f''(x)=\lim_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}.

    This should answer your question.
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  3. #3
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    Quote Originally Posted by Rapha View Post
    Hello,

    it is
    \frac{\partial u}{\partial t} = k*\frac{\partial^2 u}{\partial x^2} (as known as heat equation)

    Show that

    \frac{\partial^2 u}{\partial x^2} \approx \frac{u(t,x_{i+1})-2u(t,x_i) + u(t,x_{i-1})}{h^2}

    Anyone knows how to show this?

    Best Regards,
    Rapha
    This is a forward difference approximation.

    Read this: Solution of the Diffusion Equation by Finite Differences
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  4. #4
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    Hi

    thank you Laurent,

    thank you mr fantastic
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