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</math>
 
</math>
  
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If we assume that the extreme are at zero temperature, the system of equations for u is given by
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<math>
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\left\{ \begin{array}{l}
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u_t-u_{xx}=0, \qquad x\in(0,L), \qquad t>0, \\
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u(0,t)=0, \qquad t>0, \\
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u(L,t)=0, \qquad t>0, \\
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u(x,0)=u^0(x), \qquad x\in(0,L).
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\end{array} \right.
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</math>
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where <math>u_0(x)</math> is a function that describes the initial temperature of the bar.
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 +
We describe below the Fourier method to approximate the solutions of this system.
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 +
The main point is to observe that, if <math>\varphi(x)</math> is solution of the eigenvalue problem
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<math>
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\left\{ \begin{array}{l}
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\varphi''(x)+\lambda \varphi(x)=0, \\
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\varphi(0)=0, \quad \varphi(L)=0, \end{array} \right.
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</math>
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for some <math>\lambda</math>, then
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<math>
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u(x,t)=\varphi(x) e^{-\lambda t}
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</math>
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is a solution of the heat equation <math>u_t-u_{xx}=0</math> and the boundary conditions <math>u(0,t)=u(L,t)=0.</math>
  
  

Revisión del 10:15 26 abr 2016

Let [math]{\bf u(x_1,x_2,x_3,t)}[/math] be the temperature at the point [math](x_1,x_2,x_3)\in [/math] bar, and time [math]t\gt0[/math].

  • Bar.jpg

Assume that the cross section is so small that we can consider the bar as an unidimensional object in the interval [math]x\in [0,L][/math], [math] u(x_1,x_2,x_3,t)=u(x_1,t)=u(x,t). [/math]

If we assume that the extreme are at zero temperature, the system of equations for u is given by [math] \left\{ \begin{array}{l} u_t-u_{xx}=0, \qquad x\in(0,L), \qquad t\gt0, \\ u(0,t)=0, \qquad t\gt0, \\ u(L,t)=0, \qquad t\gt0, \\ u(x,0)=u^0(x), \qquad x\in(0,L). \end{array} \right. [/math] where [math]u_0(x)[/math] is a function that describes the initial temperature of the bar.

We describe below the Fourier method to approximate the solutions of this system.

The main point is to observe that, if [math]\varphi(x)[/math] is solution of the eigenvalue problem [math] \left\{ \begin{array}{l} \varphi''(x)+\lambda \varphi(x)=0, \\ \varphi(0)=0, \quad \varphi(L)=0, \end{array} \right. [/math] for some [math]\lambda[/math], then [math] u(x,t)=\varphi(x) e^{-\lambda t} [/math] is a solution of the heat equation [math]u_t-u_{xx}=0[/math] and the boundary conditions [math]u(0,t)=u(L,t)=0.[/math]

Obtenido de «https://mat.caminos.upm.es/w/index.php?title=Approximation_of_the_heat_equation:_Fourier_method&oldid=34254»