Approximation of the heat equation: Fourier method

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].

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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]