Mathematical Methods in the Physical Sciences · Problem 5.9

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · cube-temperature-laplace

Problem

Find the steady-state distribution of temperature in a cube of side 10 if the temperature is 100∘100^\circ on the face z=0z = 0 and 0∘0^\circ on the other five faces. HintHint: Separate Laplace's equation in three dimensions in rectangular coordinates, and follow the methods of Section 2. You will want to expand 100 in the double Fourier series ∑n=1∞∑m=1∞anmsin⁡nπxlsin⁡mπyl.\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} a_{nm}\sin\frac{n\pi x}{l}\sin\frac{m\pi y}{l}. The coefficients anma_{nm} are determined by making use of the orthogonality of the functions sin⁡(nπx/l)sin⁡(mπy/l)\sin(n\pi x/l)\sin(m\pi y/l) over the square, that is, ∫0l∫0lsin⁡nπxlsin⁡mπylsin⁡pπxlsin⁡qπyl dx dy=0unless{n=p,m=q.\int_0^l\int_0^l \sin\frac{n\pi x}{l}\sin\frac{m\pi y}{l}\sin\frac{p\pi x}{l}\sin\frac{q\pi y}{l}\,dx\,dy = 0 \qquad \text{unless} \begin{cases} n = p, \\ m = q. \end{cases}

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