Chem Differential Eq HW Solutions Fall 2011 48

Chem Differential Eq HW Solutions Fall 2011 48 - 48 Chapter...

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Unformatted text preview: 48 Chapter 3 Partial Differential Equations in Rectangular Coordinates Solutions to Exercises 3.7 5. We proceed as in Exercise 3. We have ∞ ∞ ∗ (Bmn cos λmn t + Bmn sin λmn t) sin mπx sin nπy, u(x, y, t) = n=1 m=1 √ where λmn = m2 + n2 , Bmn = 0, and ∗ Bmn = = 4 2 + n2 m 4 √ 2 + n2 m √ = 1 1 √ 0 sin mπx sin nπy dx dy 0 0 1 1 sin mπx dx 0 16 m2 +n2 (mn)π 2 sin nπy dy 0 if m and n are both odd, otherwise. Thus u(x, y, t) = ∞ ∞ k =0 l=0 16 sin((2k + 1)πx) sin((2l + 1)πy ) (2k + 1)2 + (2l + 1)2 (2k + 1)(2l + 1)π 2 sin (2k + 1)2 + (2l + 1)2t ...
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