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Unformatted text preview: nd let φ : ∂D → R be a continuous function. Find a harmonic function u : D → R such that for (x, y ) ∈ ∂D. u(x, y ) = φ(x, y ), Our physical intuition suggests that the Dirichlet problem will always have a unique solution. This is proven mathematically for many choices of boundary in more advanced texts on complex variables and partial differential equations. The mathematical proof that a unique solution exists provides evidence that the mathematical model we have constructed for heat flow may in fact be valid. Our goal here is to find the explicit solutions in the case where the region D is sufficiently simple. Suppose, for example, that D = {(x, y ) ∈ R 2 : 0 ≤ x ≤ a, 0 ≤ y ≤ b}. 119 Suppose, moreover that the function φ : ∂D → R vanishes on three sides of ∂D, so that φ(0, y ) = φ(a, y ) = φ(x, 0) = 0, while φ(x, b) = f (x), where f (x) is a given continuous function which vanishes when x = 0 and x = a. In this case, we seek a function u(x, y ), defined for 0 ≤ x ≤ a and 0 ≤ y ≤ b, such that 1. u(x, y ) satisfies L...
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