2.3. HEAT EQUATION 45 2.3.1. Fundamental solution. a. Derivation of the fundamental solution. As noted in §2.2.1 an important first step in studying any PDE is often to come up with some specific solutions. We observe that the heat equation involves one derivative with respect to the time variable t, but two derivatives with respect to the space vari- ables xi (i = 1,... , n). Consequently we see that if u solves (1), then so does u(λx, λ2t) for λ R. This scaling indicates the ratio r2 t (r = |x|) is important for the heat equation and suggests that we search for a solution of (1) having the form u(x, t) = v( r2 t ) = v( |x|2 t ) (t 0, x Rn), for some function v as yet undetermined. Although this approach eventually leads to what we want (see Problem 13), it is quicker to seek a solution u having the special structure (4) u(x, t) = 1 v x (x Rn, t 0), where the constants α, β and the function v : Rn R must be found. We come to (4) if we look for a solution u of the heat equation invariant under the dilation scaling u(x, t) λαu(λβx, λt). That is, we ask that u(x, t) = λαu(λβx, λt) for all λ 0, x Rn, t 0. Setting λ = t−1, we derive (4) for v(y) := u(y, 1). Let us insert (4) into (1) and thereafter compute (5) αt−(α+1)v(y) + βt−(α+1)y · Dv(y) + t−(α+2β)Δv(y) = 0 for y := t−βx. In order to transform (5) into an expression involving the variable y alone, we take β = 1 2 . Then the terms with t are identical, and so (5) reduces to (6) αv + 1 2 y · Dv + Δv = 0. We simplify further by guessing v to be radial that is, v(y) = w(|y|) for some w : R R. Thereupon (6) becomes αw + 1 2 rw + w + n 1 r w = 0,
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