Explicit Solutions of the Heat Equation

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1 LECTURE 6 Eplicit Solutions of the Heat Equation Recall the -dimensional homogeneous) Heat Equation: ) u t a 2 u. In this lecture our goal is to construct an eplicit solution to the Heat Equation ) on the real line, satisfying a given initial temperture distribution 2) u, ) f ) < < +, t +. We ll begin, however, by describing some special functional properties of solutions of ).. Four Ways of Generating Me Solutions Lemma 6.. Suppose u, t) is a solution of the homogeneous heat equation ). Then f any fied constant y, ϕ y, t) : u y, t) is also a solution of ). Proof. Indeed, suppose u, t) is a solution of ) and set ũ, t) u X ), t), where X ) y Then ũ u, t), t) t t ũ u X X X y u X u X y, t) Y y and similarly 2 ũ 2 u XX y, t) ũ t, t) k2 ũ, t) u t y, t) u XX y, t) because 6) holds at all points, y). Lemma 6.2. Suppose u, t) is a solution of ). Then any partial derivative of u, t) is also a solution. Proof. Let u, t) be a solution and set v, t) u t, t). Then v t k 2 v u tt k 2 u t ut k 2 ) u t t ). Similarly, if w, t) u, t), then w t k 2 w ut k 2 ) u ) 24

2 . FOUR WAYS OF GENERATING MORE SOLUTIONS 25 Lemma 6.3. Suppose u, t) is a solution of ) and suppose g ) is any differentiable function such that the integral converges. Then is also a solution of ). φ,t : u, t) g ) d u y, t) g y) dy Proof. We have v, t) k2 v, t) t 2 u t y, t) g y) dy k 2 u y, t) g y) dy ut y, t) k 2 u y, t) ) g y) dy g y) dy Lemma 6.4. Suppose u, t) is a solution of ) and let a be a real number. Then the dilated function v a, t) : u a, a 2 t ) f any a > is also a solution of ). Proof. We have t v, t) a2 u t a, a 2 t ) v, t) au a, a 2 t ) 2 2 v, t) a au a, a 2 t )) a2 u a, a 2 t ) v t, t) k 2 v, t) a 2 u t a, a 2 t ) k 2 a 2 u a, a 2 t ) a 2 u t a, a 2 t ) k 2 u a, a 2 t )) a 2 Upshot. Once we have a solution of ) we have at least four different ways of generating me solutions. In fact, our basic strategy f solving the Cauchy problem u t k 2 u 7a) u, ) f ) 7b) will be to find a basic solution of the heat equation 7a) and then use it and Theem 8.2 to construct a solution that satisfies the initital condition. u, ) f)

3 2. A BASIC SOLUTION OF THE HEAT EQUATION A basic solution of the heat equation Lemma 9.4 says that if u, t) is a solution then so is u a, a 2 t ) f any positive constant a. This property suggests looking first f a solution that doesn t change at all when a, t a 2 t. F eample, suppose u, t) g q), where q : t Then we d automatically have u, t) u a, a 2 t ). However, with an prescient eye towards a nmalization that will prove convenient later on, let s look instead f a solution of the fm 8) u, t) g p), where p Such a solution will the same property u, t) u a, a 2 t ), and yet, in the end, yield a tidier fmula. Thus, we suppose that ) u, t) g satisfies 6), and we ll try to find a choice of g that makes this assumption wk. We have u t dg p dp t 4k t) 3/2 g p) 2t g p) 2t pg t) u dg dp dp g p) u dg dp dp 4k 2 t g p), the requirement u t k 2 u on u, t) will lead to the following condition on g p) 2t pg p) k 2 4k 2 t g p) 2t 9) g + 2pg. pg p) ) 2 g p) Let G p) g p). Then G + 2pG dg dp 2pG dg G dp 2p Thus, the equation f G is separable. And its solution is given implicitly by G dg 2pdp + C ln G p 2 + C We now fmally integrate both sides of to get G e p2 +C c e p2 g p) G p) c e p2 ) g p) c e p2 + c 2

4 2. A BASIC SOLUTION OF THE HEAT EQUATION 27 as the general solution to 9). In conclusion, the function ) Q, t) c will be a solution of the Heat Equation. e p2 dp + c 2 At this point, we ll employ another bit of fesight and make an especially convenient choice f the constants c and c 2 ; namely, c π, c 2 2 Then Now Thus, lim Q, t) π t + lim L sgn)l π lim L e p2 dp 2) lim t + Q, t) 2k τ e p2 dp + 2 sgn)l e p2 dp + π 2 π 2 e p2 dp e p2 dp + 2 { if > if < { if > if < We ll now use Theem 8.2 to manipulate this basic solution further. The integral in ) is a nasty one; it has no closed epression in terms of elementary functions although it is a very imptant integral). Fact 8.3 says we can get another solution by differentiating ). So if we set S, t) Q, t) π S, t) will also be a solution of the heat equation. S, t) 4πk2 t e e 4k 2 t Lastly, we ll use Theem 8.2 ii), to produce yet another solution 3) u, t) I ll now show that That is, the solution 3) will satisfy S y, t) φ y) dy the general Cauchy problem f the heat equation. 4πk2 t lim u, t) φ ) t + u t k 2 u u, ) φ ) e y)2 / 4k 2t φ y) dy

5 2. A BASIC SOLUTION OF THE HEAT EQUATION 28 We have u, t) S y, t) φ y) dy Q y, t) φ y) dy Q y, t) φ y) dy y Q y) φ y) + + Q y, t) φ y) dy where we have used integration by parts. Assuming lim y ± φ y) otherwise the integral in 3) is unlikely to converge). we have At t we have from 2) Therefe, u, t) Q y, t) φ y) dy { if > lim Q, t) lim t + t + if < u, ) Q y, t) φ y) dy φ y) dy + φ y) dy φ y) φ ) φ y) dy