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Unformatted text preview: Math 124A – November 02, 2010 Viktor Grigoryan 11 Comparison of wave and heat equations In the last several lectures we solved the initial value problems associated with the wave and heat equa tions on the whole line x ∈ R . We would like to summarize the properties of the obtained solutions, and compare the propagation of waves to conduction of heat. Recall that the solution to the wave IVP on the whole line ( u tt c 2 u xx = 0 , u ( x, 0) = φ ( x ) , u t ( x, 0) = ψ ( x ) , (1) is given by d’Alambert’s formula u ( x,t ) = 1 2 [ φ ( x + ct ) + φ ( x ct )] + 1 2 c Z x + ct x ct ψ ( s ) ds. (2) Most of the properties of this solution can be deduced from the solution formula, which can be un derstood fairly well, if one thinks in terms of the characteristic coordinates. This is how we arrived at the properties of finite speed of propagation, propagation of discontinuities of the data along the characteristics, and others. On the other hand, the solution to the heat IVP on the whole line u t ku xx = 0 , u ( x, 0) = φ ( x ) , (3) is given by the formula u ( x,t ) = 1 √ 4 πkt Z ∞∞ e ( x y ) 2 / 4 kt φ ( y ) dy. (4) We saw some of the properties of the solutions to the heat IVP, for example the smoothing property, in the case of the fundamental solution or the heat kernel S ( x,t ) = 1 √ 4 πkt e x 2 / 4 kt , (5) which had the Dirac delta function as its initial data. The solution u given by (4) can be written in terms of the heat kernel, and we use this to prove the properties for solutions to the general IVP (3). In terms of the heat kernel the solution is given by u ( x,t ) = Z ∞∞ S ( x y,t ) φ ( y ) dy = Z ∞∞ S ( z,t ) φ ( x z ) dz, where we made the change of variables z = x y to arrive at the last integral. Making a further change of variables p = z/ √ kt , the above can be written as u ( x,t ) = 1 √ 4 π Z ∞∞ e p 2 / 4 φ ( x p √ kt ) dp. (6) This last form of the solution will be handy when proving the smoothing property of the heat equation, the precise statement of which is contained in the following. Theorem 11.1. Let φ ( x ) be a bounded continuous function for∞ < x < ∞ . Then (4) defines an infinitely differentiable function u ( x,t ) for all x ∈ R and t > , which satisfies the heat equation, and lim t → 0+ u ( x,t ) = φ ( x ) , ∀ x ∈ R . 1 The proof is rather straightforward, and amounts to pushing the derivatives of u ( x,t ) onto the heat kernel inside the integral. All one needs to guarantee for this procedure to go through is the uniform convergence of the resulting improper integrals. Let us first take a look at the solution itself given by (4). Notice that using the form in (6), we have  u ( x,t )  ≤ 1 √ 4 π Z ∞∞ e p 2 / 4 φ ( x p √ kt ) dp ≤ 1 √ 4 π (max  φ  ) Z ∞∞ e p 2 / 4 dp = max  φ  , which shows that u , given by the improper integral, is welldefined, since φ is bounded. One can also see the maximum principle in the above inequality. We will use similar logic to show that the impropersee the maximum principle in the above inequality....
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 Fall '08
 Ponce
 Differential Equations, Equations, Derivative, Partial Differential Equations, Partial differential equation, Riemann integral, Dirac delta function

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