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Unformatted text preview: Math 124A – October 28, 2010 « Viktor Grigoryan 10 Heat equation: interpretation of the solution Last time we considered the IVP for the heat equation on the whole line u t- ku xx = 0 (-∞ < x < ∞ , < t < ∞ ) , u ( x, 0) = φ ( x ) , (1) and derived the solution formula u ( x,t ) = Z ∞-∞ S ( x- y,t ) φ ( y ) dy, for t > , (2) where S ( x,t ) is the heat kernel, S ( x,t ) = 1 √ 4 πkt e- x 2 / 4 kt . (3) Substituting this expression into (2), we can rewrite the solution as u ( x,t ) = 1 √ 4 πkt Z ∞-∞ e- ( x- y ) 2 / 4 kt φ ( y ) dy, for t > . (4) Recall that to derive the solution formula we first considered the heat IVP with the following particular initial data Q ( x, 0) = H ( x ) = 1 , x > , , x < . (5) Then using dilation invariance of the Heaviside step function H ( x ), and the uniqueness of solutions to the heat IVP on the whole line, we deduced that Q depends only on the ratio x/ √ t , which lead to a reduction of the heat equation to an ODE. Solving the ODE and checking the initial condition (5), we arrived at the following explicit solution Q ( x,t ) = 1 2 + 1 √ π Z x/ √ 4 kt e- p 2 dp, for t > . (6) The heat kernel S ( x,t ) was then defined as the spatial derivative of this particular solution Q ( x,t ), i.e. S ( x,t ) = ∂Q ∂x ( x,t ) , (7) and hence it also solves the heat equation by the differentiation property. The key to understanding the solution formula (2) is to understand the behavior of the heat kernel S ( x,t ). To this end some technical machinery is needed, which we develop next. 10.1 Dirac delta function Notice that, due to the discontinuity in the initial data of Q , the derivative Q x ( x,t ), which we used in the definition of the function S in (7), is not defined in the traditional sense when t = 0. So how can one make sense of this derivative, and what is the initial data for S ( x,t )? It is not difficult to see that the problem is at the point x = 0. Indeed, using that Q ( x, 0) = H ( x ) is constant for any x 6 = 0, we will have S ( x, 0) = 0 for all x different from zero. However, H ( x ) has a jump discontinuity at x = 0, as is seen in Figure 1, and one can imagine that at this point the rate of growth of H is infinite. Then the “derivative” δ ( x ) = H ( x ) (8) is zero everywhere, except at x = 0, where it has a spike of zero width and infinite height. Refer to Figure 2 below for an intuitive sketch of the graph of δ . Of course, δ is not a function in the traditional sense, 1 1 x H Figure 1: The graph of the Heaviside step function. x Δ Figure 2: The sketch of the Dirac δ function. but is rather a generalized function , or distribution . Unlike regular functions, which are characterized by their finite values at every point in their domains, distributions are characterized by how they act on regular functions....
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