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**Unformatted text preview: **Math 124A October 26, 2010 Viktor Grigoryan 9 Heat equation: solution Equipped with the uniqueness property for the solutions of the heat equation with appropriate auxiliary conditions, we will next present a way of deriving the solution to the heat equation u t- ku xx = 0 . (1) Considering the equation on the entire real line x R simplifies the problem by eliminating the effect of the boundaries, we will first concentrate on this case, which corresponds to the dynamics of the temperature in a rod of infinite length. We want to solve the IVP { u t- ku xx = 0 (- < x < , < t < ) , u ( x, 0) = ( x ) . (2) Since the solution to the above IVP is not easy to derive directly, unlike the case of the wave IVP, we will first derive a particular solution for a special simple initial data, and try to produce solutions satisfying all other initial conditions by exploiting the invariance properties of the heat equation. 9.1 Invariance properties of the heat equation The heat equation (1) is invariant under the following transformations (a) Spatial translations: If u ( x, t ) is a solution of (1), then so is the function u ( x- y, t ) for any fixed y . (b) Differentiation: If u is a solution of (1), then so are u x , u t , u xx and so on. (c) Linear combinations: If u 1 , u 2 , . . . , u n are solutions of (1), then so is u = c 1 u 1 + c 2 u 2 + . . . c n u n for any constants c 1 , c 2 , . . . , c n . (d) Integration: If S ( x, t ) is a solution of (1), then so is the integral v ( x, t ) = - S ( x- y, t ) g ( y ) dy for any function g ( y ), as long as the improper integral converges (we will ignore the issue of the convergence for the time being). (e) Dilation (scaling): If u ( x, t ) is a solution of (1), then so is the dilated function v ( x, t ) = u ( ax, at ) for any constant a > 0 (compare this to the scaling property of the wave equation, which is invariant under the dilation u ( x, t ) 7 u ( ax, at ) for all a R )....

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