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**Unformatted text preview: **Math 124A October 21, 2010 Viktor Grigoryan 8 Heat equation: properties We would like to solve the heat (diffusion) equation u t- ku xx = 0 , (1) and obtain a solution formula depending on the given initial data, similar to the case of the wave equa- tion. However the methods that we used to arrive at dAlamberts solution for the wave IVP do not yield much for the heat equation. To see this, recall that the heat equation is of parabolic type, and hence, it has only one family of characteristic lines. If we rewrite the equation in the form ku xx + = 0 , where the dots stand for the lower order terms, then you can see that the coefficients of the leading order terms are A = k, B = C = 0 . The slope of the characteristic lines are then dt dx = B 2 A = B 2 A = 0 . Consequently, the single family of characteristic lines will be given by t = c. These characteristic lines are not very helpful, since they are parallel to the x axis. Thus, one cannot trace points in the xt plane along the characteristics to the x axis, along which the initial data is defined. Notice that there is also no way to factor the heat equation into first order equations, either, so the methods used for the wave equation do not shed any light on the solutions of the heat equation. Instead, we will study the properties of the heat equation, and use the gained knowledge to devise a way of reducing the heat equation to an ODE, as we have done for every PDE we have solved so far. 8.1 The maximum principle The first properties that we need to make sure of, are the uniqueness and stability for the solution of the problem with certain auxiliary conditions. This would guarantee that the problem is wellpossed, and the chosen auxiliary conditions do not break the physicality of the problem. We begin by establishing the following property, that will be later used to prove uniqueness and stability. Maximum Principle. If u ( x,t ) satisfies the heat equation (1) in the rectangle R = { x l, t T } in space-time, then the maximum value of u ( x,t ) over the rectangle is assumed either initially ( t = 0), or on the lateral sides ( x = 0, or x = l ). Mathematically, the maximum principle asserts that the maximum of u ( x,t ) over the three sides must be equal to the maximum of the u ( x,t ) over the entire rectangle. If we denote the set of points com- prising the three sides by = { ( x,t ) R | t = 0 or x = 0 or x = l } , then the maximum principle can be written as max ( x,t ) { u ( x,t ) } = max ( x,t ) R { u ( x,t ) } . (2) If you think of the heat conduction phenomena in a thin rod, then the maximum principle makes physical sense, since the initial temperature, as well as the temperature at the endpoints will dissipate through conduction of heat, and at no point the temperature can rise above the highest initial or end- point temperature. In fact, a stronger version of the maximum principle holds, which asserts that the maximum over the rectangle...

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