Lecture 8 - 8 Boundary Value Problems for PDEs Before we...

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8 Boundary Value Problems for PDEs Before we specialize to boundary value problems for PDEs — which only make sense for elliptic equations — we need to explain the terminology “elliptic”. 8.1 Classification of Partial Differential Equations We therefore consider general second-order partial differential equations (PDEs) of the form Lu = au tt + bu xt + cu xx + f = 0 , (75) where u is an unknown function of x and t , and a , b , c , and f are given functions. If these functions depend only on x and t , then the PDE (75) is called linear . If a , b , c , or f depend also on u , u x , or u t , then the PDE is called quasi-linear . Remark 1. The notation used in (75) suggests that we think of one of the variables, t , as time, and the other, x , as space. 2. In principle, we could also have second-order PDEs involving more than one space dimension. However, we limit the discussion here to PDEs with a total of two independent variables. 3. Of course, a second-order PDE can also be independent of time, and contain two space variables only (such as Laplace’s equation). These will be the elliptic equations we are primarily interested in. There are three fundamentally different types of second-order quasi-linear PDEs: If b 2 - 4 ac > 0, then L is hyperbolic . If b 2 - 4 ac = 0, then L is parabolic . If b 2 - 4 ac < 0, then L is elliptic . Example 1. The wave equation u tt = α 2 u xx + f ( x, t ) is a second-order linear hyperbolic PDE since a 1, b 0, and c ≡ - α 2 , so that b 2 - 4 ac = 4 α 2 > 0 . 2. The heat or diffusion equation u t = ku xx is a second-order quasi-linear parabolic PDE since a = b 0, and c ≡ - k , so that b 2 - 4 ac = 0 . 89
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3. For Poisson’s equation (or Laplace’s equation in case f 0) u xx + u yy = f ( x, y ) we use y instead of t . This is a second-order linear elliptic PDE since a = c 1 and b 0, so that b 2 - 4 ac = - 4 < 0 .
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  • Spring '07
  • Greg Fasshauer
  • Differential Equations, Equations, UK, Boundary value problem, Partial differential equation, Green's function, Hyperbolic partial differential equation

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