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Unformatted text preview: 12.14 Review of Chapter 12 12.14.1 Basic Concepts (12.1) A partial differential equation of order k N for an unknown function u : R , R d open, d > 1 (the total number of variables), is an expression of the form F ( ( D k u )( x ) , ( D k 1 u )( x ) ,..., ( Du )( x ) ,u ( x ) ,x ) = 0 , x , (432) with a function F : R d k R d k 1 R d R R . A typical mathemat- ical model is given by (432), together with auxiliary conditions on , such as initial and/or boundary conditions. We have mainly looked at ho- mogeneous, linear PDEs, which are of the form summationdisplay | | k a ( x )( D u )( x ) = 0 , (433) with coefficients a : R , | | k . For these PDEs, the superposition principle (Theorem 1) is valid. 12.14.2 Examples We have considered problems in two variables ( d = 2): 1 spatial variable + time: 1D wave equation, u tt = c 2 u xx , c > (12.3, 12.4), 1D heat equation u t = c 2 u xx , c > 0 (12.5, 12.6) 2 spatial variables: 2D Laplace equation, u = 0 (steady heat problems, 12.5) problems in three variables ( d = 3): 2 spatial variables + time: 2D wave equation, u tt = c 2 u , c > (12.8, 12.9) 3 spatial variables: 3D Laplace equation, u = 0 (12.10) 12.14.3 Separation of Variables (12.3, 12.5, 12.8, 12.9, 12.10) The method of separating variables aims at transforming a PDE into a system of differential equations involving fewer independent variables, which may be solved more easily than the original PDE. 79 1. Separation of Variables For problems involving time, we write the dependent variable in the form u ( x ,t ) = F ( x ) G ( t ). In the case of wave and heat equations, respectively, this leads to u tt- c 2 u = F G- c 2 FG = 0 , in (0 , ) , (434) u t- c 2 u = F G- c 2 FG = 0 , in (0 , ) , (435) where R d 1 denotes the spatial domain only, from now on. After division by c 2 FG we obtain G c 2 G = F F , in (0 , ) , (436) G c 2 G = F F , in (0 , ) . (437) Because the left-hand side of (436), (437) depends only on the time variable t > 0, and the right-hand side of (436), (437) depends only on the spatial variable(s) x , both sides must be equal to a constant, the separation constant k R . So we obtain two separate equations, one in space and one in time (which is an ODE): F = kF, in , (+ boundary conditions) , (438) and either G = c 2 kG, in (0 , ) , (+ initial conditions) , (439) or G = c 2 kG, in (0 , ) , (+ initial conditions) . (440) The separation constant k R is part of the solution of the spatial eigenvalue problem (438), which involves d- 1 variables. The Laplace equation, u = 0, is just a special case of (438), with k = 0. Therefore, the same solution strategies apply....
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- Spring '08