Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 11 - Initial-Boundary Valu

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 11 - Initial-Boundary Valu

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CHAPTER 11 Initial-Boundary Value Problems T he infnite spatial domains considered in the previous chapter give insight regarding the behavior oF waves and diFFusions. However, since such domains are not physically realistic, we need to develop new techniques For solving pde s on bounded domains. As a frst step towards solving the heat and wave equations over fnite spatial domains (such as the interval 0 x L in one space dimension), we will solve these equations on “semi-infnite” domains whose boundaries consist oF one point. 11.1. Heat and Wave Equations on a Half-Line We begin by solving the homogeneous Dirichlet problem For the heat equation on the interval 0 x < ; that is, u t = κ u xx ( 0 < x < ) ( 11 . 1 ) u ( x ,0 )= φ ( x )( 0 < x < ) ( 11 . 2 ) u ( 0, t 0 ( t 0 ) .( 11 . 3 ) The homogeneous boundary condition is quite important For the solution tech- nique that Follows. In the context oF heat transFer within a “one-dimensional” wire, this Dirichlet boundary condition is analogous to immersing the x = 0 end oF the wire in a bath oF ice water with temperature zero degrees Celsius. We will solve the homogeneous Dirichlet problem ( 11 . 1 )–( 11 . 3 ) using a refection method , temporarily extending our spatial domain to the entire real line and solving a Cauchy problem instead. By quoting the Formula For the solution 297
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298 heat and wave equations on a half - line φ odd (x) φ (x) xx Figure 11 . 1 . Illustration of the odd extension of a function φ ( x ) . of the Cauchy problem in the preceding chapter, we will obtain the solution of ( 11 . 1 )–( 11 . 3 ) by restricting ourselves to the original spatial domain. First, recall that a function f ( x ) of a single variable is called odd if it has the property that f ( - x )= - f ( x ) for all real x . Examples of odd functions include sin ( x ) and x 3 . If f is an odd function, notice that f ( 0 f ( - 0 - f ( 0 ) , which implies that f ( 0 0. Now, referring to the initial condition ( 11 . 2 ) above, we de±ne the odd extension of φ ( x ) as φ odd ( x φ ( x ) if x > 0 - φ ( - x ) if x < 0 0 if x = 0. By construction, φ odd is an odd function and is de±ned for all real x (see Fig- ure 11 . 1 ). Now consider the Cauchy problem v t = κ v ( - < x < ) v ( x ,0 φ odd ( x )( - < x < ) . From the previous chapter, we know that the solution is given by the convolution of the heat kernel S ( x , t ) with the initial condition: v ( x , t ) = ( S ± φ odd x , t # - S ( x - y , t ) φ odd ( y ) d y . We claim that the restriction of v ( x , t ) to the domain x 0 is the solution of the Dirichlet problem ( 11 . 1 )–( 11 . 3 ). To see why, we need to verify that all three conditions of our Dirichlet problem are satis±ed. Certainly v ( x , t ) satis±es the
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initial - boundary value problems 299 same pde as u ( x , t ) on the domain x > 0. The initial conditions also match on that domain, because v ( x ,0 )= φ ( x u ( x ) whenever x > 0. Checking the boundary condition requires a bit more care. As an exercise, you should verify that since the initial condition for v ( x , t ) is odd, then the solution v ( x , t ) will remain odd for all t > 0. That is, v ( - x , t - v ( x , t ) for all t 0. By our earlier remarks on odd functions, this implies that v ( 0, t 0 for all t 0. It follows
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This note was uploaded on 02/27/2012 for the course MATH 532 taught by Professor Reynolds during the Fall '11 term at VCU.

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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 11 - Initial-Boundary Valu

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