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Unformatted text preview: Math 108 Eigenfunction Expansions November 4, 2006 Eigenfunction expansions can be used to solve partial differential equations, such as the heat equation and the wave equation. In particular, we can use eigenfunction expansions to treat bound ary conditions with inhomogeneities that change in time, or partial differential equation inhomo geneities that change in time. The technique will be to use the Lagrange identity (on p. 667 of the book) to develop ordinary differential equations for the coefficients in the eigenfunction expansion. We will examine this approach through several examples, then explain the general idea. For our first example, let us consider an easy problem that we previously solved by separation of variables. Problem 7 on page 610: ∂u ∂t 100 ∂ 2 u ∂x 2 = 0 , < x < 1 , < t u (0 , t ) = 0 , < t u (1 , t ) = 0 , < t u ( x, 0) = sin(2 πx ) sin(5 πx ) , < x < 1 The easy way to solve this equation is to use separation of variables, as in section 10.5, and that is the approach we would suggest that you use to solve this problem on a test. Separation of variables works because the differential equation and the boundary conditions are all homogeneous. That is what allows us to take a linear combination of functions X n ( x ) T n ( t ) satisfying the (homogeneous) boundary conditions, and get a general function satisfying the (homogeneous) boundary conditions. Below, we will show how the eigenfunction expansion approach works. First, let us motivate the determination of a SturmLiouville problem for this heat equation. We would find that a solution of the heat equation of the form X n ( x ) T n ( t ) would satisfy T n T n = 100 X 00 n X n = λ n with X n (0) = 0 = X n (1) . The associated SturmLiouville problem is (100 X n ) = λ n X n , X n (0) = 0 = X n (1) . In the book’s description of SturmLiouville problems on p. 666, we have p ( x ) = 1 , q ( x ) = 0 and r ( x ) = 1. We solve the differential equation for the eigenfunctions to get X n ( x ) = sin( nπx ) , λ n = (10 nπ ) 2 . Next, let us solve the original partial differential equation by using eigenfunction expansions. Theorem 11.2.4 on p. 672 of the book shows us that we can write u ( x, t ) = ∞ X n =1 c n ( t ) X n ( x ) 1 for some coefficients c n ( t ) that are yet to be determined. We will develop ordinary differential equations to determine c n ( t ). The orthogonality of the eigenfunctions implies that Z 1 u ( x, t ) r ( x ) X m ( x ) dx ≡ ( u, rX m ) = ∞ X n =1 c n ( t ) ( X n , rX m ) = c m ( t ) ( X m , rX m ) ≡ c m ( t ) Z 1 r ( x ) X m ( x ) 2 dx We note that r ( x ) = 1 in this example, and solve for c m ( t ) to get c m ( t ) = R 1 u ( x, t ) X m ( x ) dx R 1 X m ( x ) 2 dx . (1) This does not solve the problem, because we do not know what u ( x, t ) is, so we can’t compute c m ( t ) directly from this formula. However, we can use this equation to determine an ordinary differential equation for c m ( t )....
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This note was uploaded on 03/31/2008 for the course MATH 108 taught by Professor Trangenstein during the Fall '07 term at Duke.
 Fall '07
 Trangenstein
 Differential Equations, Equations, Partial Differential Equations

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