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Unformatted text preview: Week 7 Lectures, Math 716, Tanveer 1 Introduction Recall we discussed completeness of Fourier Series. This is relevant for constant coefficient partial differential equations in simple rectangular geometries or sometimes in circular geometry as well. Separation of variable procedure leads to a series representation of solution. The undetermined constants are Fourier Coefficients of the series evaluated initially or on certain segments of the boundary. As an example, recall solution to u = 0 for x 2 + y 2 < 1 was in the form u ( r, ) = a 2 + X n =1 ( a n r n cos n + b n r n sin n ) If we use boundary condition u (1 , ) = f ( ), then the constants a n , b n are simply the Fourier coefficients of f ( ). In more complicated situations, separation of variable is not possible and even if it is possible, it does not lead to a simple Fourier Series. For instance, in one of your homework problems involving wave equation in a circular geometry, Bessel function arose and one needed to use some properties of Bessel functions. We seek to construct a more general theory on completeness of series representation based on eigenfunctions of some differential operators with homogenous boundary conditions. Note that the Fourier Sine Series sin nx l n =1 is a special example in 1-D as it arises from eigenfunction of the operator- d 2 dx 2 satisfying- d 2 u dx 2 = u for < x < l ; with BC u (0) = 0 = u ( l ) 2 General Eigen Value Problem: We introduce L 2 () inner-product ( f, g ) = Z f ( x ) [ g ( x )] * d x (1) Definition 1 An operator A : L 2 () L 2 () is symmetric (or self-adjoint) if for any u , v in the domain of A , ( v, Au ) = ( Av, u ) Definition 2 A symmetric operator is positive if ( u, Au ) > for any nonzero u Domain ( A ) . A symmetric operator is semi-postive if ( u, Au ) for any u Domain ( A ) . Lemma 3 If A : L 2 () L 2 () is symmetric (self-adjoint), then its eigenvalues are real. Further, eigenvectors corresponding to two distinct eigenvalues are orthogonal, with respect to the inner-product defined above. Further, if A is positive, any eigenvalue > and for A semi-positive, . 1 Proof. If u is an eigen function of A corresponding to eigenvalue , then it follows from definition of inner-product ( u, u ) = ( Au, u ) = ( u, Au ) = * ( u, u ) Hence = * since ( u, u ) = k u k 2 6 = 0 for an eigenfunction Further, if u and v are eigenfunctions corresponding to unequal eigenvalues and it follows that ( u, v ) = ( Au, v ) = ( u, Av ) = ( u, v ) = * ( u, v ) = ( u, v ) Therefore, ( - )( u, v ) = 0 implying ( u, v ) = 0, i.e. orthogonal. If A is positive, ( u, u ) = ( Au, u ) > Therefore, > 0. For semi-positive A , the above clearly gives Remark 1 There is no loss of generality assuming orthogonality of eigenvectors ( u, v ) = 0 even when corresponding eigenvalues = . This is because a Gram-Schmidt orthogonalization procedure can be employed.procedure can be employed....
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