# SUPPLEMENT ORTHOGONALITY OF EIGENFUNCTIONS We now develop some properties of eigenfunctions to be used in Chapter for Fourier Series and Partial Dierential Equations PDF document - DocSlides 2014-12-14 142K 142 0 0

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8 SUPPLEMENT ORTHOGONALITY OF EIGENFUNCTIONS We now develop some properties of eigenfunctions to be used in Chapter 9 for Fourier Series and Partial Di64256erential Equations 1 De64257nition of Orthogonali ID: 24010

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3.8 (SUPPLEMENT) � ORTHOGONALITY OF EIGENFUNCTIONS We now develop some properties of eigenfunctions, to be used in Chapter 9 for Fourier Series and Partial Diﬀerential Equations. 1. Deﬁnition of Orthogonality We say functions ) and ) are orthogonal on a if dx = 0 [Motivation: Let�s approximate the integral with a Riemann sum, as follows. Take a large integer , put = ( /N and partition the interval a < x < b by deﬁning h,x + 2 h,...,x Nh . Then dx ��� = ( where = ( ,...,f )) and = ( ,...,g )) are vectors containing the values of and . The vectors and are said to be orthogonal (or perpendicular) if their dot product equals zero ( = 0), and so when we let in the above formula it makes sense to say the functions and are orthogonal when the integral dx equals zero.] Example. sin and cos are orthogonal on , since sin cos xdx sin = 0. 2. Integration Lemma Suppose functions ) and ) satisfy the diﬀerential equations 00 = 0 , a 00 = 0 , a for some numbers , . Then dx = [ )] Proof. LHS = [( )] dx by taking the �s inside the integral 00 00 dx since 00 and 00 dx as you can check by diﬀerentiating! = RHS by the Fundamental Theorem of Calculus.
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3. Boundary Conditions Consider a function ) for a . Four boundary condition (�BC�) types are: Dirichlet BC ) = ) = 0 Neumann BC ) = ) = 0 Mixed1 BC ) = ) = 0 Mixed2 BC ) = ) = 0 Periodic BC ) = , X ) = (The two varieties of Mixed BC are very similar, diﬀering only as to which endpoint has = 0 and which has = 0.) 4. Orthogonality of Eigenfunctions Theorem: Eigenfunctions corresponding to distinct eigenvalues must be orthogonal. Precise statement: suppose 00 = 0 and 00 = 0 on a , and that and both satisfy the same type of BC . If then and are orthogonal: dx = 0 Proof. By the Integration Lemma, we have dx )] = 0 under Dirichlet BCs, because ) = ) = 0 and ) = ) = 0 = 0 under Neumann BCs, because ) = ) = 0 and ) = ) = 0 = 0 under Mixed BCs, for similar reasons. For periodic BCs, we use that ) = ) and ) = ) and so on, to see )] = [ )] )] = 0 5. Example Show that sin and sin 2 are orthogonal for 0 Solution. We could just show sin( ) sin(2 dx = 0 by using trigonometric identities to evaluate the integral. But it is easier to notice that both ) = sin( ), with = 1 , and ) = sin(2 ), with = 2 , are eigenfunctions for 00 λX = 0 (0) = ) = 0 (Dirichlet BC)
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Since , the Orthogonality Theorem implies dx = 0, so that sin( ) and sin(2 ) are orthogonal for 0 6. Orthogonality of sines and cosines for The following formulas will be essential for our study of Fourier series, in Chapter 9. Let n,m 1 be integers. Then: cos( nx ) cos( mx dx = 0 if (1) if m, (2) sin( nx ) sin( mx dx = 0 if (3) if m, (4) cos( nx ) sin( mx dx = 0 if (5) = 0 if m. (6) Proof. For formula (1), we let ) = cos( nx ) and ) = cos( mx ), so that and are eigenfunctions of 00 λX = 0 on , with eigenvalues and respectively. Also, and satisfy periodic boundary conditions (since they are both -periodic). If then , and so the Orthogonality Theorem implies and are orthogonal, which is equation (1). Equations (3) and (5) are proved similarly. [Exercise.] To get equation (2), we substitute and evaluate the integral explicitly as cos mx dx (1 + cos(2 mx )) dx π. Equation (4) proceeds similarly. [Exercise.] And ﬁnally, equation (6) is true since we can substitute and evaluate the integral explicitly as cos( mx ) sin( mx dx sin mx = 0 Remark. Here we have used the Orthogonality Theorem to evaluate integrals (1), (3) and (5). They can also be evaluated using trigonometric identities (or the integration formulas inside the back cover of the text). But �orthogonality of eigenfunctions� is the best way to think about these results, and we will use orthogonality in Chapter 9 for other applications as well.