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Unformatted text preview: Partial Differential Equations – Math 442 C13/C14 Fall 2009 Homework 5 Solutions 1. (Strauss 5.2.2.) Show that cos( x ) + cos( αx ) is periodic if α is a rational number and compute its period. What happens if α is not rational? Solution: Let us first notice that if f, g are both periodic with period p , then so is their sum: ( f + g )( x + p ) = f ( x + p ) + g ( x + p ) = f ( x ) + g ( x ) = ( f + g )( x ) . So we simply have to show that cos( x ) , cos( αx ) share a period. Note that the periods of cos( x ) are 2 nπ with n ∈ Z , and the periods of cos( αx ) are 2 mπ/α , where m ∈ Z . The question is then, do these two sets of numbers share an element, i.e. is there an n and an m so that 2 nπ = 2 mπ α ? If α = p/q , then we have 2 nπ = 2 mqπ p , or n = mq p . This has many solutions, choose, for example, m = p, n = q . If α is irrational, then this does not work; in fact, solving the first equation gives α = m n which is only possible if α is rational. 2. Define f ( x ) = x 3 on the interval [0 , 1]. Compute its Fourier sine series and its Fourier cosine series. Solution: The Fourier sine series coefficients are given by B n = 2 L integraldisplay L f ( x ) sin( nπx/L ) dx, and in this case L = 1, so we need to compute B n = 2 integraldisplay 1 x 3 sin( nπx ) dx. Integrating by parts several times gives B n = ( − 1) n 2(6 − n 2 π 2 ) n 3 π 3 , and then the F.S.S. is ∑ ∞ n =1 B n sin( nπx ). The Fourier cosine series coefficients are given by A n = 2 integraldisplay 1 x 3 cos( nπx ) dx....
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 Spring '09
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 Equations, Fourier Series, Periodicity, Periodic function, Partial differential equation, Joseph Fourier

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