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Mathematics
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MATH 251
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Spring

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Second Order Linear Partial Differential Equations Part II Fourier series; Euler▯Fourier formulas; Fourier Convergence Theorem; Even and odd functions; Cosine and Sine Series Extensions; Particular solution of the heat conduction equation Fourier Series Suppose f is a periodic function with a period T = 2L. Then ther series representation of f is a trigonometric series (that is, it is an infinite series consists of sine and cosine terms) of the form a ∞  nπ x nπ x  f (x)= 0+ ∑ a nos +b nin  2 n=1 L L  Where the coefficients are given by the Euler▯Fourier formulas: L a = 1 f (x)cosmπ x dx m L ∫ L , m = 0, 1, 2, 3, … −L 1 L nπ x bn= ∫ f( )sin dx, n = 1, 2, 3, … L −L L The coefficients a’s are called the Fourier cosine coefficients (i0cluding a , the constant term, which is in reality the 0▯th cosine term), and b’s are called the Fourier sine coefficients. © 2008, 2012 Zachary S Tseng E▯2 ▯ 1 Note 1: Thus, every periodic function can be decomposed into a sum of one or more cosine and/or sine terms of selected frequencies determined solely by that of the original function. Conversely, by superimposing cosines and/ or sines of a certain selected set of frequencies we can reconstruct any periodic function. Note 2: If f is piecewise continuous, then the definite integrals in the Euler▯ Fourier formulas always exist (i.e. even in the cases where they are improper integrals, the integrals will converge). On the other hand, f needs not to be piecewise continuous to have a Fourier series. It just needs to be periodic. However, if f is not piecewise continuous, then there is no guarantee that we could find its Fourier coefficients, because some of the integrals used to compute them could be improper integrals which aredivergent. Note 3: Even though that the “=” sign is usually used to equate a periodic function and its Fourier series, we need to be a little careful. The function f and its Fourier series “representation” are only equal to each other if, and whenever, f is continuous. Hence, if f is continuous for −∞ < x < ∞, then f is exactly equal to its Fourier series; but if f is piecewise continuous, then it disagrees with its Fourier series at every discontinuity. (See the Fourier Convergence Theorem below for what happens to the Fourier series at a discontinuity of f .) Note 4: Recall that a function f is said to be periodic if there exists a positive number T, such that f (x + T) = f(x), for all x in its domain. In such a case the number T is called a period of f. A period is not unique, since if f(x + T) = f(x), then f(x + 2T) = f(x) and f(x + 3T) = f(x) and so on. That is, every integer▯multiple of a period is again another period. The smallest such T is called the fundamental period of the given function f. A special case is the constant functions. Every constant function is clearly a periodic function, with an arbitrary period. It, however, has no fundamental period, because its period can be an arbitrarily small real number. The Fourier series representation defined above is unique for each function with a fixed period T = 2L. However, since a periodic function has infinitely many (non▯ fundamental) periods, it can have many different Fourier series by using different values of L in the definition above. The difference, however, is really in a technical sense. After simplification they would look the same. © 2008, 2012 Zachary S Tseng E▯2 ▯ 2 Therefore, technically at least, a Fourier series of a periodic function depends both on the function as well as its chosen period. Note 5: The definite integrals in the Euler▯Fourier formulas can be found be integrating over any interval of length 2L. However, from −L to L is the convention, and is often the most convenient interval to use. Note 6: Since the Fourier coefficients are calculated by definite integrals, which are insensitive to the value of the function at finitely many points. Consequently, piecewise continuous functions of the same period that differ from each other at finitely many points (notably, at isolated discontinuities) per period will have the same Fourier series. Note 7: The constant term in the Fourier series, which has expression a0 1 1 L 1 L = ⋅ ∫ f (x)cos(0)dx= ∫ f (x)dx , 2 2 L −L 2L −L is just the average or mean value of f(x) on the interval [−L, L]. Since f is periodic, this average value is the same for everyperiod of f. Therefore, the constant term in a Fourier series represents the average value of the function f over its entire domain. © 2008, 2012 Zachary S Tseng E▯2 ▯ 3 Example: Find a Fourier series for f(x) = x, −2 < x < 2, f(x + 4) = f(x). First note that T = 2L = 4, hence L = 2. The constant term is one half of: L 2 2 1 mπ x 1 1 x2 1 a0 = ∫ f x)cos dx = ∫ xdx = = (2− 2) = 0 L−L L 2 −2 2 2 −2 2 The rest of the cosine coefficients, for n = 1, 2, 3, …, are L 2 1 nπ x 1 nπx an= L ∫ f (x)cos L dx = 2 ∫xcos 2 dx −L −2  2 2  = 1 x sinn x − 2 sinn x dx  2 n π 2 −2 nπ −2 2    1 2x n x 4 n x 2  =  sin + 2 2 cos  2  π 2 n π 2 −2 1  4   4  = 2 0 + n2π 2cos(nπ )  0 + n2π 2cos(−nπ) = 0   Hence, there is no nonzero cosine coefficient for this function. That is, its Fourier series contains no cosine terms at all. (We shall see the significance of this fact a little later.) © 2008, 2012 Zachary S Tseng E▯2 ▯ 4 The sine coefficients, for n = 1, 2, 3, …, are L 2 b = 1 f (x)sinnπ xdx = 1 xsinnπ x dx n L −L L 2 −2 2 1 − 2x nπx 2 −2 2 nπx  =  cos − ∫ cos dx 2 n π 2 −2 nπ −2 2  2 1 − 2x nπx 4 nπx  =  cos + 2 2 sin  2 n π 2 nπ 2 −2  = 1 − 4cos(nπ )−0 − 4 cos(−nπ )−0  2 nπ  n π  = − 2(cos(nπ) +os(nπ) = − 4)cos(nπ) nπ nπ  4 , n = odd n+1 = nπ = (−1) 4  −4 nπ .  nπ , n =even 4 ∞ ( 1)n+1 nπ x Therefore,( )= ∑ sin . π n=1 n 2 © 2008, 2012 Zachary S Tseng E▯2 ▯ 5 Figure : the graph of the partial sum of the first 30 terms of the Fourier series ∞ n+1 f (x) =4 (−1) sin nπ x π n=1 n 2 . Compare it against the graph of the actual function the series represents the function f(x) = x, −2 < x < 2, f(x + 4) = f(x), seen earlier. © 2008, 2012 Zachary S Tseng E▯2 ▯ 6 Example: Find a Fourier series for f(x) = x, 0 < x < 4, f(x + 4) = f(x). How will it be different from the series above? 4 2 4 1 1 x 1 a0= 2 ∫ xdx = 2 2 = 2 (8−0) = 4 0 0 For n = 1, 2, 3, … : 4 a = 1 xcos nπx dx n 2 ∫ 2 0 1 2x n x 4 n x 4 =  sin + 2 2cos  2  π 2 n π 2 0 = 1 0+ 4 cos(2nπ) − 0+ 4 cos(0)= 0 2  n π 2   n π 2    1 4 nπx bn= ∫ xsin dx 2 0 2 4 = 1 − 2x cosn x + 4 sinn x  2  nπ 2 n2π2 2   0  1 − 8   − 4 =  cos(2n π 0− − (0 0 = 2 nπ   nπ Consequently, a0 ∞  nπx nπ x  − 4 ∞ 1 nπx f( )= 2 + ∑  ancos L +bn sin L  =2+ π ∑ nsin 2 . n=1  n=1 © 2008, 2012 Zachary S Tseng E▯2 ▯ 7 Example: Find a Fourier series for f(x) = | x|, −2 < x < 2, f(x + 4) = f(x). 8 ∞ 1 (2n −1) x Answer : f( ) 1− ∑ cos π 2 n=12 n −1)2 2 Example: Find a Fourier series for f (x)= − 2, −1≤ x < 0  2, 0 ≤ x <1 , f(x + 2) = f(x).  8 ∞ 1 Answer: f (x) = ∑ sin((2n −1)π x) π n=1(2n −1) © 2008, 2012 Zachary S Tseng E▯2 ▯ 8 Comment: Just because a Fourier series could have infinitely many (nonzero) terms does not mean that it will always have that many terms. If a periodic function f can be expressed by finitely many terms normally found in a Fourier series, then the expression must be the Fourier series of f. (This is analogous to the fact that the Maclaurin series of any polynomial function is just the polynomial itself, which is a sum of finitely many powers of x.) Example: The Fourier series (period 2π) representing f (x) = 5 + cos(4x) − sin(5x) is just f(x) = 5 + cos(4x) − sin(5x). Example: The Fourier series (period 2π) representing f (x) = 6cos(x)sin(x) is not exactly itself as given, since the product cos(x)sin(x) is not a term in a Fourier series representation. However, we can use the double▯angle formula of sine to obtain the result: 6cos(x)sin(x) = 3sin(2x). Consequently, the Fourier series is f(x) = 3sin(2x). © 2008, 2012 Zachary S Tseng E▯2 ▯ 9 The Fourier Convergence Theorem Here is a theorem that states a sufficient condition for the convergence of a given Fourier series. It also tells us to what value does the Fourier series converge to at each point on the real line. Theorem: Suppose f and f ′ are piecewise continuous on the interval −L ≤ x ≤ L. Further, suppose that f is defined elsewhere so that it is periodic with period 2L. Then f has a Fourier series as stated previously whose coefficients are given by the Euler▯Fourier formulas. The Fourier series converge to f (x) at all points where f is continuous, and to lim f (x) lim f (x) /2  x→c − x→c+  at every point c where f is discontinuous. Comment: As seen before, the fact that f is piecewise continuous guarantees that the Fourier coefficients can be found. The condition that f ′ is also piecewise continuous is a sufficient condition to guarantee that the series thusly found will be convergent everywhere on the real line. As well, recall that, suppose f is continuous at c, then by definition f(c) equals both one▯ sided limits of f(x) as x approaches c. Therefore, the second part of the theorem could be even more succinctly stated as that the Fourier series representing f will always converge to   x→c − (x)+ lix→c+(x) /2   at every point c (and not just at discontinuities of f ). A consequence of this theorem is that the Fourier series of f will “fill in” any removable discontinuity the original function might have. A Fourier series will not have any removable▯type discontinuity. © 2008, 2012 Zachary S Tseng E▯2 ▯ 10 Example: Let us revisit the earlier calculation of the Fourier series representing f(x) = x, −2 < x < 2, f(x + 4) = f(x). The Fourier series, as we have found, is ∞ n+1 f (x) = 4 (−1) sinnπ x π n=1 n 2 . The following figures are the graphs of various finite n▯th partial sums of the series above. n = 3 n = 10 © 2008, 2012 Zachary S Tseng E▯2 ▯ 11 n = 20 n = 30 n = 50 © 2008, 2012 Zachary S Tseng E▯2 ▯ 12 Note that superimposed sinusoidal curves take on the general shape of the piecewise continuous periodic function f (x) almost immediately. As well, for the parts of the curve where f(x) is continuous (where the Fourier Convergence Theorem predicts a perfect match) the composite curve of the Fourier series converges rapidly to that of f (x), as predicted. The convergence is not as rapidly near the jump discontinuities. Indeed, for all but the lowest partial sums of the Fourier series, the curve seems to “overshoot” that of f (x) near each jump discontinuity by a noticeable margin. Further more, this discrepancy does not fade away for any finitely larger n. That is, the convergence of a Fourier series, while predictable, is not uniform. (That is a small price we pay for approximating a piecewise continuous periodic function by sinusoidal curves. It can be done, but the Fourier series does not converge uniformly to the actual function.) This behavior is known as the Gibbs Phenomenon. It further states that the partial sums of a Fourier series will overshoot a jump discontinuity by an amount approximately equal to 9% of the jump. That is, near each jump discontinuity, the overshoot amounts to about 0.09 lim f +x)− lim f (x)− , x →c x →c for large n. Further, this overshoot does not go away for any finitely large n. © 2008, 2012 Zachary S Tseng E▯2 ▯ 13 Question: Sketch the graph of the Fourier series of f(x) = x, −2 < x < 2, f(x + 4) = f(x). We have seen a few graphs of its partial sums. But what will the graph of the actual Fourier series look like? Example: Sketch the graph of the Fourier series of f(x) = | x|, −2 < x < 2, f(x + 4) = f(x). Example: Sketch the graph of the Fourier series of  − 2, −1≤ x < 0 f (x)=  , f(x + 2) = f(x).  2, 0 ≤ x <1 © 2008, 2012 Zachary S Tseng E▯2 ▯ 14 Even and Odd Functions Recall that an even function is any function f such that f(−x) = f(x), for all x in its domain. 2 4 6 −2 −4 Examples: cos(x), sec(x), any constant function, x , x , x , …
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