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In this Section we look at a typical application of Fourier series Vibration problems are often modelled by ordinary differential equations with constant 

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Some of these problems can be solved by use of Fourier series (see Problem 13 24) EXAMPLE The classical problem of a vibrating string may be idealized in the 

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problems in infinite domains where the solution is required to be bounded The formula for the Fourier series of a periodic function f with period 2a will be given ( see the reference Know how to derive and use formulas for the sine/cosine

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6 avr 2020 · Problems 1 (a) What is the Fourier representation of f (t) = 1, −π < t < π? (b) Use Maple to create a graph of f (t) and a partial Fourier series 2

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with other metamathematical questions, caused nineteenth-century The applications of Fourier analysis outside of mathematics continue to multiply

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Section 1: Theory 7 A more compact way of writing the Fourier series of a function f(x), with period 2π, uses the variable subscript n = 1, 2, 3, f(x) = a0 2 + ∞

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We have seen in Chapter 1 that nonhomogeneous differential equations with constant coeffi- cients containing sinusoidal input functions (e.g., A sin t can be solved quite easily for any input frequency There are many examples, however, of periodic input functions that are not sinusoidal.

Figure 7.1

illustrates four common ones. The voltage input to a circuit or the force on a spring-mass system may be periodic but possess discontinuities such as those illustrated. The object of this chapter is to present a technique for solving such problems and others con- nected to the solution of certain boundary-value problems in the theory of partial differential equations. The technique of this chapter employs series of the form a 0 2 n=1 a n cos nt T +b n sinnt T (7.1.1) the so-called trigonometricseries.unlike power series, such series present many pitfalls and subtleties. A complete theory of trigonometric series is beyond the scope of this text and most works on applications of mathematics to the physical sciences. We make our task tractable by narrowing our scope to those principles that bear directly on our interests. Let f(t)be sectionally continuous in the interval ŠThas at most a Þnite number of discontinuities. at each point of discontinuity the right- and left-

hand limits exist; that is, at the end points ŠTand Tof the interval ŠT7.1 INTRODUCTION 7

Fourier Seriesf(t)

t f(t) t f(t) t f(t) tFigure 7.1Some periodic input functions. M. C. Potter et al., Advanced Engineering Mathematics 7

413© Springer Nature Switzerland AG 2019

f (-T )and f(T )as limits from the right and left, respectively, according to the following expressions: f(-T )=lim t→-T t>-T f(t),f(T )=lim t→T tFrobenius series. In this chapter we will use Maplecommands from Appendix C, assumefrom Chapter 3, and dsolvefrom Chapter 1. New commands include: sumand simplify/trig.

7.1.1MapleApplications

It will be useful to compare a function to its Fourier series representation. Using Maple,we can create graphs to help us compare. For example, in order to compare Eq. 7.1.5 with f(t)=t, we

414?CHAPTER7 /FOURIER SERIES

can start by deÞning a partial sum in

Maple:

>fs:=(N, t) -> sum(2*(-1)^(n+1)* sin(n*t)/n, n=1..N); fs:=(N,t)π N n=1

2(Š1)

(n+1) sin(nt) n In this way, we can use whatever value of Nwe want and compare the Nth partial sum with the function f(t): >plot({fs(4, t), t}, t=-5..5); Observe that the Fourier series does a reasonable job of approximating the function only on the interval Š? t 2 2 4 4 44

7.1 INTRODUCTION?415

Problems

1.(a) What is the Fourier representation of f(t)=1,

Š? (b) Use Mapleto create a graph of f(t)and a partial

Fourier series.

2.Verify the representation, Eq. 7.1.5, by using Eqs. 7.1.3

and 7.1.4.

3.Does the series (Eq. 7.1.5) converge if tis exterior to

Š?

4.Show that the Fourier series representation given as

Eq. 7.1.4 may be written

f(t)? 1 2T T ŠT f(t)dt 1 T n=1 T ŠT f(s)cos n?t T (sŠt)dt

5.Explain how

4 =1Š 1 3 1 5 1 7 follows from Eq. 7.1.5. Hint:Pick t=?/2. Note that this result also follows from tan Š1 x=xŠ x 3 3 x 5 5 x 7 7 +∑∑∑,Š16.What is the Fourier series expansion of f(t)=Š1,

ŠT

7.Create a graph of tan

Š1 xand a partial sum, based on the equation in Problem 5.

8.One way to derive Eqs. 7.1.3 is to think in terms of a least

squares fit of data (see Section 5.4). In this situation, we let g(t)be the Fourier series expansion of f(t), and we strive to minimize:quotesdbs_dbs14.pdfusesText_20