10.4 Fourier Cosine and Sine Series PDF

CHAPTER 4 FOURIER SERIES AND INTEGRALS

Term by term we are “projecting the function onto each axis sin kx.” Fourier Cosine Series. The cosine series applies to even functions with C(−x) = C(x):.

10.4 Fourier Cosine and Sine Series

To solve a partial differential equation typically we represent a function by a trigonometric series consisting of only sine functions or only cosine

MATH 2280 - LECTURE 24 1. Fourier Sine and Cosine Series In this

Using just these basic facts we can figure out some important prop- erties of the Fourier series we get for odd or even functions.

Sec 3.3: Fourier sine and Cosine Series

b) But the func. f in the LHS of (7) may not be an odd func. e.g. f(x) = 100°C. i. The RHS of (7) is odd. (7) ii. After comparing the series.

Unit 4 (Fourier Series & PDE with Constant Coefficient)

2020年5月4日 required Hay range sine series is given by. 00 f(x) = { bn. Sin (nx) n=1. ㅈ where bn= I f(x)· Sin (nx)dx. 2. 大. Half- Range. Cosine Series!.

Fourier Series

Fourier Sine Series. • Thm. The Fourier series of an odd function of period 2L is obtain a Fourier cosine series: • If we apply odd periodic extension we.

APPLIED MATHEMATICS

10.10Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . 36. 10.11 • Fourier Sine Series (for odd periodic function f(x) with period p = 2L) f ...

Chapter 3: Fourier series

▻ Odd periodic extension to sketch ˜f. Section 3.3 Fourier cosine and sine series. 14. Page 15. f(x)

Fourier series

In various engineering problems it will be necessary to express a function in a series of sines and cosines which are periodic functions. Most of the single

Chapter 13 Fourier Series (Def) 週期函數設函數( ) f x 定義在區間( 內

(sine series)；若( ). f x 為偶函數，則它的傅里葉級數形如. 0. 1 cos. 2 n n a a nx ... (cosine series). Ex73: 已知週期為2π 的函數( ). f x 在一個週期內的表達式為.

10.4 Fourier Cosine and Sine Series

To solve a partial differential equation typically we represent a function by a trigonometric series consisting of only sine functions or only cosine

CHAPTER 4 FOURIER SERIES AND INTEGRALS

This section explains three Fourier series: sines cosines

5. fourier series

In various engineering problems it will be necessary to express a function in a series of sines and cosines which are periodic functions. Most of the single

FOURIER COSINE AND SINE SERIES 11.3

11 mars 2011 1 and 11.3.2. The trigonometric cosine and sine functions are even and odd functions respectively

MATH 2280 - LECTURE 24 1. Fourier Sine and Cosine Series In this

1. Fourier Sine and Cosine Series. In this lecture we'll develop some of our machinery for using Fourier series and see how we can use these Fourier series

Sine and Cosine Series (Sect. 10.4). Even odd functions.

Sine and cosine series. Theorem (Cosine and Sine Series). Consider the function f : [?LL] ? R with Fourier expansion f (x) =.

1 Fourier Sine and Cosine Series 2 Heat Equation

10 déc. 2019 1 Fourier Sine and Cosine Series. The reason we are studying Fourier Series in this course is to introduce a way to solve Heat Equations ...

File Type PDF Fourier Series Problems And Solutions Copy

30 août 2022 [PDF]Fourier series (based) multiscale method for computational analysis in ... 559 9.3 Fourier Sine and Cosine Series on 0 x L 568 9.4.

LECTURE 6 - Fourier Series

The Fourier series is named in honour of Jean-Baptiste Joseph of cosines and sines such as ... half range Fourier sine and cosine series.

Unit 4 (Fourier Series & PDE with Constant Coefficient)

4 mai 2020 Since Sin(x) = Sin (x+2x) = sin(x+4x). ... Fourier Series of a function ... Thus

CHAPTER 4 FOURIER SERIES AND INTEGRALS - MIT Mathematics

This section explains three Fourier series: sines cosines and exponentials eikx Square waves (1 or 0 or ?1) are great examples with delta functions in the derivative We look at a spike a step function and a ramp—and smoother functions too

Images

To solve a partial di erential equation typically we represent a function by a trigonometric series consisting of only sine functions or only cosine functions Recall that the Fourier series for an odd function de ned on [ L;L] consists entirely of sine terms Thus we might achieve f(x) = X1 n=1 a

10.4 Fourier Cosine and Sine Series

To solve a partial dierential equation, typically we represent a function by a trigonometric series consisting

of only sine functions or only cosine functions.

Recall that the Fourier series for an odd function dened on [L;L] consists entirely of sine terms. Thus

we might achieve f(x) =1X n=1a nsinnxL (1)

by articially extending the functionf(x);0< x < Lto the interval (L;L) in such a way that the extended

function is odd. That is, f o(x) =f(x);0< x < L; f(x);L < x <0; and extendingfo(x) to allxusing 2L-periodicity.fo(x) is an extension off(x) becausefo(x) =f(x) on

(0;L). This extension is called theodd2L-periodic extensionoff(x). The resulting Fourier series expansion

is called a half-range expansion forf(x) because it represents the functionf(x) on (0;L). Similarly, theeven2L-periodic extensionoff(x) as the function f e(x) =f(x);0< x < L; f(x);L < x <0; withfe(x+ 2L) =fe(x). To illustrate the various extensions, let's consider the functionf(x) =x;0< x < . If we extendf(x) to the interval (;) using-periodicity, then the extensionfis given by e f(x) =x;0< x < x+; < x <0; with ef(x+ 2) =ef(x). In the previous quiz we saw that the Fourier series foref(x) is e f(x)2 1X k=11k sin2kx;

which consists of both odd functions (the sine terms) and even functions (the constant term), because the

-periodic extensionef(x) is neither an even nor an odd function. The odd 2-periodic extension off(x) is

justfo(x) =x; < x < , which has the Fourier series expansion f o(x)21X n=1(1)n+1n sinnx:(2) Becausefo(x) =f(x) on the interval (0;), the expansion in (2) is a half-range expansion forf(x). The even 2-periodic extension off(x) is the functionfe(x) =jxj; < x < , which has the Fourier series expansion f e(x) =2 4 1 X k=11(2k1)2cos(2k1)x(3) (see Example 2 inx10.3 lecture notes). The preceding three extensions, the-periodic functionef(x), the odd 2-periodic functionfo(x), and

the even 2-periodic functionfe(x), are natural extensions off(x). The Fourier series expansions forfo(x)

andfe(x), given in (2) and (3) equalf(x) on the interval (0;). This motivates the following denitions.

1 Denition.Letf(x)be piecewise continuous on the interval[0;L]. The Fourier cosine series off(x)on [0;L]is a 02 +1X n=1a ncosnxL ;(4) where a n=2L Z L 0 f(x)cosnxL dx; n= 0;1;::::(5)

The Fourier sine series off(x)on[0;L]is

1 X n=1b nsinnxL ;(6) where b n=2L Z L 0 f(x)sinnxL dx; n= 1;2;::::(7)

The trigonometric series in (4) is the Fourier series forfe(x), the even 2L-periodic extension off(x). The

trigonometric series in (6) is the Fourier series forfo(x), the odd 2L-periodic extension off(x). These are

calledhalf-range expansionsforf(x). Example 1.Determine (a) the-periodic extensionef, (b) the odd2-periodic extensionfo, and (c) the even2-periodic extensionfe, forf(x) =x;0< x < . Example 2.Compute the Fourier sine series forf(x) =x;0< x < . Example 3.Compute the Fourier cosine series forf(x) =ex;0< x <1.

A mathematical model for source-less the heat

ow in a uniform wire whose ends are kept at constant

temperature 0 is the following initial value problem, whereu(x;t) is the temperature in the wire at location

xand timet: @u@t (x;t) =@2u@x

2(x;t);0< x < L;t >0 (8)

u(0;t) =u(L;t) = 0; t >0 (9) u(x;0) =f(x);0< x < L:(10) Using the method of separation of variables, we may derive the following solution: u(x;t) =1X n=1c ne(n=L)2tsinnxL :(11)

Example 4.Find the solution to the heat problem

@u@t = 5@2u@x

2;0< x < ;t >0

u(0;t) =u(;t) = 0; t >0 u(x;0) =x(x);0< x < : 2quotesdbs_dbs8.pdfusesText_14

[PDF] 10.4 Fourier Cosine and Sine Series

10.4 Fourier Cosine and Sine Series

The Fourier sine series off(x)on[0;L]is

A mathematical model for source-less the heat

2(x;t);0< x < L;t >0 (8)

Example 4.Find the solution to the heat problem

2;0< x < ;t >0