Partial Differential Equations with MATLAB
86 An Introduction to Partial Differential Equations with MATLAB©R Therefore, (3.7) becomes 〈 sin nπx L ,sin mπx L 〉 = ∫L 0 [ co ...
Fourier Series 87 − (^1) − 3 − 2 − 1 0 1 2 3 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 FIGURE 3.5 MATLAB graph showing the orthogo ...
88 An Introduction to Partial Differential Equations with MATLAB©R c) Iff(x)andg(x) are periodic with fundamental periodT,then f ...
Fourier Series 89 This is called theinner product with respect to the weight functionw(x) and, if〈f,g〉=0,wesaythatf andgareortho ...
90 An Introduction to Partial Differential Equations with MATLAB©R Again, let us proceed formally, assuming that there are const ...
Fourier Series 91 where the coefficients are given by an= 1 L ∫L −L f(x)cos nπx L dx, n=0, 1 , 2 ,... bn= 1 L ∫L −L f(x)sin nπx ...
92 An Introduction to Partial Differential Equations with MATLAB©R where we have integrated by parts = 2 π [ − 1 n πcosnπ ] =− 2 ...
Fourier Series 93 Exercises 3.3 In Exercises 1–12, calculate the Fourier series off(x)onthegiveninterval. 1.f(x)= { 2 , if− 1 ≤x ...
94 An Introduction to Partial Differential Equations with MATLAB©R b) Show that, if a 0 2 + ∑∞ n=1 ( ancos nπx L +bnsin nπx L ) ...
Fourier Series 95 It turns out that, when complex functions are involved, we must change the definition of inner product, so tha ...
96 An Introduction to Partial Differential Equations with MATLAB©R y 3 1 2 x f(2−) = 1 f(2+) = 3 FIGURE 3.6 A typical jump disco ...
Fourier Series 97 (^01) 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1 2 3 4 5 6 FIGURE 3.7 MATLAB graph of the function from Example 3. ...
98 An Introduction to Partial Differential Equations with MATLAB©R − (^2) − 2 −1.5 − 1 −0.5 0 0.5 1 1.5 2 −1.5 − 1 −0.5 0 0.5 1 ...
Fourier Series 99 Now we are ready to state the big theorem. Theorem 3.1Iff(x)is piecewise smooth‡on−L≤x≤L,thenitsFourier series ...
100 An Introduction to Partial Differential Equations with MATLAB©R We note that the Fourier series is “blind” to the discontinu ...
Fourier Series 101 and f(L−)=fp(−L−). We then have the following corollary. Corollary 3.1Iff(x)is piecewise smooth on−L≤x≤L,andf ...
102 An Introduction to Partial Differential Equations with MATLAB©R 11.f(x)=| 3 −x|on− 1 ≤x≤ 1 12.f(x)= ⎧ ⎪⎨ ⎪⎩ 2+x, if− 3 ≤x< ...
Fourier Series 103 family is to music. The most well-known of the Bernoullis are James; his brother John who, incidentally, had ...
104 An Introduction to Partial Differential Equations with MATLAB©R a)f(x)=x,−π≤x≤π b)f(x)=x^2 ,− 1 ≤x≤ 1 c) f(x)= { 2 , if− 2 ≤ ...
Fourier Series 105 [1] Rewritesn(x)as sn(x)= 1 2 π ∫π −π [ 1+2 ∑n k=1 cosk(t−x) ] f(t)dt. [2] Show that 1+2 ∑n k=1 cosk(t−x)= si ...
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