Partial Differential Equations with MATLAB
106 An Introduction to Partial Differential Equations with MATLAB©R Therefore, sn(x)= a 0 2 + ∑n k=1 (akcoskx+bksinkx) = 1 2 π ∫ ...
Fourier Series 107 Then, 1+2 ∑n k=1 coskθ=1+ ∑n k=1 (eikθ+e−ikθ) =e−inθ+e−i(n−1)θ+...+e−iθ+1+eiθ+...+einθ =e−inθ(1 +eiθ+e^2 iθ+. ...
108 An Introduction to Partial Differential Equations with MATLAB©R PROOF of[[[ 4 ]]] ∫π −π sin^2 n 2 +1u sinu/ 2 du= ∫π −π [ 1+ ...
Fourier Series 109 wheregi(x)=g(x)on(xi− 1 ,xi)andgiis continuous on [xi− 1 ,xi],i=1,...,n. It follows that eachgiisuniformlycon ...
110 An Introduction to Partial Differential Equations with MATLAB©R Now, asgis uniformly continuous on [xi− 1 ,xi], we know that ...
Fourier Series 111 Now, what happens at a pointxwherefisnotcontinuous? Life is a bit more complicated, since at least one of lim ...
112 An Introduction to Partial Differential Equations with MATLAB©R which exists asfis piecewise smooth.‡So we are done! Now tha ...
Fourier Series 113 −0.2− 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 a −0.2− 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 0 0.2 0 ...
114 An Introduction to Partial Differential Equations with MATLAB©R −0.2− 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1. ...
Fourier Series 115 Next, we may use an argument similar to the proof of [2], above (see Exer- cise 1), to show that g′n(x)= 2 π ...
116 An Introduction to Partial Differential Equations with MATLAB©R Show that the function ∑n k=1 1 2 k− 1 sin (2k−1)π 2 n has ...
Fourier Series 117 a)f(x)=x^3 +5x b) f(x)= { 2 x+3,ifx< 0 x− 1 ,ifx≥ 0 c)f(x)= { x^2 , ifx≤ 0 x^2 +1,ifx> 0 In Exercise 7 ...
118 An Introduction to Partial Differential Equations with MATLAB©R What we do is this: Iff is piecewise smooth, then we extend ...
Fourier Series 119 Let us calculate the Fourier series ofgandh, with an eye toward getting everything in terms off.Forgwe get th ...
120 An Introduction to Partial Differential Equations with MATLAB©R Corollary 3.2Iffis piecewise smooth on 0 ≤x≤L, we have Fc(x) ...
Fourier Series 121 Also, Fc(x)= a 0 2 + ∑∞ n=1 ancosnx,wherean= 2 π ∫π 0 3cosnx dx. Now, wecoulddo the calculation and find that ...
122 An Introduction to Partial Differential Equations with MATLAB©R −3 −2 −1 1 2 3 y 1 −3 x 1 1 y 1 −2 −1 1 2 3 x (a) y = f(x) ( ...
Fourier Series 123 8.MATLAB:Following Example 1, Section 3.5, use MATLAB to graph the truncated Fourier sine and cosine series, ...
124 An Introduction to Partial Differential Equations with MATLAB©R 3.7 Completeness............................ So far, in our ...
Fourier Series 125 and got (3.15) as our set of eigenfunctions. Similarly, (3.16) are the eigen- functions of the problem y′′+λy ...
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