MATH 414 Lecture 24

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Homework Questions

#13

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u'' + 2u' + 2u = 3 \cos{5t}}

by MoUC, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = A\,u'' + B\,u' + C\,u}

Convert to a "discrete" version of the equation - get difference equation: (backward difference)

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u''(t) \approx \frac{u'(t) - u'(t-h)}{h} = \frac{u(t) - 2u(t-h) + u(t-2h)}{h^2} \quad\overset{\rightarrow}{t=t+h}\quad \frac{u(t+h) - 2u(t) + u(t-h)}{h^2}}

Let's assume that the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} describes a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} -periodic function (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t+2\pi) = f(t)} )

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) \approx A \, \left( \frac{u(t+h) - 2u(t) + u(t-h)}{h^2} \right) + B \, \left( \frac{u(t) - u(t-h)}{h} \right) + C \, u(t)}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_k = \frac{2\pi}{n} \, k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_k = u(t_k)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \in 0, 1, \ldots, n} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t_k) = f_k \approx A \, \left( \frac{u_{k+1} - 2u_k + u_{k-1}}{h^2} \right) + B \, \left( \frac{u_k - u_{k-1}}{h} \right) + C \, u_k = \alpha u_{k+1} + \beta u_k + \gamma u_{k-1}}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} -periodic, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_k} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -periodic, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}_n[u_{k+1}]_j = \omega^j \, \mathcal{F}_n[u_k]_j = \omega^j \, \hat{u}_j} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}_n[u_{k-1}]_j = \omega^{-j} \, \mathcal{F}_n[u_k]_j = \omega^{-j} \, \hat{u}_j} .

Now we can solve the following for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{u}_j} and take the inverse discrete fourier transform:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}_j = \frac{1}{h^2} \, \left( \alpha \, \omega^j + \beta + \gamma \omega^{-j} \right) \, \hat{u}_j}


From Last Time

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\lambda) = \frac{1}{\sqrt{2\pi}} \, \int_a^b f(t) \, \mathrm{e}^{-i \, \lambda \, t} \,\mathrm{d}t} .

We change variables to go to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ 0, 2\pi \right]} instead of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ a,b \right]} : let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \frac{(t-a) \cdot 2\pi}{b-a}} , then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\lambda) = \frac{b-a}{2\pi} \cdot \frac{1}{\sqrt{2\pi}} \, \int_{0}^{2\pi} f \left( a + \frac{b-a}{2\pi} \, \theta \right) \, \mathrm{e}^{-i \, \lambda \, \left( a + \frac{b-a}{2\pi} \, \theta \right)} \,\mathrm{d}\theta}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\theta) = f \left( a + \frac{b-a}{2\pi} \, \theta \right)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\lambda) = \frac{b-a}{\sqrt{2\pi}} \, \frac{1}{2\pi} \, \int_{0}^{2\pi} F(\theta) \, \mathrm{e}^{-i\,\lambda \, a} \, \mathrm{e}^{-i \, \left( \frac{b-a}{2\pi} \right) \, \lambda \, \theta} \,\mathrm{d}\theta}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta = \frac{b-a}{2\pi} \, \lambda} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = \frac{2\pi}{b-a} \, \eta} . Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{f} \left( \frac{2\pi}{b-a} \, \eta \right) &= \frac{b-a}{\sqrt{2\pi}} \, \frac{\mathrm{e}^{-i \, \left( \frac{2\pi}{b-a} \, \eta \right) \, a}}{2\pi} \, \int_{0}^{2\pi} F(\theta) \, \mathrm{e}^{-i \, \eta \, \theta} \,\mathrm{d}\theta \\ &= \frac{1}{\sqrt{2\pi}} \, \left( b-a \right) \, \mathrm{e}^{-i \, \frac{2\pi\,a}{b-a} \, \eta} \, \frac{1}{2\pi} \, \int_{0}^{2\pi} F(\theta) \, \mathrm{e}^{-i\,\eta\,\theta} \,\mathrm{d}\theta \end{align}}


Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_j = a + \frac{b-a}{n} \, j} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_j = \frac{(t_j - a) \, 2\pi}{b-a} = \frac{2\pi}{n} \, j}

We have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i = f(t_i) = F \left( \frac{2\pi}{n} \, j \right)} . We know Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le j \le n-1}

Take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_k = \frac{2\pi}{b-a} \, k}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f} \left( \frac{2\pi}{b-a} \, k \right) = (\mbox{stuff}) \, \underbrace{\frac{1}{2\pi} \, \int_{0}^{2\pi} F(\theta) \, \mathrm{e}^{-i \, \theta \, k} \,\mathrm{d} \theta}_{c_k}}

Observe that since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_k = \frac{1}{n} \, \hat{y}_k} , we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f} \left( \frac{2\pi}{b-a} \, k \right) = \frac{b-a}{n \, \sqrt{2\pi}} \, \mathrm{e}^{-\frac{2\pi \, i \, k \, a}{b-a}} \hat{y}_k}

This is an approximation to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\lambda)} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_k = \frac{2\pi}{b-a} \, k}

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