Solve Laplace's equation
Solve Laplace's equation, \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,0<x<a, 0<y<b\), (see (1) in Section 12.5) for a rectangular plate subject to the given boundary conditions.
$$ \begin{gathered} \left.\frac{\partial u}{\partial x}\right|_{x=0}=u(0, y), \quad u(\pi, y)=1 \\ u(x, 0)=0, \quad u(x, \pi)=0 \\ u(x, y)=\square+\sum_{n=1}^{\infty}(\square \end{gathered} $$
Homework Answers
Solve Laplace's equation on
Solve Laplace's equation on \(-\pi \leq x \leq \pi\) and \(0 \leq y \leq 1\),$$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 $$subject to periodic boundary conditions in \(x\),$$ \begin{aligned} u(-\pi, y) &=u(\pi, y) \\ \frac{\partial u}{\partial x}(-\pi, y) &=\frac{\partial u}{\partial x}(\pi, y) \end{aligned} $$and the Dirichlet conditions in \(y\),$$ u(x, 0)=h(x), \quad u(x, 1)=0 $$
14. Solve the P.D.E.
14. Solve the P.D.E.$$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 ; u(x, 0)=u(0, y)=0, u(x, 1)=\pi, u(1, y)=2 \pi $$$$ \text { 令 } u(x, y)=\phi(x, y)+\varphi(x, y) $$$$ u(x, y)=\sum_{n=1}^{\infty} \frac{2}{n \sinh n \pi}\left[1-(-1)^{n}\right](\sinh n \pi y \sin n \pi x+2 \sinh n \pi x \sin n \pi y) $$
Ou(x.y)@uxy)o for the temperature 2. Solve Laplace's equation distribution in a rectangular plate...
ou(x.y)@uxy)o for the temperature 2. Solve Laplace's equation distribution in a rectangular plate 0sx s1, 0sysl subject to the following conditions. (a) u(0,y)-0, uy)-0, u(x,0)-fx), u(x,I)-0 au (x,y) x, y y- o
ou(x.y)@uxy)o for the temperature 2. Solve Laplace's equation distribution in a rectangular plate 0sx s1, 0sysl subject to the following conditions. (a) u(0,y)-0, uy)-0, u(x,0)-fx), u(x,I)-0 au (x,y) x, y y- o
Consider the heat conduction problem
5. If \(f(x)=\left\{\begin{array}{cc}0 & -2<x<0 \\ x & 0<x<2\end{array} \quad\right.\)is periodio of period 4 , and whose Fourier series is given by \(\frac{a_{0}}{2}+\sum_{n=1}^{2}\left[a_{n} \cos \left(\frac{n \pi}{2} x\right)+b_{n} \sin \left(\frac{n \pi}{2} x\right)\right], \quad\) find \(a_{n}\)A. \(\frac{2}{n^{2} \pi^{2}}\)B. \(\frac{(-1)^{n}-1}{n^{2} \pi^{2}}\)C. \(\frac{4}{n^{2} \pi^{2}}\)D. \(\frac{2}{n \pi}\)\(\mathbf{E}_{1} \frac{2\left((-1)^{n}-1\right)}{n^{2} \pi^{2}}\)F. \(\frac{4}{n \pi}\)6. Let \(f(x)-2 x-l\) on \([0,2]\). The Fourier sine series for \(f(x)\) is \(\sum_{w}^{n} b_{n} \sin \left(\frac{n \pi}{2} x\right)\), What is \(b, ?\)A. \(\frac{4}{3 \pi}\)B. \(\frac{2}{\pi}\)C. \(\frac{4}{\pi}\)D. \(\frac{-4}{3 \pi}\)E. \(\frac{-2}{\pi}\)F. \(\frac{-4}{\pi}\)7. Let \(f(x)\) be periodic...
3. Consider the Laplace's equation on a rectangular domain subject to the following boundary conditions that...
3. Consider the Laplace's equation on a rectangular domain subject to the following boundary conditions that represents the steady-state heating of a plate. A temperature probe shows that (1/2, 1/4) = 0. Solve this problem using the method of separation of variables. (7) byllyy = 0 0 <I<41 and O y <21 U-(0,y)=0, 1-(41, y) = cos(2), 4(1,0) = cos(2), 4(1,2)=0. (total 25 marks
Solve the boundary value problem
Solve the boundary value problem $$ \begin{gathered} y^{\prime \prime \prime}=-\frac{1}{x} y^{\prime \prime}+\frac{1}{x^{2}} y^{\prime}+0.1\left(y^{\prime}\right)^{3} \\ y(1)=0 \quad y^{\prime \prime}(1)=0 \quad y(2)=1 \end{gathered} $$Use difference equations method. You can get help from matlab for solving the system.
Wave problem
Find the solution \(\boldsymbol{u}(\boldsymbol{x}, \boldsymbol{t})\) for the wave problem on a string of length \(\boldsymbol{L}=\pi\) with \(c^{2}=1\) and conditions given by:\(\left\{\begin{array}{l}u(0, t)=0, u(\pi, t)=0, \quad t>0 \\ u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=\sin x, 0<x<\pi\end{array}\right.\)
Frequency-domain sampling using Fourier series
Frequency-domain sampling. Consider the following discrete-time signal$$ x(n)= \begin{cases}a^{|n|}, & |n| \leq L \\ 0, & |n|>L\end{cases} $$where \(a=0.95\) and \(L=10\).(a) Compute and plot the signal \(x(n)\).(b) Show that$$ X(\omega)=\sum_{n=-\infty}^{\infty} x(n) e^{-j \omega n}=x(0)+2 \sum_{n-1}^{L} x(n) \cos \omega n $$Plot \(X(\omega)\) by computing it at \(\omega=\pi k / 100, k=0,1, \ldots, 100\).(c) Compute$$ c_{k}=\frac{1}{N} X\left(\frac{2 \pi}{N} K\right), \quad k=0,1, \ldots, N-1 $$for \(N=30\).(d) Determine and plot the signal$$ \tilde{x}(n)=\sum_{k=0}^{N-1} c k e^{j(2 \pi / N) k n} $$What is the...
7. Use the Alternating Series Test to determine the convergence or divergence of the series
7. Use the Alternating Series Test to determine the convergence or divergence of the series a) \(\sum_{n=1}^{\infty} \frac{(-1)^{n} \sqrt{n}}{2 n+1}\)b) \(\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{2 n-1}\)8. Use the Ratio Test or the Root Test to determine the convergence or divergence of the seriesa) \(\sum_{n=0}^{\infty}\left(\frac{4 n-1}{5 n+7}\right)^{n}\)b) \(\sum_{n=0}^{\infty} \frac{\pi^{n}}{n !}\)
10. [18 Marks] Using separation of variables, solve Laplace's equation for {(x,y): 0 < x < 2,0 < y < 2)...
10. [18 Marks] Using separation of variables, solve Laplace's equation for {(x,y): 0 < x < 2,0 < y < 2), subject to the boundary conditions 0 (0, y) = d(x, 2) 6 + cos(nz) = In your solution, you must consider all three cases for the separation constant λ.
10. [18 Marks] Using separation of variables, solve Laplace's equation for {(x,y): 0
ou(x.y)@uxy)o for the temperature 2. Solve Laplace's equation distribution in a rectangular plate 0sx s1, 0sysl subject to the following conditions. (a) u(0,y)-0, uy)-0, u(x,0)-fx), u(x,I)-0 au (x,y) x, y y- o
ou(x.y)@uxy)o for the temperature 2. Solve Laplace's equation distribution in a rectangular plate 0sx s1, 0sysl subject to the following conditions. (a) u(0,y)-0, uy)-0, u(x,0)-fx), u(x,I)-0 au (x,y) x, y y- o
3. Consider the Laplace's equation on a rectangular domain subject to the following boundary conditions that represents the steady-state heating of a plate. A temperature probe shows that (1/2, 1/4) = 0. Solve this problem using the method of separation of variables. (7) byllyy = 0 0 <I<41 and O y <21 U-(0,y)=0, 1-(41, y) = cos(2), 4(1,0) = cos(2), 4(1,2)=0. (total 25 marks
Solve the boundary value problem $$ \begin{gathered} y^{\prime \prime \prime}=-\frac{1}{x} y^{\prime \prime}+\frac{1}{x^{2}} y^{\prime}+0.1\left(y^{\prime}\right)^{3} \\ y(1)=0 \quad y^{\prime \prime}(1)=0 \quad y(2)=1 \end{gathered} $$Use difference equations method. You can get help from matlab for solving the system.
10. [18 Marks] Using separation of variables, solve Laplace's equation for {(x,y): 0 < x < 2,0 < y < 2), subject to the boundary conditions 0 (0, y) = d(x, 2) 6 + cos(nz) = In your solution, you must consider all three cases for the separation constant λ.
10. [18 Marks] Using separation of variables, solve Laplace's equation for {(x,y): 0
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