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 \(y\),
$$ u(x, 0)=h(x), \quad u(x, 1)=0 $$
Homework Answers
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.$$ \begin{gathered} \left.\frac{\partial u}{\partial x}\right|_{x=0}=u(0, y), \quad u(\pi, y)=1 \\ u(x, 0)=0, \quad u(x, \pi)=0 \\ u(x, y)=\square+\sum_{n=1}^{\infty}(\square \end{gathered} $$
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...
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
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
Use the Laplace transform to solve the given system of differential equations.
Use the Laplace transform to solve the given system of differential equations.$$ \begin{aligned} &\frac{d x}{d t}=x-2 y \\ &\frac{d y}{d t}=5 x-y \\ &x(0)=-1, \quad y(0)=5 \end{aligned} $$
Problem 2. (15 points) Solve the following Laplace's equation in a cube as outlined below. au...
Problem 2. (15 points) Solve the following Laplace's equation in a cube as outlined below. au au au 2,2 + a2 + a2 = 0, on 0<x<1, 0<y<1, 0<?<1, (0, y, z) = (1, y, z) = 0, (x, 0, 2) = u(x, 1, ) = 0, (x, y,0) = 0, u(x,y, 1) = x. (a) Seek a solution of the form u(x, y, z) = F(x) G(v) H(-). Show that with the appropriate choice of separation constants, you can...
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a...
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a given periodic function with satisfying f (0) = f (2π) and Moreover, u(r,0 is bounded for r s a Which of the following is the (general) solution Select one: A. where for B. where )cos(n)de and for C. where and 2m for n- 1,2,3, D. where Co E R f(0) cos(n0)de and for
Let a >0 Solve the following Laplace's equation...
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:\(\left\{\begin{array}{l}u(0, t)=0, u(\pi, t)=0, \quad t>0 \\ u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=\sin x, 0<x<\pi\end{array}\right.\)
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
Problem 2 (Chapter 7; 60 points) Solve the following Laplace's equation in a parallelepiped, care...
Problem 2 (Chapter 7; 60 points) Solve the following Laplace's equation in a parallelepiped, carefully explaining all steps. Make sure to check for zero eigenvalues. Note that the single non-homogeneous boundary condition is on the y-plane (y D). The boundary conditions are simple enough so that all Fourier coefficient integrals are easily calculated. You don't have to give details in the solution of the two LHBVPs, but make sure to check all boundary conditions after you find the solution: ou...
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
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
Problem 2. (15 points) Solve the following Laplace's equation in a cube as outlined below. au au au 2,2 + a2 + a2 = 0, on 0<x<1, 0<y<1, 0<?<1, (0, y, z) = (1, y, z) = 0, (x, 0, 2) = u(x, 1, ) = 0, (x, y,0) = 0, u(x,y, 1) = x. (a) Seek a solution of the form u(x, y, z) = F(x) G(v) H(-). Show that with the appropriate choice of separation constants, you can...
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a given periodic function with satisfying f (0) = f (2π) and Moreover, u(r,0 is bounded for r s a Which of the following is the (general) solution Select one: A. where for B. where )cos(n)de and for C. where and 2m for n- 1,2,3, D. where Co E R f(0) cos(n0)de and for
Let a >0 Solve the following Laplace's equation...
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
Problem 2 (Chapter 7; 60 points) Solve the following Laplace's equation in a parallelepiped, carefully explaining all steps. Make sure to check for zero eigenvalues. Note that the single non-homogeneous boundary condition is on the y-plane (y D). The boundary conditions are simple enough so that all Fourier coefficient integrals are easily calculated. You don't have to give details in the solution of the two LHBVPs, but make sure to check all boundary conditions after you find the solution: ou...
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