Podcast
Questions and Answers
Which method is most suitable for solving the given Laplace equation with a source term and zero boundary conditions on a square?
Which method is most suitable for solving the given Laplace equation with a source term and zero boundary conditions on a square?
- Direct integration
- Finite element method
- Using Green's functions
- Separation of variables with Fourier series expansion (correct)
Given the equation $-4u + 3u_{xx} + u_{yy} + sin(2x) = 0$ and assuming a solution of the form $u(x, y) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn}sin(mx)sin(ny)$, what equation must $A_{mn}$ satisfy?
Given the equation $-4u + 3u_{xx} + u_{yy} + sin(2x) = 0$ and assuming a solution of the form $u(x, y) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn}sin(mx)sin(ny)$, what equation must $A_{mn}$ satisfy?
- $(-4 - 3m^2 - n^2)A_{mn} + \int_{0}^{\pi} \int_{0}^{\pi} sin(2x) dxdy = 0$
- $(-4 + 3m^2 + n^2)A_{mn} = 0$
- $(-4 - 3m^2 - n^2)A_{mn} + \int_{0}^{\pi} \int_{0}^{\pi} sin(2x)sin(mx)sin(ny) dxdy = 0$ (correct)
- $(-4 - 3m^2 - n^2)A_{mn} = 0$
In solving $-4u + 3u_{xx} + u_{yy} + sin(2x) = 0$ with $u(x, y) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn}sin(mx)sin(ny)$, for what values of $m$ and $n$ will $\int_{0}^{\pi} sin(2x)sin(mx) dx$ be non-zero?
In solving $-4u + 3u_{xx} + u_{yy} + sin(2x) = 0$ with $u(x, y) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn}sin(mx)sin(ny)$, for what values of $m$ and $n$ will $\int_{0}^{\pi} sin(2x)sin(mx) dx$ be non-zero?
- For all integer values of $m$
- Only when $n = 0$
- Only when $m = 0$
- Only when $m = 2$ (correct)
Given the equation $-4u + 3u_{xx} + u_{yy} + sin(2x) = 0$ with zero boundary conditions, and the solution form $u(x, y) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn}sin(mx)sin(ny)$, which expression correctly describes $A_{2n}$?
Given the equation $-4u + 3u_{xx} + u_{yy} + sin(2x) = 0$ with zero boundary conditions, and the solution form $u(x, y) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn}sin(mx)sin(ny)$, which expression correctly describes $A_{2n}$?
If the solution to $-4u + 3u_{xx} + u_{yy} + sin(2x) = 0$ is given by $u(x, y) = \sum_{n=1}^{\infty} A_{2n}sin(2x)sin(ny)$ and $A_{2n} = \frac{c}{(-16 - n^2)}$, what happens to the amplitude of higher modes as n increases?
If the solution to $-4u + 3u_{xx} + u_{yy} + sin(2x) = 0$ is given by $u(x, y) = \sum_{n=1}^{\infty} A_{2n}sin(2x)sin(ny)$ and $A_{2n} = \frac{c}{(-16 - n^2)}$, what happens to the amplitude of higher modes as n increases?
Flashcards
Laplace Equation
Laplace Equation
A second-order partial differential equation that describes the steady-state distribution of a potential, such as temperature or voltage.
Solution to Laplace Equation
Solution to Laplace Equation
A function, u(x,y), whose Laplacian (sum of second-order partial derivatives) equals zero.
Boundary Conditions
Boundary Conditions
Conditions specifying the value of the unknown function on the boundary of the domain.
Given Problem
Given Problem
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Particular Solution
Particular Solution
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Study Notes
- The task is to solve the Laplace equation: -4u + 3u_xx + u_yy + sin(2x) = 0
- The equation needs to be solved on a square where 0 < x, y < π.
- The boundary conditions are zero, meaning u = 0 on the top, bottom, left, and right sides of the square.
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