Solved examples: Laplace's Equation on a square plate with Dirichlet conditions.

📚 Quick Study Guide

• 📐 Laplace's Equation: $\nabla^2 u = 0$, where $u$ is a scalar function. In 2D Cartesian coordinates: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$.
• 🔥 Dirichlet Boundary Conditions: Specifies the value of the function $u$ on the boundary of the domain. For a square plate, this means knowing the temperature (or potential) on all four sides.
• ➗ Separation of Variables: A common technique to solve Laplace's equation. Assume a solution of the form $u(x, y) = X(x)Y(y)$, substitute into the equation, and separate variables to obtain two ordinary differential equations.
• ➕ Superposition Principle: If $u_1$ and $u_2$ are solutions to Laplace's equation, then $c_1u_1 + c_2u_2$ is also a solution, where $c_1$ and $c_2$ are constants. This allows us to combine solutions to satisfy complex boundary conditions.
• 💡 Solution Form: For a square plate $0 \le x \le a$, $0 \le y \le a$, the general solution often takes the form: $u(x,y) = \sum_{n=1}^{\infty} A_n \sin(\frac{n\pi x}{a}) \sinh(\frac{n\pi y}{a})$, where $A_n$ are coefficients determined by the boundary conditions.

Practice Quiz

1. Question 1: What is the general form of Laplace's equation in two dimensions?
1. A. $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$
2. B. $\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0$
3. C. $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
4. D. $\frac{\partial u}{\partial x} - \frac{\partial u}{\partial y} = 0$


2. Question 2: What type of boundary condition specifies the value of the function on the boundary?
1. A. Neumann condition
2. B. Robin condition
3. C. Dirichlet condition
4. D. Cauchy condition


3. Question 3: Which method is commonly used to solve Laplace's equation?
1. A. Integration by parts
2. B. Separation of variables
3. C. Euler's method
4. D. Newton's method


4. Question 4: What principle allows us to combine multiple solutions of Laplace's equation?
1. A. Principle of least squares
2. B. Superposition principle
3. C. Uncertainty principle
4. D. Principle of inclusion-exclusion


5. Question 5: Consider a square plate with side length $a$. Which of the following is a possible solution form for Laplace's equation with Dirichlet conditions?
1. A. $u(x,y) = \sum_{n=1}^{\infty} A_n \cos(\frac{n\pi x}{a}) \cos(\frac{n\pi y}{a})$
2. B. $u(x,y) = \sum_{n=1}^{\infty} A_n \sin(\frac{n\pi x}{a}) \cosh(\frac{n\pi y}{a})$
3. C. $u(x,y) = \sum_{n=1}^{\infty} A_n \sin(\frac{n\pi x}{a}) \sinh(\frac{n\pi y}{a})$
4. D. $u(x,y) = \sum_{n=1}^{\infty} A_n \cos(\frac{n\pi x}{a}) \sinh(\frac{n\pi y}{a})$


6. Question 6: If the boundary conditions on a square plate are all zero, what is the solution to Laplace's equation?
1. A. $u(x,y) = x + y$
2. B. $u(x,y) = xy$
3. C. $u(x,y) = 0$
4. D. $u(x,y) = 1$


7. Question 7: What does $\nabla^2$ represent in Laplace's equation?
1. A. Gradient
2. B. Curl
3. C. Divergence
4. D. Laplacian