2D Laplace equation

By: Prof. Dr. Fazal Rehman Shamil | Last updated: December 17, 2024

[latex]
\[
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
\]
\[
\text{where:}
\]
\begin{aligned}
& u(x, y) \text{ is the unknown function}, \\
& \frac{\partial^2 u}{\partial x^2} \text{ is the second partial derivative of } u \text{ with respect to } x, \\
& \frac{\partial^2 u}{\partial y^2} \text{ is the second partial derivative of } u \text{ with respect to } y.
\end{aligned}

Applications

1. Electrostatics: Solving for electric potential in charge-free regions.
2. Fluid Dynamics: Modeling incompressible and irrotational fluid flow.
3. Heat Transfer: Steady-state heat distribution in a 2D region.
4. Structural Mechanics: Analyzing stress distributions.

Boundary Conditions

The Laplace equation is solved using boundary conditions, which could be:

1. Dirichlet Boundary Condition: Specifies the value of
   $u$ 

   $u$ on the boundary.

2. Neumann Boundary Condition: Specifies the normal derivative of
   $u$ 

   $u$ on the boundary.

3. Mixed Boundary Condition: Combines both Dirichlet and Neumann conditions.

Solution Techniques

1. Separation of Variables: Used for simple geometries with well-defined boundaries.
2. Finite Difference Method (FDM): Discretizes the domain into a grid and approximates derivatives.
3. Finite Element Method (FEM): Breaks the domain into elements for complex geometries.
4. Fourier Transform Methods: Useful for periodic boundary conditions.