Laplaces Equation Calculator

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Example

Created on 2024-06-20Asked by Abigail Perez (Solvelet student)

Solve Laplace's equation

\nabla^2 u = 0

in two dimensions for a rectangular domain with boundary conditions

u(0, y) = 0

u(a, y) = 0

u(x, 0) = 0

, and

u(x, b) = U

Solution

We solve Laplace's equation

\nabla^2 u = 0

in a rectangular domain

0 \leq x \leq a

0 \leq y \leq b

with the given boundary conditions using separation of variables. Assume

u(x, y) = X(x)Y(y)

. Then:

\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = X''(x)Y(y) + X(x)Y''(y) = 0

Divide by

X(x)Y(y)

\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0

This implies:

\frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = \lambda

We solve the ordinary differential equations:

X''(x) - \lambda X(x) = 0, \quad Y''(y) + \lambda Y(y) = 0

For

X(x)

X(x) = A \sin\left(\frac{n\pi x}{a}\right)

For

Y(y)

Y(y) = B \sinh\left(\frac{n\pi y}{a}\right)

Thus, the solution is:

u(x, y) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{a}\right) \sinh\left(\frac{n\pi y}{a}\right)

Using the boundary condition

u(x, b) = U

U = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{a}\right) \sinh\left(\frac{n\pi b}{a}\right)

Therefore, the solution to Laplace's equation is:

u(x, y) = \sum_{n=1}^{\infty} \frac{2U \sinh\left(\frac{n\pi y}{a}\right)}{\sinh\left(\frac{n\pi b}{a}\right)} \sin\left(\frac{n\pi x}{a}\right)

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Sebastian Young on Solvelet

1. Solve Laplace's equation

\nabla^2 \phi = 0

subject to the boundary conditions

\phi(0, y) = 0

and

\phi(a, y) = \sin\left(\frac{\pi y}{a}\right)

,2. Determine the smallest positive integer that is divisible by both

15

and

20

DefinitionThe second is Laplaces equation, which is a second-order partial differential equation: ∇2ϕ=0, where the laplacian is ∇2. It models steady-state heat conduction, electrostatics, and incompressible fluid flow. Harmonic functions- solutions to Laplaces equation- occur in the study of various phenomena in physics and engineering. For example, in two dimensions (x and y), Laplaces equation is ∂ϕ∂x2+∂ϕ∂y2=0. A solution is ϕ(x,y)=x2−y2.

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