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Laplaces Equation Calculator

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Example
Created on 2024-06-20Asked by Abigail Perez (Solvelet student)
Solve Laplace's equation 2u=0 \nabla^2 u = 0 in two dimensions for a rectangular domain with boundary conditions u(0,y)=0 u(0, y) = 0 , u(a,y)=0 u(a, y) = 0 , u(x,0)=0 u(x, 0) = 0 , and u(x,b)=U u(x, b) = U .

Solution

We solve Laplace's equation 2u=0 \nabla^2 u = 0 in a rectangular domain 0xa 0 \leq x \leq a , 0yb 0 \leq y \leq b with the given boundary conditions using separation of variables. Assume u(x,y)=X(x)Y(y) u(x, y) = X(x)Y(y) . Then: 2u=2ux2+2uy2=X(x)Y(y)+X(x)Y(y)=0 \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) X(x)Y(y) : X(x)X(x)+Y(y)Y(y)=0 \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0 This implies: X(x)X(x)=Y(y)Y(y)=λ \frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = \lambda We solve the ordinary differential equations: X(x)λX(x)=0,Y(y)+λY(y)=0 X''(x) - \lambda X(x) = 0, \quad Y''(y) + \lambda Y(y) = 0 For X(x) X(x) : X(x)=Asin(nπxa) X(x) = A \sin\left(\frac{n\pi x}{a}\right) For Y(y) Y(y) : Y(y)=Bsinh(nπya) Y(y) = B \sinh\left(\frac{n\pi y}{a}\right) Thus, the solution is: u(x,y)=n=1Bnsin(nπxa)sinh(nπya) 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(x, b) = U : U=n=1Bnsin(nπxa)sinh(nπba) 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)=n=12Usinh(nπya)sinh(nπba)sin(nπxa) 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 2ϕ=0\nabla^2 \phi = 0 subject to the boundary conditions ϕ(0,y)=0\phi(0, y) = 0 and ϕ(a,y)=sin(πya)\phi(a, y) = \sin\left(\frac{\pi y}{a}\right),2. Determine the smallest positive integer that is divisible by both 1515 and 2020.,
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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