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Elliptic Equations Calculator

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Example
Created on 2024-06-20Asked by Benjamin Miller (Solvelet student)
Solve the elliptic equation ∂2u∂x2+∂2u∂y2=0 \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 .

Solution

To solve the elliptic equation ∂2u∂x2+∂2u∂y2=0 \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 , we need to find a function u(x,y) u(x, y) that satisfies this partial differential equation. This is the Laplace equation, and its solutions are harmonic functions. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Sofia Hall on Solvelet
1. Solve the Laplace's equation ∇2u=0 \nabla^2 u = 0 in two dimensions subject to the boundary conditions u(x,0)=0 u(x, 0) = 0 , u(x,1)=1 u(x, 1) = 1 , u(0,y)=0 u(0, y) = 0 , and u(1,y)=0 u(1, y) = 0 .2. Find the solution to the Poisson's equation ∇2u=−1 \nabla^2 u = -1 with the boundary conditions u(x,0)=0 u(x, 0) = 0 , u(x,1)=0 u(x, 1) = 0 , u(0,y)=0 u(0, y) = 0 , and u(1,y)=0 u(1, y) = 0 in two dimensions.
DefinitionThe partial differential equations known as elliptic equations represent an inheritance of the behaviour of harmonic functions or steady-state phenomena in general, evolving smooth and at the same time decaying spatially at infinity. Elliptic Equations - they are known for their elliptic operators, the most frequently appearing of which is the Laplacian, written usually as Laplaces equation or Poissons equation. Key applications of elliptic equations include heat conduction, electrostatics, fluid flow, and potential fields that are used in physics, engineering, and mathematical modelling. These are solved by way of boundary value problems (BVP) and numerical approximation techniques such as finite differences and finite elements. In this equation set, u is the temperature function and theequations describe: (i) Steady-state temperature distributions in a homogeneous medium withouth internal heat sources (Laplaces equation),.
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