Question

In two dimensions suppose u=u(r,θ) satisfies Laplaces equation Δu=0 in the disk 0≤ r<2, and on the boundary it satisfies u(2,θ)=3sin2θ+1,0≤θ<2π. What is the value of u at the origin? Where do the maximum and minimum of u occur in the closed domain 0≤r≤2?

Answer 1

To find the value of u at the origin and determine the maximum and minimum of u in the given domain, we need to solve Laplace's equation in polar coordinates using the method of separation of variables. The boundary condition u(2,θ) = 3sin^2θ + 1 provides a relationship between θ and the value of u at r = 2. By superimposing the solutions to the separate differential equations, we can determine the value of u at the origin and the locations of the maximum and minimum of u.

Step-by-step explanation:

To find the value of u at the origin, we need to use the method of separation of variables to solve Laplace's equation in polar coordinates. We assume that u can be written as a product of two functions, R(r) and Θ(θ), such that u(r, θ) = R(r)Θ(θ). Substituting this into Laplace's equation, we obtain separate differential equations for R(r) and Θ(θ).

For the boundary condition u(2,θ) = 3sin^2θ + 1, we substitute r = 2 into the expression for u and set it equal to the boundary condition. This gives us a relationship between θ and the value of u at r = 2.

Since the Laplace equation is linear, we can superimpose the solutions to the separate differential equations to get the solution to the Laplace equation. Therefore, the value of u at the origin is determined by finding the coefficients of the individual solutions and substituting the origin coordinates into the resulting expression for u.