2. Find the solution to the following Laplace equation Vu -0, 0
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![2. Find the solution to the following Laplace equation
Vu = 0, 0<r< 3, 0<y<1
where the boundary conditions are given by
du (0. y) = 0,
du
(3, y) = 0,
(x,0) = 0, u(x, 1) =1- cos (z2).
ду
(a) Use separation of variables u(z, y) = X(2)Y(y) to write ordinary differential equations for X(r)
and Y(y) in terms of a separation constant p.
(b) The homogeneous boundary conditions for the problem in the r-direction constrain the possible
values of p and functional forms of X(x). What are these possible values and functions?
(c) Caleulate the form of Y (y), and write down the general form of u(z, y) as a superposition of the
eigenfunetions.
(d) Enforce the boundary condition at y = 1, and determine what the coefficients must be using
orthogonality relations. Express your final answer in terms of u(r, y).
[Hint: There will only be two terms in the final answer.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F405e1fa1-2ca4-40d3-9b13-49de31f46477%2F875e2c50-c1ba-492e-b864-68b52dae4f23%2Fb6wd9hj_processed.png&w=3840&q=75)
Transcribed Image Text:2. Find the solution to the following Laplace equation
Vu = 0, 0<r< 3, 0<y<1
where the boundary conditions are given by
du (0. y) = 0,
du
(3, y) = 0,
(x,0) = 0, u(x, 1) =1- cos (z2).
ду
(a) Use separation of variables u(z, y) = X(2)Y(y) to write ordinary differential equations for X(r)
and Y(y) in terms of a separation constant p.
(b) The homogeneous boundary conditions for the problem in the r-direction constrain the possible
values of p and functional forms of X(x). What are these possible values and functions?
(c) Caleulate the form of Y (y), and write down the general form of u(z, y) as a superposition of the
eigenfunetions.
(d) Enforce the boundary condition at y = 1, and determine what the coefficients must be using
orthogonality relations. Express your final answer in terms of u(r, y).
[Hint: There will only be two terms in the final answer.]
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