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Engineering Error Questions Matlab Maple
Question 1
Approximate the solution to Laplace's equation on [0, 1] × [0, 1] with the Dirichlet boundary condition:
with h = 0.25.
Answer:
u(0.25, 0.25) ≈ 0.5
u(0.25, 0.5) ≈ 0.625
u(0.25, 0.75) ≈ 0.5
u(0.5, 0.25) ≈ 0.375
u(0.5, 0.5) ≈ 0.5
u(0.5, 0.75) ≈ 0.375
u(0.75, 0.25) ≈ 0.5
u(0.75, 0.5) ≈ 0.625
u(0.75, 0.75) ≈ 0.5
Question 2
Given the same boundary conditions as in Question 1, approximate the solution to Poisson's equation with g = 16.
Answer:
u(0.25, 0.25) ≈ -0.1875
u(0.25, 0.5) ≈ -0.25
u(0.25, 0.75) ≈ -0.1875
u(0.5, 0.25) ≈ -0.5
u(0.5, 0.5) ≈ -0.625
u(0.5, 0.75) ≈ -0.5
u(0.75, 0.25) ≈ -0.1875
u(0.75, 0.5) ≈ -0.25
u(0.75, 0.75) ≈ -0.1875
Question 3
Approximate the solution to Poisson's equation uxx + uyy = 1 given the boundary condition ∂u = 0 on the square [3, 5] × [4, 6] with h = 0.25.
Hint: use symmetry to reduce this to a system with 10 equations. Be sure to use Matlab.
Answer:
u(3.25, 4.25) ≈ -0.017779
u(3.25, 4.5) ≈ -0.027746
u(3.25, 4.75) ≈ -0.032916
u(3.25, 5) ≈ -0.034524
u(3.5, 4.5) ≈ -0.044663
u(3.5, 4.75) ≈ -0.053768
u(3.5, 5) ≈ -0.056641
u(3.75, 4.75) ≈ -0.065229
u(3.75, 5) ≈ -0.068876
u(4, 5) ≈ -0.072783
Question 4
Given the boundary conditions in Question 3, what is the solution if we were solving Laplace's equation?
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