Consider the equation Δυ = = Uxx + Uyy = f on D = [0, 1] × [0,1] u = 0 on aD. Approximate values of u at the set of grid points (xi, Yj), i, j = 0,...,n+1 will be denoted by Ui, j ≈ u(xi, yj). The grid points cover the domain D, so that, calling h the mesh-size, we have (n + 1)h = 1. A common second-order approximation of the Laplacian is (1) Ui+1,j + Ui−1,j + Ui, j+1 +Ui, j-1 - 4ui,j Au≈ (Au)i,j == h² Setting ui,o the form = Ui,n+1 = uo,j = Un+1,0 = where uT = (u1,1, U2,1, , un, 1, u1,2, An is an n² x n² matrix. (2) = 0, the discrete analog of equation (1) can be expressed in Anu = f, Un,n), ƒT = (f1,1, f2,1, · · ·, fn,1, f1,2, . . ., fn,n) and where For this assignment take n = 10, 20, 40 and some other values you consider interesting. For each value form the matrix An and use appropriate MATLAB commands to evaluate the eigenvalues of An. Consider the following questions: (1) What are the eigenvalues of the continuous Laplacian with vanishing Dirichlet boundary conditions? (The eigenvectors are of the form sin(kлx) sin(lπy).) (2) What are the minimum and maximum eigenvalues of An. How do they change with n? Do they seem to be converging to the eigenvalues of the continuous Laplacian? If so, at what rate? Is this expected?
Consider the equation Δυ = = Uxx + Uyy = f on D = [0, 1] × [0,1] u = 0 on aD. Approximate values of u at the set of grid points (xi, Yj), i, j = 0,...,n+1 will be denoted by Ui, j ≈ u(xi, yj). The grid points cover the domain D, so that, calling h the mesh-size, we have (n + 1)h = 1. A common second-order approximation of the Laplacian is (1) Ui+1,j + Ui−1,j + Ui, j+1 +Ui, j-1 - 4ui,j Au≈ (Au)i,j == h² Setting ui,o the form = Ui,n+1 = uo,j = Un+1,0 = where uT = (u1,1, U2,1, , un, 1, u1,2, An is an n² x n² matrix. (2) = 0, the discrete analog of equation (1) can be expressed in Anu = f, Un,n), ƒT = (f1,1, f2,1, · · ·, fn,1, f1,2, . . ., fn,n) and where For this assignment take n = 10, 20, 40 and some other values you consider interesting. For each value form the matrix An and use appropriate MATLAB commands to evaluate the eigenvalues of An. Consider the following questions: (1) What are the eigenvalues of the continuous Laplacian with vanishing Dirichlet boundary conditions? (The eigenvectors are of the form sin(kлx) sin(lπy).) (2) What are the minimum and maximum eigenvalues of An. How do they change with n? Do they seem to be converging to the eigenvalues of the continuous Laplacian? If so, at what rate? Is this expected?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 13AEXP
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help me with part (1) and (2) please. please also provide the matlab code
![Consider the equation
Δυ
= = Uxx + Uyy = f
on
D = [0, 1] × [0,1]
u = 0
on
aD.
Approximate values of u at the set of grid points (xi, Yj), i, j = 0,...,n+1 will be denoted by
Ui, j ≈ u(xi, yj).
The grid points cover the domain D, so that, calling h the mesh-size, we have (n + 1)h = 1.
A common second-order approximation of the Laplacian is
(1)
Ui+1,j + Ui−1,j + Ui, j+1
+Ui, j-1
-
4ui,j
Au≈ (Au)i,j
==
h²
Setting ui,o
the form
= Ui,n+1 = uo,j = Un+1,0
=
where uT
=
(u1,1, U2,1, , un, 1, u1,2,
An is an n² x n² matrix.
(2)
= 0, the discrete analog of equation (1) can be expressed in
Anu = f,
Un,n), ƒT = (f1,1, f2,1, · · ·, fn,1, f1,2, . . ., fn,n) and where
For this assignment take n = 10, 20, 40 and some other values you consider interesting. For each
value form the matrix An and use appropriate MATLAB commands to evaluate the eigenvalues of
An.
Consider the following questions:
(1) What are the eigenvalues of the continuous Laplacian with vanishing Dirichlet boundary
conditions? (The eigenvectors are of the form sin(kлx) sin(lπy).)
(2) What are the minimum and maximum eigenvalues of An. How do they change with n? Do
they seem to be converging to the eigenvalues of the continuous Laplacian? If so, at what
rate? Is this expected?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fafade1a9-1f29-4d84-b55a-a57e5a303c8b%2F23de29fa-cff4-4b22-bb0b-1874bdce9727%2Fwfa4sij_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the equation
Δυ
= = Uxx + Uyy = f
on
D = [0, 1] × [0,1]
u = 0
on
aD.
Approximate values of u at the set of grid points (xi, Yj), i, j = 0,...,n+1 will be denoted by
Ui, j ≈ u(xi, yj).
The grid points cover the domain D, so that, calling h the mesh-size, we have (n + 1)h = 1.
A common second-order approximation of the Laplacian is
(1)
Ui+1,j + Ui−1,j + Ui, j+1
+Ui, j-1
-
4ui,j
Au≈ (Au)i,j
==
h²
Setting ui,o
the form
= Ui,n+1 = uo,j = Un+1,0
=
where uT
=
(u1,1, U2,1, , un, 1, u1,2,
An is an n² x n² matrix.
(2)
= 0, the discrete analog of equation (1) can be expressed in
Anu = f,
Un,n), ƒT = (f1,1, f2,1, · · ·, fn,1, f1,2, . . ., fn,n) and where
For this assignment take n = 10, 20, 40 and some other values you consider interesting. For each
value form the matrix An and use appropriate MATLAB commands to evaluate the eigenvalues of
An.
Consider the following questions:
(1) What are the eigenvalues of the continuous Laplacian with vanishing Dirichlet boundary
conditions? (The eigenvectors are of the form sin(kлx) sin(lπy).)
(2) What are the minimum and maximum eigenvalues of An. How do they change with n? Do
they seem to be converging to the eigenvalues of the continuous Laplacian? If so, at what
rate? Is this expected?
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