(i) Consider the LU factorization 0 0 [b, C1 a2 b2 C2 0 0 a3 bz C3 as b, 14 0 0 01[1 u 0 0 01 U2 0 1 U3 1 (1.1) %3D 0 a3 l3 lo o 0 0 a, 4JL0 o 0 Work out the product on the right hand side to show that the result has the same non-zero structure as the left hand side. Hence, show that the elements ly and uu are given by the following formulae. l1 = b1, ui =7, 4+1 = b+1 - a¿+1U¡, i = 1,2,3. (1.2) %3D
(i) Consider the LU factorization 0 0 [b, C1 a2 b2 C2 0 0 a3 bz C3 as b, 14 0 0 01[1 u 0 0 01 U2 0 1 U3 1 (1.1) %3D 0 a3 l3 lo o 0 0 a, 4JL0 o 0 Work out the product on the right hand side to show that the result has the same non-zero structure as the left hand side. Hence, show that the elements ly and uu are given by the following formulae. l1 = b1, ui =7, 4+1 = b+1 - a¿+1U¡, i = 1,2,3. (1.2) %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Question 1 [25 marks]
(i) Consider the LU factorization
[b C1
0 0
0 0
0 a3
[1 u1
1
01
U2 0
1
a2 b2
C2 0
az l2
(1.1)
bz C3
0 0
0 az
lo o
Uz
1.
Work out the product on the right hand side to show that the result has the same
non-zero structure as the left hand side. Hence, show that the elements l, and uu
are given by the following formulae.
l1 = b1, uj =4, l+1 = bi+1– ai+1ui, i = 1,2,3.
(1.2)
(ii) Consider the differential equation
y"(x) = y(x)
(1.3)
with domain 0 <x< 2 and boundary values y(0) = 1, y(2) = 0.25. If we use the
central finite difference formula for the second derivative to discretise the differential
equation and take the step size to be h = 0.4, show the discretization results in
following system of equations:
图-L
[2.25
-1
0 0
1
2.25
-1 0
2.25
y2
-1
0 -1
0 0
(1.4)
-1
Уз
-1
2.25ly4
L0.0820850I
Use the formulas (1.2) to find the LU factorisation of the matrix in (1.4). Use 4
significant figures for your calculations.
Then, solve the system using LU factorization and show that
y1 = 0.609427,y2 = 0.371211, y3 = 0.225799, y4 = 0.136837.
(1.5)
(ii) Show that the exact solution to equation (1.3) which also satisfies the boundary
conditions y(0) and y(2) is given by y(x) = e-x. Use this result to determine the
discretization error in the numerical solution obtained in part (ii).
kel](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d9b2fdc-aca5-49e4-9c03-ab242757ddbb%2F61b100a2-c088-49ca-aeb2-bb36ce3e7e6f%2F3ymj5uw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 1 [25 marks]
(i) Consider the LU factorization
[b C1
0 0
0 0
0 a3
[1 u1
1
01
U2 0
1
a2 b2
C2 0
az l2
(1.1)
bz C3
0 0
0 az
lo o
Uz
1.
Work out the product on the right hand side to show that the result has the same
non-zero structure as the left hand side. Hence, show that the elements l, and uu
are given by the following formulae.
l1 = b1, uj =4, l+1 = bi+1– ai+1ui, i = 1,2,3.
(1.2)
(ii) Consider the differential equation
y"(x) = y(x)
(1.3)
with domain 0 <x< 2 and boundary values y(0) = 1, y(2) = 0.25. If we use the
central finite difference formula for the second derivative to discretise the differential
equation and take the step size to be h = 0.4, show the discretization results in
following system of equations:
图-L
[2.25
-1
0 0
1
2.25
-1 0
2.25
y2
-1
0 -1
0 0
(1.4)
-1
Уз
-1
2.25ly4
L0.0820850I
Use the formulas (1.2) to find the LU factorisation of the matrix in (1.4). Use 4
significant figures for your calculations.
Then, solve the system using LU factorization and show that
y1 = 0.609427,y2 = 0.371211, y3 = 0.225799, y4 = 0.136837.
(1.5)
(ii) Show that the exact solution to equation (1.3) which also satisfies the boundary
conditions y(0) and y(2) is given by y(x) = e-x. Use this result to determine the
discretization error in the numerical solution obtained in part (ii).
kel
Transcribed Image Text:(iv) Using centred-differences, express the partial differential equation
a'u
x² + y?, for 0 < x < 0.75, 0 < y < 0.75
(1.6)
ax2
ду?
0, u(0, y)
0, u(x, 0.75) =
subject to the boundary conditions u(x,0)
0.75x and u(0.75, y) = y² in finite difference form with h = 0.25. Make sure
%3D
%3D
your system of equations has the form:
01 ru11
U21
U12
(Each "*" represents a number and these numbers can be different.) Draw a
diagram of the grid, label the grid points, and put the boundary values on the grid.
Note that you do not have to solve the system.
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