(a) Consider the following system of linear equations 2x +6y - z= 18 x+ y+5z =-1.5 7x +2y + z=11 Write the equations in diagonally dominant form. Hence use 3 iterations of the Gauss-Seidel method to find approximate solutions for x, y, z. Use initial estimates of x, = y, = z, =0 and work to 4 decimal places of accuracy throughout.
(a) Consider the following system of linear equations 2x +6y - z= 18 x+ y+5z =-1.5 7x +2y + z=11 Write the equations in diagonally dominant form. Hence use 3 iterations of the Gauss-Seidel method to find approximate solutions for x, y, z. Use initial estimates of x, = y, = z, =0 and work to 4 decimal places of accuracy throughout.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Q4. (a)
Consider the following system of linear equations
2x +6y - z= 18
X+ y+5z =-1.5
7x+2y + z=11
Write the equations in diagonally dominant form. Hence use 3
iterations of the Gauss-Seidel method to find approximate solutions for
x, y, z. Use initial estimates of x, = Yo = z, = 0 and work to 4 decimal
places of accuracy throughout.
(b)
The heat conduction equation which describes the temperature in °C,
T(x,t), of a particular metal rod at position x and time t is
ат
0.15
at
The rod is of length 2m and is held at 0°C until timet =0. Two heat
sources are introduced at time t = 0, one of 5°C at the end x = 0 and
one of 70°C at the end x = 2m.
(1)
Approximate the above heat conduction equation with finite
differences, using a uniform time step of k and uniform spacing h
in the x-direction. Show that with a spacing of h=1m and time
steps of k = 2s, the difference scheme is
T.j = 0.3 (T + T,)+ 0.47,
(ii)
x = 1m when t= 4s.
Use this difference scheme to estimate the temperature at](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbedbf054-b02b-435a-b47b-fc754fa6df7c%2Fa7a7029c-70ca-42de-9a5e-51a6cc28cfa4%2Fd5y0r9h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q4. (a)
Consider the following system of linear equations
2x +6y - z= 18
X+ y+5z =-1.5
7x+2y + z=11
Write the equations in diagonally dominant form. Hence use 3
iterations of the Gauss-Seidel method to find approximate solutions for
x, y, z. Use initial estimates of x, = Yo = z, = 0 and work to 4 decimal
places of accuracy throughout.
(b)
The heat conduction equation which describes the temperature in °C,
T(x,t), of a particular metal rod at position x and time t is
ат
0.15
at
The rod is of length 2m and is held at 0°C until timet =0. Two heat
sources are introduced at time t = 0, one of 5°C at the end x = 0 and
one of 70°C at the end x = 2m.
(1)
Approximate the above heat conduction equation with finite
differences, using a uniform time step of k and uniform spacing h
in the x-direction. Show that with a spacing of h=1m and time
steps of k = 2s, the difference scheme is
T.j = 0.3 (T + T,)+ 0.47,
(ii)
x = 1m when t= 4s.
Use this difference scheme to estimate the temperature at
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