Problem 1: Consider an infinite slab of unit width. The differential equation describing the system is d?T dT dT 0 = +g(z) = 0 = 0 dz? dz dz z=1 g(z) = 10 – 20z 1) Show that the following function are eigenfunctions of the linear transformation: $,(z) = H, cos(@,z), where @, = (i – 1)x , H1 1, Н. 12. Determine the corresponding eignenvalues. 2) Using 4 eigenfunctions , (z) = H, cos(@,z), where a, = (i– 1)T , H1 = 1, H¡ = /2, i 2...4, approximate the differential equation with a set of algebraic equations using the Galerkin method. Report your answer in the form of Ax = b by identifying x, A and b
Problem 1: Consider an infinite slab of unit width. The differential equation describing the system is d?T dT dT 0 = +g(z) = 0 = 0 dz? dz dz z=1 g(z) = 10 – 20z 1) Show that the following function are eigenfunctions of the linear transformation: $,(z) = H, cos(@,z), where @, = (i – 1)x , H1 1, Н. 12. Determine the corresponding eignenvalues. 2) Using 4 eigenfunctions , (z) = H, cos(@,z), where a, = (i– 1)T , H1 = 1, H¡ = /2, i 2...4, approximate the differential equation with a set of algebraic equations using the Galerkin method. Report your answer in the form of Ax = b by identifying x, A and b
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
![Problem 1: Consider an infinite slab of unit width. The differential equation describing the
system is
d?T
dT
dT
= 0
dz? +8(z)
dz
dz
z=0
Iz=1
g(z) = 10– 20z
Show that the following function are eigenfunctions of the linear transformation:
$,(z) = H, cos(@,z), where @, = (i – 1)7, H1 = 1, H;
12. Determine the
corresponding eignenvalues.
2) Using 4 eigenfunctions o, (z) = H, cos(@,z), where w, = (i– 1)7, H1 = 1, H¡ = /2, i =
%3D
%3D
2...4, approximate the differential equation with a set of algebraic equations using the
Galerkin method. Report your answer in the form of Ax =b by identifying x, A and b
and indicating how an approximation of T(z) can be constructed from x. You should
find the following identity useful
cos(@z), z sin(@ z)
Szcos(@2)dz =
3) Show that the solution to the system of equations from part 1 is
T(z) = a, +(b, / n' )¢,(z)+(b, / 4n² )¢,(z)+(b, /9n² )¢,(z)
%3D
where b; are element of the vector b and a, is any arbitrary number. Using MATLAB,
plot T(z) with a, = 303.
4) Repeat part 1 with g(z) = 10 and show that the resulting set of algebraic equations has
no solution. Discuss this conclusion in the context of Example 2.13 from the book
(Graham and Rawlings).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcaa02cc4-b1fc-43d6-8c81-ebdaf1dd21d1%2F30340d90-c63d-401e-99f2-27eeb62f9d7b%2F4nm8t3l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1: Consider an infinite slab of unit width. The differential equation describing the
system is
d?T
dT
dT
= 0
dz? +8(z)
dz
dz
z=0
Iz=1
g(z) = 10– 20z
Show that the following function are eigenfunctions of the linear transformation:
$,(z) = H, cos(@,z), where @, = (i – 1)7, H1 = 1, H;
12. Determine the
corresponding eignenvalues.
2) Using 4 eigenfunctions o, (z) = H, cos(@,z), where w, = (i– 1)7, H1 = 1, H¡ = /2, i =
%3D
%3D
2...4, approximate the differential equation with a set of algebraic equations using the
Galerkin method. Report your answer in the form of Ax =b by identifying x, A and b
and indicating how an approximation of T(z) can be constructed from x. You should
find the following identity useful
cos(@z), z sin(@ z)
Szcos(@2)dz =
3) Show that the solution to the system of equations from part 1 is
T(z) = a, +(b, / n' )¢,(z)+(b, / 4n² )¢,(z)+(b, /9n² )¢,(z)
%3D
where b; are element of the vector b and a, is any arbitrary number. Using MATLAB,
plot T(z) with a, = 303.
4) Repeat part 1 with g(z) = 10 and show that the resulting set of algebraic equations has
no solution. Discuss this conclusion in the context of Example 2.13 from the book
(Graham and Rawlings).
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)