1. Suppose that a metal rod of length 40 cm is heated to a temperature of 80 degrees Celsius, and the ends are kept at 0 degrees. Assume that the diffusion coefficient a = 1 You will get the following set of equations: ди ²u at əx² u(x,0) = 80 u(0, t) = 0, u(40, t) = 0 a. Jse separation of variables to solve this equation. Show all your work. b. Write the solution as an infinite sum. c. Sketch a graph of the solutions at t=0, t=1, t=10, t=50. d. Which point/points on the metal rod are the hottest as time increases. e. Approximately how long will it take for the entire rod to cool below 5 degrees Celsius? You may use a single term expansion of the series in your approximation. f. Describe following limit and find the solution: lim u(x, t). t→∞
1. Suppose that a metal rod of length 40 cm is heated to a temperature of 80 degrees Celsius, and the ends are kept at 0 degrees. Assume that the diffusion coefficient a = 1 You will get the following set of equations: ди ²u at əx² u(x,0) = 80 u(0, t) = 0, u(40, t) = 0 a. Jse separation of variables to solve this equation. Show all your work. b. Write the solution as an infinite sum. c. Sketch a graph of the solutions at t=0, t=1, t=10, t=50. d. Which point/points on the metal rod are the hottest as time increases. e. Approximately how long will it take for the entire rod to cool below 5 degrees Celsius? You may use a single term expansion of the series in your approximation. f. Describe following limit and find the solution: lim u(x, t). t→∞
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
CDEF PLease
Expert Solution
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 with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,