A temperature field T(x,t) in a one-dimensional structure (x = [0, L]) is determined by 1...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
A temperature field T(x,t) in a one-dimensional structure (x = [0, L]) is determined by 1 ar 8²T the following governing equation Based on previous experience, one a êt proposes that the temperature field T(x,t) can be constructed as a form of expansion of cosine function cos(x): T(x,1)= cos(2x)T(2,1). (1) K-0 where An = (2n+1)z/(2L) (n=0, 1, 2, ...) are known. T(,1) can be viewed as the corresponding coefficients of the expansion in terms of cos(x). Using the orthogonality property of cos(x) in domain x = [0, L], one obtains [T(x,1) cos(2x)dx T(2,1)= (2) cos² (2,x)dx Shown in Eq. 1, the only thing unknown at this moment is the coefficient inside the expansion. One needs to solve for T(21) to finally determine the temperature field T(x, t). Please follow the steps described below: dT(2,1) dt inside the integration in Eq. 2. (B) Then based on the governing equation (Eq. 1), replace inside the integration (A) Derive the expression for ar from Eq. 2, you should end up with a ar ôt 0²T in Eq. 2 with a You should end up with an integration with inside. (Please note that ôt 2²T du (C) You then rearrange it using integration by parts twice to the derived integration in (B). In procedures of integration by parts, you will see an integration is modified as fudx = uvla-fv- dx, leave the first term without further derivation, continue to work on the 2nd term via performing integration by parts again. dx (D) Eventually, you should end up with an ordinary differential equation for T(2,1). If not, continue to check your derivations. at (x,t) ox may not be equal to - A sin(x)7(2)!!! Just follow the guideline mentioned, even if you may not understand completely why I ask you to do A temperature field T(x,t) in a one-dimensional structure (x = [0, L]) is determined by 1 ar 8²T the following governing equation Based on previous experience, one a êt proposes that the temperature field T(x,t) can be constructed as a form of expansion of cosine function cos(x): T(x,1)= cos(2x)T(2,1). (1) K-0 where An = (2n+1)z/(2L) (n=0, 1, 2, ...) are known. T(,1) can be viewed as the corresponding coefficients of the expansion in terms of cos(x). Using the orthogonality property of cos(x) in domain x = [0, L], one obtains [T(x,1) cos(2x)dx T(2,1)= (2) cos² (2,x)dx Shown in Eq. 1, the only thing unknown at this moment is the coefficient inside the expansion. One needs to solve for T(21) to finally determine the temperature field T(x, t). Please follow the steps described below: dT(2,1) dt inside the integration in Eq. 2. (B) Then based on the governing equation (Eq. 1), replace inside the integration (A) Derive the expression for ar from Eq. 2, you should end up with a ar ôt 0²T in Eq. 2 with a You should end up with an integration with inside. (Please note that ôt 2²T du (C) You then rearrange it using integration by parts twice to the derived integration in (B). In procedures of integration by parts, you will see an integration is modified as fudx = uvla-fv- dx, leave the first term without further derivation, continue to work on the 2nd term via performing integration by parts again. dx (D) Eventually, you should end up with an ordinary differential equation for T(2,1). If not, continue to check your derivations. at (x,t) ox may not be equal to - A sin(x)7(2)!!! Just follow the guideline mentioned, even if you may not understand completely why I ask you to do
Expert Answer:
Answer rating: 100% (QA)
To solve for the coefficient T11 in the given problem lets go through the steps again A Derive the e... View the full answer
Related Book For
Discrete Mathematics and Its Applications
ISBN: 978-0073383095
7th edition
Authors: Kenneth H. Rosen
Posted Date:
Students also viewed these accounting questions
-
The functions cos x and sin x are both unit vectors in C[-Ï, Ï] with inner product defined by (a) Show that cos x ¥ sin x. (b) Determine the value of ||cos x + sin x||2. = | f(x)g(x)dx
-
x = t sin t y = t 2 + t
-
A linear function f: X Y is continuous if and only if it is continuous at 0. A linear function f: X Y is bounded if there exists a constant M such that || f(x) || M ||x|| for every x X Note that...
-
Estimate the errors involved in Exercise 63, parts (a) and (b). How large should be in each case to guarantee an error of less than 0.00001?
-
As discussed in Chapter 1, the International Accounting Standards Board (IASB) develops accounting standards for many international companies. The IASB also has developed a conceptual framework to...
-
What interest rate will make $7,500 per year accumulate to $279,600 in 15 years?
-
A company starts in business on 1 January 19X1. You are to write up the motor vans account and the provision for depreciation account for the year ended 31 December 19X1 from the information given...
-
Presidio, Inc. produces one model of mountain bike. Partial information for the company follows: Required: 1. Complete Presidios cost data table. 2. Calculate Presidios contribution margin ratio and...
-
Discuss how accounting for human rights can help in giving visibility to the plight of the marginalised people and how it can display the social cost of corporate profits. Discuss how accounting...
-
Huimin Ltd. had the following account balances at December 2021 1 3 25 29 32 30 33 Huimin Ltd. had the following account balances at Decemb Account Name Dividends Prepaid expenses Interest...
-
Lester Company, as lessee, enters into a non-cancelable lease agreement for equipment on January 1, 2021. The following information about the lease is provided: 1. The term of the non-cancelable...
-
1. Suppose you plan to have $60,000 in 15 years from now and you can invest your savings at 6% compounded continuously. Assuming you can save the same amount of money each year, how much do you need...
-
How does the operating system manage memory allocation and deallocation in a multi-programming environment with considerations for virtual memory and paging strategies ?
-
What the process is for creating an Azure SQL database? How do you configure Elastic Pooling? Why would you want to configure Geo-Replication for an Azure SQL database?
-
What are the mechanisms employed by modern operating systems to enforce process synchronization and concurrency control, ensuring the integrity of shared resources in a multiprocessing environment ?
-
Deed Co. owns 2% of Beck Cosmetic Retailers. A property dividend by Beck consisted of merchandise with a fair value lower than the listed retail price. Deed in turn gave the merchandise to its...
-
Using Lean to Improve My Processes Many who learn the continuous improvement concepts of Lean state that they are built on common sense. Furthermore, we begin to see processes that we follow every...
-
If a and b are positive numbers, find the maximum value of f ( x ) = x a (9 x ) b on the interval 0 x 9.
-
Use resolution to show that the compound proposition (p q) (p q) (p q) (p q) is not satisfiable.
-
Show that every forest can be colored using two colors.
-
A says "I am not the spy," B says "I am not the spy," and C says "I am not the spy." Exercises 32-38 are puzzles that can be solved by translating statements into logical expressions and reasoning...
-
48 Use divisibility rules to determine if each of the following is divisible by 3.
-
24 Use divisibility rules to determine if each of the following is divisible by 2 .
-
Construct a truth table or Venn diagram to prove whether the following argument is valid. If the argument is valid, determine whether it is sound. If John Mayer played MTV unplugged, then some...
Study smarter with the SolutionInn App