Reference no: EM131027821
Let u(x,t) describe the temperature of a thin metal ring with circumference 2pi. For convenience, let's orient the ring so that x spans the interval |-pi, pi|. Suppose that the ring has some internal heating that is angle-dependent, so that u(x, t) satisfies the inhomogeneous heat equation
u_t = ku_zz + f(x),
where k is the thermal diffusivity and f(x) describes the internal heating. Furthermore, assume that the temperature of the ring is initially zero.
a) Because the temperature u(x, t) is parameterized by the angle x, the temperature must be a 2pi-periodic function. This suggests that we should write the temperature as a complex Fourier expansion with time-dependent coefficients,
u(x, t) = SUM (A_n)e^(inz).
Substitue this expression into the heat equation, and obtain a differential equation for each (A_n)(t). Using the correct initial conditions, solve each ODE and write down the final solution.
b) Suppose f(x) = cos^2 (x). Find an explicit solution for u(x, t). (Hint: Expand cos^2(x) into a finite number of complex Fourier modes, and show that all but three of the A_n are always zero.)
|
Write a half page about race and iq
: Please read the articles and write a half page about it single space. Let's Talk about Race and IQ
|
|
Any perceived challenges in their hiring practices
: South Florida is a very diverse metropolitan area. As a Human Resource professional, what are the benefits of diversity to a South Florida organization? Are there any perceived challenges in their hiring practices? Why or why not? How would this dive..
|
|
Forecasted sales and actual sales and have bonus calculated
: The bonus, B, paid to a company’s franchise managers is based on the following formulae: Can you model the formulae above in Excel…so that your “employees” can enter their Forecasted Sales and their Actual Sales and have their bonuses calculated (in ..
|
|
Calculate the total annual cost of the system
: ABC Company has a demand for 2000 units each year. The cost of each unit is $100. It costs ABC $200 to place an order and $4 to carry a unit in the stock pear year. Further, the supplier needs week and a half to deliver their order. Assume 50 week..
|
|
Describe the temperature of a thin metal ring
: Suppose f(x) = cos^2 (x). Find an explicit solution for u(x, t). (Hint: Expand cos^2(x) into a finite number of complex Fourier modes, and show that all but three of the A_n are always zero.)
|
|
Solutions for smith organization
: Develop and provide some solutions for Smith's organization of how he can improve service-related issues as experienced by his customers. Be sure to include in your answer the following issues:
|
|
Leadership performance on concept of triple bottom line
: Evaluate Seth Goldman’s leadership performance on the concept of the “Triple Bottom Line” using at least two (2) examples from both resources and some examples to support your evaluation.
|
|
How the notions of leadership as meaning-making
: Consider the problems that arise from the case and how the notions of leadership as meaning-making, or sense-making, and resilience might be applied
|
|
What is the marginal rate of transformation impact
: What would be the production possibility frontiers for Brazil and the United States - Without trade, the United States produces AND CONSUMES 32,500 units of clothing and 125,000 cans of soda.
|