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3. (20 points) Let an aluminum rod of length 20cm be initially at the uniform temper- ature of 25°C. Assume the thermal diffusivity of the aluminum is a² Suppose that at time t = 0, the end x = = 0.8(cm²/s). 0 of the rod is cooled to 10°C while the end x = 20 is heated to 60°C, and both are thereafter maintained at those temperatures. (a) (5 points) According to the description in the problem, set up the corresponding heat equation of the temperature u = u(x,t) with suitable initial and boundary conditions. (b) (15 points) Solve the heat equation to find the temperature u.

3. (20 points) Let an aluminum rod of length 20cm be initially at the uniform temper- ature of 25°C. Assume the thermal diffusivity of the aluminum is a² Suppose that at time t = 0, the end x = = 0.8(cm²/s). 0 of the rod is cooled to 10°C while the end x = 20 is heated to 60°C, and both are thereafter maintained at those temperatures. (a) (5 points) According to the description in the problem, set up the corresponding heat equation of the temperature u = u(x,t) with suitable initial and boundary conditions. (b) (15 points) Solve the heat equation to find the temperature u.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 18EQ
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3. (20 points) Let an aluminum rod of length 20cm be initially at the uniform temper- ature of 25°C. Assume the thermal diffusivity of the aluminum is a² Suppose that at time t = 0, the end x = = 0.8(cm²/s). 0 of the rod is cooled to 10°C while the end x = 20 is heated to 60°C, and both are thereafter maintained at those temperatures. (a) (5 points) According to the description in the problem, set up the corresponding heat equation of the temperature u = u(x,t) with suitable initial and boundary conditions. (b) (15 points) Solve the heat equation to find the temperature u.
Transcribed Image Text:3. (20 points) Let an aluminum rod of length 20cm be initially at the uniform temper- ature of 25°C. Assume the thermal diffusivity of the aluminum is a² Suppose that at time t = 0, the end x = = 0.8(cm²/s). 0 of the rod is cooled to 10°C while the end x = 20 is heated to 60°C, and both are thereafter maintained at those temperatures. (a) (5 points) According to the description in the problem, set up the corresponding heat equation of the temperature u = u(x,t) with suitable initial and boundary conditions. (b) (15 points) Solve the heat equation to find the temperature u.
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