Heat Equation Time dependant heat loss

In summary, the problem involves solving the heat equation, given specific boundary and initial conditions. The steady state solution is zero and to find the time-dependent solution, you can try finding functions f(t) and g(x) that satisfy the homogeneous heat equation. Then, use the boundary conditions to solve for the time-dependent solution.
  • #1
PAR
30
0

Homework Statement


Solve the heat equation: [tex]u_{t} = u_{xx} - u - x*e^{-t}[/tex]
BC: u(0,t) = 0, u(1,t) = 0
IC: u(x,0) = x


Homework Equations





The Attempt at a Solution



The only progress I've made so far is figuring out that the steady state solution is zero. Other than that I don't know where to start with the time dependent solution. Basically I could use some help starting this problem, thanks!
 
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  • #2
Try finding functions [itex]f(t)[/itex] and [itex]g(x)[/itex] such that [itex]w(x,t) \equiv f(t)u(x,t)+g(x)[/itex] satisfies the homogeneous heat equation [itex]w_t=w_{xx}[/itex]...then use your boundary conditions for u(x,t) to find corresponding BCs for w(x,t) and solve for w(x,t) and use that to find u(x,t).
 

1. What is the heat equation?

The heat equation is a mathematical formula that describes how heat is distributed and transferred in a given system or material over time. It takes into account factors like temperature, heat capacity, and thermal conductivity to determine the rate of heat transfer.

2. How is the heat equation used in science?

The heat equation is used in various scientific fields, including physics, engineering, and meteorology, to study and solve problems related to heat transfer. It is especially useful in analyzing systems with changing temperatures over time, such as in heat loss calculations.

3. What is time-dependent heat loss?

Time-dependent heat loss refers to the gradual loss of heat from a system or material over time. This can be caused by factors like temperature differences, thermal conductivity, and surface area. The heat equation can be used to calculate the rate of heat loss and predict how much heat will be lost at a certain time.

4. How does the heat equation account for time in heat loss calculations?

The heat equation includes a time variable, which allows for the calculation of heat loss at different points in time. This is important because the rate of heat loss can vary over time, especially in systems with changing temperatures or thermal conductivity.

5. What are some real-world applications of the heat equation in time-dependent heat loss?

The heat equation is used in many real-world scenarios, such as designing efficient heating and cooling systems, predicting the spread of heat in buildings or structures, and understanding the effects of climate change on global temperature patterns. It is also used in industries like manufacturing and transportation to optimize energy usage and reduce heat loss.

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