Finding u(x) for the 1-D Heat Equation

In summary, the problem deals with finding u(x) using the 1-D heat equation for a rod of length L with heat radiation, where u and F are independent of time t and u_t = 0. The solution involves finding a particular solution and a homogeneous solution to fit the given boundary conditions.
  • #1
clope023
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Homework Statement



If there is heat radiation in a rod of length L, then the 1-D heat equation might take the form:

u_t = ku_xx + F(x,t)

exercise deals with the steady state condition => temperature u and F are independent of time t and that u_t = 0.

u_t = partial derivative with respect to t
u_xx = 2nd partial derivative with respect to x

Problem: find u(x) if F(x) = sinx, k = 2, u(0) = u'(0), u(L) = 1

Homework Equations




e^(itheta) = cos(theta) + isin(theta)
e^(-itheta) = cos(theta) - isin(theta)
e^(itheta) + e^(-itheta) = 2cos(theta)
e^(itheta) - e^(-itheta) = 2isin(theta)

The Attempt at a Solution



u_t = ku_xx + F(x,t)
u_t = ku_xx + F(x)G(t)
2u_xx + sin(x)G(t) = 0
characteristic equation: m^2+1=0, m^2 = -1, m = +-i
general soltn: u(x) = c1cos(x) + c2sin(x)
u(0) = c1cos(0) + c2sin(0)
u'(x) = -c1sin(x) + c2cos(x)
u'(0) = -c1sin(0) + c2cos(0)
=> c2 = c1
u(x) = c1cos(x) + c1sin(x)
u(L) = c1cos(L) + c1sin(L) = 1

from here I'm actually a little lost, I'm not sure what to do with u(L) = 1 portion; do I perhaps need to find a particular solution? any help would be appreciated.
 
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  • #2
If everything is time independent then it's really just an ODE with boundary conditions, right? Yes, find a particular solution. Then find a homogeneous solution and adjust the constants to fit the boundary conditions.
 

Related to Finding u(x) for the 1-D Heat Equation

1. What is the 1-D Heat Equation and why is it important?

The 1-D Heat Equation is a partial differential equation that describes the flow of heat in a one-dimensional system. It is important because it allows us to mathematically model and understand heat transfer in various physical systems, such as in buildings, engines, and other engineering applications.

2. How can u(x) be found for the 1-D Heat Equation?

The solution to the 1-D Heat Equation, u(x), can be found using various methods such as analytical, numerical, or computational techniques. These methods involve solving the equation using initial and boundary conditions to obtain the temperature distribution along the one-dimensional domain.

3. What are the common boundary conditions for the 1-D Heat Equation?

The most common boundary conditions for the 1-D Heat Equation are the Dirichlet boundary condition, where the temperature at the boundary is specified, and the Neumann boundary condition, where the heat flux at the boundary is specified. Other boundary conditions, such as Robin boundary condition, may also be used depending on the specific problem being solved.

4. How does the 1-D Heat Equation change with different materials?

The 1-D Heat Equation remains the same for different materials, as it is a fundamental equation that describes heat transfer. However, the thermal conductivity and specific heat of the materials will affect the equation's coefficients, which will impact the solution and temperature distribution.

5. What are some real-world applications of the 1-D Heat Equation?

The 1-D Heat Equation has numerous real-world applications, including predicting the temperature distribution in buildings, designing heat exchangers in chemical processing plants, and analyzing the thermal performance of electronic devices. It is also used in weather forecasting, studying the Earth's climate, and understanding the thermal behavior of materials in manufacturing processes.

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