Non-Homogeneous Heat Equation (Insulated Bar Question)

In summary, the conversation discusses finding U(x,t) given the differential equation dU/dt = d2U/dx2 + sin x and the boundary conditions dU/dx (0,t) = 0 and U(1,t) = 0, with the initial condition U(x,0) = cos 7*π*x. The conversation suggests using the method of separation of variables and finding the eigenfunctions and eigenvalues of the homogenous boundary value problem. One possible solution is U_n = cos(n \pi x) and \lambda _n = -(n \pi)^2. However, it is unclear if this satisfies the boundary conditions.
  • #1
speedycatz
2
0

Homework Statement


Find U(x,t)

dU/dt = d2U/dx2 + sin x

Boundary Conditions:
dU/dx (0,t) = 0

and

U(1,t) = 0

Initial Condition: U(x,0) = cos 7*π*x

2. The attempt at a solution

I start off with:
d2(Un)/dx2 = λnUn (as an initial value problem)

[d(Un)/dx](0) = 0; [d(Un)/dx](1) = 0

My teacher told me to use the orthogonality later on, but at this point I'm stuck.
Can anyone enlighten me? Thanks!
 
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  • #2
ok, so so far you have assumed separatino of variables, that is u(x,t) = U(x)T(t) and

so now you'll need to find the eigenfunctions of the homogenous boundary value problem
[tex] U_n''(x) = \lambda_n U_n(x) [/tex]
[tex] U_n(0) = 0 [/tex]
[tex] U_n'(1) = 0 [/tex]

now solve the ODE above for 3 separate cases λn <0, =0, >0 and see which values of λn are allowable for the given boundary conditions. This yields the eigenvalues & eigenfunctions for the boundary value problem.
 
  • #3
lanedance said:
ok, so so far you have assumed separatino of variables, that is u(x,t) = U(x)T(t) and

so now you'll need to find the eigenfunctions of the homogenous boundary value problem
[tex] U_n''(x) = \lambda_n U_n(x) [/tex]
[tex] U_n(0) = 0 [/tex]
[tex] U_n'(1) = 0 [/tex]

now solve the ODE above for 3 separate cases λn <0, =0, >0 and see which values of λn are allowable for the given boundary conditions. This yields the eigenvalues & eigenfunctions for the boundary value problem.

I get like [tex] U_n = cos(n \pi x) [/tex] and [tex] \lambda _n = -(n \pi)^2 [/tex]
Is that correct?
 
  • #4
do they satisfy you boundary conditions?

look ok for the 2nd, not so sure about the first
 

1. What is a Non-Homogeneous Heat Equation?

A Non-Homogeneous Heat Equation is a partial differential equation that describes the distribution of heat in a non-uniform medium over time. It takes into account both the heat transfer within the medium and any external sources of heat or cooling.

2. How is a Non-Homogeneous Heat Equation different from a Homogeneous Heat Equation?

A Homogeneous Heat Equation only considers heat transfer within a uniform medium, while a Non-Homogeneous Heat Equation also takes into account external sources of heat or cooling. This makes it a more complex equation, but also more applicable to real-world scenarios.

3. What is the purpose of an Insulated Bar Question in a Non-Homogeneous Heat Equation?

The Insulated Bar Question is a common scenario used to illustrate the application of a Non-Homogeneous Heat Equation. It involves a non-uniform bar that is insulated on all sides except for the two ends, where heat is allowed to flow in or out. This scenario is used to solve for the temperature distribution and heat flow within the bar over time.

4. How is the Non-Homogeneous Heat Equation solved for an Insulated Bar Question?

The Non-Homogeneous Heat Equation is typically solved using mathematical methods such as separation of variables, Fourier series, or Laplace transforms. These methods involve breaking down the equation into smaller, more manageable parts and using known solutions to solve for the unknown variables.

5. What are some real-world applications of the Non-Homogeneous Heat Equation?

The Non-Homogeneous Heat Equation has numerous applications in fields such as engineering, physics, and materials science. It is used to model heat transfer in various systems, such as heat exchangers, buildings, and electronic devices. It is also used to study the behavior of materials under different thermal conditions, such as in manufacturing processes or in the Earth's crust.

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