Heat Equation with movable point source

In summary, the code is to solve for the Green's function in order to find the temperature at any point on the rod. The heat source is placed at a point xo and the thermal diffusivity constant a is used. The temperature at any point on the rod can be found by multiplying the Green's function by the heat source's density and integrating over the region from 0 to L.
  • #1
brando623
3
0
1. I would like to find and plot the temperature for all points in a 1 dimensional rod of length L, due to a heat source of q placed at point xo where 0<xo<L. The ends of the rod are kept at a constant of 300 Kelvin. The thermal diffusivity constant is a. I'm also looking for the steady state solution.
2. a (del^2)u = q

Where u is the temperature at point x, the thermal diffusivity constant is a, heat source of density q.3. I'm using mathematica to do all of this, so I have included my code

Manipulate[
Plot[T[x, q, xo, L, a], {x, 0, L},
AxesLabel -> {Style["Position(m)", Bold],
Style["Temperature(K)", Bold]}, PlotRange -> {0, 10000},
PlotStyle -> {Thickness[0.0075], Hue[1]}, Filling -> Axis,
ImageSize -> {500, 300}], {{q, .1, "Heat Source Density"}, .1, 100,
Appearance -> "Labeled"}, {{a, .1,
"Conductivity(W/(m*K))"}, .1, 10000,
Appearance -> "Labeled"}, Delimiter, {{L, 1, "Region Length"}, 0.1,
5, Appearance -> "Labeled"}, {{xo, .2, "Source Point Position"}, 0, L,
Appearance -> "Labeled"}, SaveDefinitions -> True,
AutorunSequencing -> All]T[x_, L_, a_, xo_, q_] =
300*(Integrate[G[x, L, xo], {xo, 0, L},
Assumptions -> 0 < L && 0 <= xo]) + (1/a)*(Integrate[
q*G[x, L, xo], {xo, 0, L},
Assumptions -> 0 < L && 0 <= xo) // Simplify

G[x_, L_,xo_] = (1/(L))*(L*x*HeavisideTheta[L - xo] -
x*xo*HeavisideTheta[L - xo] - L*x*HeavisideTheta[x - xo] +
L*xo*HeavisideTheta[x - xo] - L*xo*HeavisideTheta[-xo] +
x*xo*HeavisideTheta[-xo]) // Simplify

For this approach, I solved for the Green's function, using the below code. After that, I multiplied it by q*KroneckerDelta[x,xo] (as the generating function) and integrated with respect to xo from 0 to L and that didn't give the desired results.
DSolve[{D[G[x], x, x] + DiracDelta[x - xo] == 0, G[0] == 0,
G[L] == 0}, G[x], x]

My problem is two things, I don't think I have the right stuff for the integral since the answer doesn't have xo in it, and I'm not sure what generating function to use, or how to develop it. Also, if there is a better way to do this, please do not hesitate to guide me in that direction.

Thanks in advance.
 
Last edited:
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  • #2
For a moving point source, you need to develop a "forcing function." Does it move back and forth, left to right, at constant speed?
 
  • #3
Once placed, the heat source doesn't move at all. The "movable" part comes in where the point source can be at any arbitrary point inside this rod. Sorry for the confusion.
 

Related to Heat Equation with movable point source

1. What is the Heat Equation with movable point source?

The Heat Equation with movable point source is a mathematical model used to describe the flow of heat in a given space or object. It takes into account factors such as the temperature, thermal conductivity, and heat generation within the system.

2. How is the Heat Equation with movable point source used in scientific research?

The Heat Equation with movable point source is used in a wide range of scientific research, from studying heat transfer in materials to analyzing the behavior of complex systems such as the Earth's atmosphere. It is also used in practical applications such as designing efficient heating and cooling systems.

3. What is the significance of the movable point source in the Heat Equation?

The movable point source in the Heat Equation represents a localized heat source that can move within the system. This allows for a more accurate representation of real-world scenarios, where heat sources may not be fixed or evenly distributed.

4. What are some limitations of the Heat Equation with movable point source?

While the Heat Equation with movable point source is a useful model, it does have some limitations. It assumes a linear relationship between temperature and heat flux, and does not take into account factors such as convection or radiation. Additionally, it may not accurately describe highly complex systems with multiple heat sources.

5. How is the Heat Equation with movable point source solved?

The Heat Equation with movable point source is typically solved using numerical methods, such as finite difference or finite element analysis. These methods involve dividing the system into smaller elements and solving for temperature at each point in time. Advanced techniques such as Green's functions or Laplace transforms can also be used for specific cases.

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