Solution with zero temperature boundaries¶
The heat equation coupled with a boundary condition can be solved to get a unique solution.
In this boundary-value problem the heat transfer within a thin rod is observed, the temperature
on both ends being zero. This restricts the number of functions
Notes:
represents initial spatial distribution of temperature.Values
are found using the boundary condition with the help of the Fourier method.The total solution
.
Conditions:
Position
.Temperature on both ends is zero:
,
Links:
- temperature¶
Solution to the heat equation corresponding to the
th mode. Seetemperature
.
- Symbol:
T_n(x, t)
- Latex:
- Dimension:
temperature
- Symbol:
B_n
- Latex:
- Dimension:
temperature
- thermal_diffusivity¶
- Symbol:
alpha
- Latex:
- Dimension:
area/time
- mode_number¶
Number of the mode. See
positive_number
.
- Symbol:
N
- Latex:
- Dimension:
dimensionless
- Symbol:
x_max
- Latex:
- Dimension:
length
- Symbol:
x
- Latex:
- Dimension:
length
- Symbol:
t
- Latex:
- Dimension:
time
- law¶
T_n(x, t) = B_n * sin(N * pi * x / x_max) * exp(-alpha * (N * pi / x_max)^2 * t)
- Latex: