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 \(f(x)\) that can satisfy the boundary equation \(f(x) = T(x, 0)\).

Notes:

1. \(f(x)\) represents initial spatial distribution of temperature.

2. Values \(B_n\) are found using the boundary condition \(f(x) = \sum_n T_n(x, 0)\) with the help of the Fourier method.

3. The total solution \(T(x, t) = \sum_n T_n(x, t)\).

Conditions:

1. Position \(x \in [0, L]\).

2. Temperature on both ends is zero: \(T_n(0, t) = 0\), \(T_n(L, t) = 0\)

Links:

1. Paul’s Online Math Notes, Example 1.

temperature

Solution to the heat equation corresponding to the \(n\)^th mode. See temperature.

Symbol:

T_n(x, t)

Latex:

\(T_{n}\)

Dimension:

temperature

scaling_coefficient

Scaling coefficient of the solution, see Notes.

Symbol:

B_n

Latex:

\(B_n\)

thermal_diffusivity

Thermal diffusivity.

Symbol:

chi

Latex:

\(\chi\)

mode_number

Number of the mode, which is a positive integer.

Symbol:

n

maximum_position

Maximum possible position.

Symbol:

L

position

Position, or spatial variable.

Symbol:

x

time

Time.

Symbol:

t

law

T_n(x, t) = B_n * sin(n * pi * x / L) * exp(-1 * chi * (n * pi / L)^2 * t)

Latex:

\[T_n(x, t) = B_n \sin \left( \frac{n \pi x}{L} \right) \exp \left[ -\chi \left( \frac{n \pi}{L} \right)^2 t \right]\]