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:

\(B_{n}\)

Dimension:

temperature

thermal_diffusivity

thermal_diffusivity.

Symbol:

alpha

Latex:

\(\alpha\)

Dimension:

area/time

mode_number

Number of the mode. See positive_number.

Symbol:

N

Latex:

\(N\)

Dimension:

dimensionless

maximum_position

Maximum possible position.

Symbol:

x_max

Latex:

\(x_\text{max}\)

Dimension:

length

position

position, or spatial variable.

Symbol:

x

Latex:

\(x\)

Dimension:

length

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

law

T_n(x, t) = B_n * sin(N * pi * x / x_max) * exp(-alpha * (N * pi / x_max)^2 * t)

Latex:

\[T_{n} = B_{n} \sin{\left(\frac{N \pi x}{x_\text{max}} \right)} \exp{\left(- \alpha \left(\frac{N \pi}{x_\text{max}}\right)^{2} t \right)}\]