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 Bn are found using the boundary condition f(x)=nTn(x,0) with the help of the Fourier method.

  3. The total solution T(x,t)=nTn(x,t).

Conditions:

  1. Position x[0,L].

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

Links:

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

temperature

Solution to the heat equation corresponding to the nth mode. See temperature.

Symbol:

T_n(x, t)

Latex:

Tn

Dimension:

temperature

scaling_coefficient

Scaling coefficient of the solution, see Notes.

Symbol:

B_n

Latex:

Bn

Dimension:

temperature

thermal_diffusivity

thermal_diffusivity.

Symbol:

alpha

Latex:

α

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:

xmax

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:
Tn=Bnsin(Nπxxmax)exp(α(Nπxmax)2t)