Solution by Separation of Variables for Heat Equation The heat equation , $$\frac{\partial u}{\partial t} - \alpha \frac{\partial^2 u}{\partial x^2} = f(x...
The heat equation,
where:
u(x,t) is the temperature at position x and time t
(\alpha) is the thermal diffusivity
(f(x,t)) is the source/sink function
has solutions that can be separated into two parts:
Substituting this into the heat equation, we get two ordinary differential equations:
Equation for X(x)
Equation for T(t)
where (\beta) is a constant related to the heat source/sink.
Solving these two equations independently, we get:
X(x) is a combination of eigenfunctions of the (\alpha)-equation, which are functions oscillating with different frequencies.
T(t) is a general function that accounts for the non-homogeneous source/sink term.
The general solution for u(x,t) is then the superposition of X(x)T(t).
Examples:
If the source function (f(x,t) = \sin(x)), the solution would involve sinusoidal functions for both X(x) and T(t).
If (\alpha = 1) and (f(x,t) = \exp(-x),) the solution would involve the exponential function for both X(x) and T(t).
Key Points:
Separation of variables allows us to solve the heat equation into two simpler, separate equations.
These equations can be solved independently, giving us the solutions for X(x) and T(t).
The general solution for u(x,t) is a linear combination of eigenfunctions of the heat equation