Heat equation. Fourier method - separation of variables.

Consider the heat equation

$$\frac{\partial }{\partial t} u(t,x) = \frac{\partial^2 }{\partial x^2}u(t,x), \;\;\;x\in (0,\;L)$$

It models the heat propagation in a thin uniform bar or wire of length $L.$ The function $u(t,x)$ describes the temperature at the point $x$ and time $t.$ The heat dynamic depends on the boundary conditions,

$$u(t,0)=0,\;\;\;u(t,L)=0$$

and initial conditions

$$u(0,x) = f(x).$$

To find the solution for the heat equation we use the Fourier method of separation of variables. First, we look for special solutions having the form

$$u(t,x)= F(t) G(x).$$

Substitution of this special type of the solution into the heat equation leads us to

$$(\frac{d}{dt}F(t))G(x) = F(t)\frac{d^2}{dx^2}G(x)$$

and after separation of variables ($x$ and $t$) we obtain

$$\frac{\frac{d}{dt}F(t)}{F(t)} = \frac{\frac{d^2}{dx^2}G(x)}{G(x)}$$

Since $t$ and $x$ are independent variables this equality can hold only when both left and right hand sides are constants. Thus we have two independent equations

$$\frac{d}{dt}F(t) = \lambda \cdot F(t)$$ (1.1)

and

$$\frac{d^2}{dx^2}G(x) = \lambda \cdot G(x)$$ (1.2)

that are related to each other only through the constant $\lambda.$ Let us first solve the equation (1.2). There are three possibilities to consider:

I.

$\lambda > 0 .$

II.

$\lambda = 0.$

III.

$\lambda < 0 .$