Initial condition

In order to find the solution of the heat equation that meets the initial condition

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

we use the following technique. Suppose one needs to find coefficients $\{ C_n \}$ such that the equality

$$f(x) = \sum_{n=1}^\infty C_n G_n(x)$$

is met. The only property we need from $\{ G_n(x) \}$ is given by

$$\int_0^L G_n(x) G_m(x) dx = 0 \mbox{ for } n \not= m$$

Then the formula for $C_n$ is

$$C_n = \frac{\int_0^L f(x) G_n(x) dx }{ \int_0^L (G_n(x))^2 dx}$$ (1.4)

Thus, the solution (1.3) satisfies the initial condition

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

as long as $\{ C_n \}$ are defined by (1.4).