Separation of Variables - Fourier Series

The idea of Separation of Variables can be used to obtain the exact solution for the Diffusion Problem:

$$\frac{\partial u(x,t)}{\partial t} = D\frac{\partial^2u(x,t)} {\partial x^2}$$ (18)

in the region $\Omega = [0;L]\times[0;\infty]$ with the boundary conditions $u(x=0,t)=0$ and $u(x=L,t)=0$ with the initial condition $u(x,t=0)=f(x)$.

The solution is a family of solutions $u_k(x,t)$ with the form

$$u_k(x,t)=X_k(x)\cdot T_k(t).$$ (19)

Since the differential equation is linear, the exact solution can be determined as a linear combination of single solutions.
By substitution of (19) into (18) we obtain (derivatives are markes with $'$)

$$X_k(x)T_k'(t) = D\cdot X_k''(x)T_k(t)$$ (20)

$$\frac{X_k(x)T_k'(t)}{D\cdot X_k(x)T_k(t)} = \frac{D\cdot X_k''(x)T_k(t)} {D\cdot X_k(x)T_k(t)}$$

$$\frac{T_k'(t)}{D\cdot T_k(t)} = \frac{X_k''(x)}{X_k(x)}$$ (21)

(21) can obviosly only be true for any arbitray $x$ and $t$ if the relation of $T$ and $T'$ equals the relation of $X$ and $X''$:

$$\frac{T_k'(t)}{D\cdot T_k(t)} = \frac{X_k''(x)}{X_k(x)} = -\lambda_k.$$

Thus, there are two separate ordinary differential equations of $x$ and $t$:

$$T_k'(t)+\lambda_kD\cdot T_k(t) = 0$$ (22)

$$\mbox{and}$$

$$X_k''(x)+\lambda_kX_k(x) = 0.$$ (23)

(22) is the well known Exponential Decay Equation with its solutions:

$$T_k(t)=e^{-\lambda_kDt}.$$ (24)

(23) can be solved with Sine- and Cosine-approaches.
Considering the boundary conditions $u(x=0,t)=0$ and $u(x=L,t)=0$, the Cosine-approaches are not applicable and we get:

$$X_k(x)=sin{\frac{k\pi x}{L}},$$ (25)

$$\lambda_k=\Big(\frac{k\pi}{L}\Big)^2, \quad k=1,2,\ldots \;.$$

By substituting both solutions in (19), we get:

$$u_k(x,t)=\exp(-\Big(\frac{k\pi}{L}\Big)^2Dt)\cdot sin{\frac{k\pi x}{L}}.$$ (26)

Since $u_k(x,t)$ is linear, all linear combinations are solutions, too:

$$u_k(x,t)=\sum_{k=1}^\infty c_k\cdot\exp(-\Big(\frac{k\pi}{L}\Big)^2Dt)\cdot sin{\frac{k\pi x}{L}},$$ (27)

for arbitrary coefficients $c_k$.
To determine the coefficients $c_k$ which fullfill the initial condition
$u(x,t=0)=f(x)$, $f(x)$ may be developed in a Fourier series.