To solve this problem, we need to apply the method of separation of variables to the given heat equation:

$\frac{\partial u}{\partial t} = 3 \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < a, \ t > 0,$

with boundary conditions:

$u(0,t) = 0, \quad u(a,t) = 0, \quad t > 0.$

Step 1: Separation of Variables

We assume the solution can be written as a product of functions in $x$ and $t$:

$u(x,t) = X(x) T(t).$

Substituting this into the heat equation:

$\frac{\partial}{\partial t}(X(x)T(t)) = 3 \frac{\partial^2}{\partial x^2}(X(x)T(t)).$

This simplifies to:

$X(x) \frac{dT(t)}{dt} = 3 T(t) \frac{d^2 X(x)}{dx^2}.$

Now, divide both sides by $X(x)T(t)$:

$\frac{1}{T(t)} \frac{dT(t)}{dt} = 3 \frac{1}{X(x)} \frac{d^2 X(x)}{dx^2}.$

Since the left side depends only on $t$ and the right side depends only on $x$, both sides must be equal to a constant, say $-\lambda$. This gives us two ordinary differential equations (ODEs):

1. For $X(x)$: $\frac{d^2 X(x)}{dx^2} + \lambda X(x) = 0.$

2. For $T(t)$: $\frac{dT(t)}{dt} = -3\lambda T(t).$

Step 2: Solving the ODE for $X(x)$

The boundary conditions are $u(0,t) = 0$ and $u(a,t) = 0$, which implies:

$X(0) = 0 \quad \text{and} \quad X(a) = 0.$

The general solution to the ODE for $X(x)$ is:

$X(x) = A \sin(\sqrt{\lambda}x) + B \cos(\sqrt{\lambda}x).$

Applying the boundary condition $X(0) = 0$:

$B \cos(0) = 0 \quad \Rightarrow \quad B = 0.$

Thus, the solution simplifies to:

$X(x) = A \sin(\sqrt{\lambda}x).$

Now applying the boundary condition $X(a) = 0$:

$A \sin(\sqrt{\lambda}a) = 0.$

For a nontrivial solution, we must have $\sin(\sqrt{\lambda}a) = 0$, which gives:

$\sqrt{\lambda}a = n\pi \quad \Rightarrow \quad \lambda = \left(\frac{n\pi}{a}\right)^2, \quad n = 1, 2, 3, \dots$

Step 3: Solving the ODE for $T(t)$

For each $\lambda_n = \left(\frac{n\pi}{a}\right)^2$, the ODE for $T(t)$ becomes:

$\frac{dT(t)}{dt} = -3\lambda_n T(t) = -3 \left(\frac{n\pi}{a}\right)^2 T(t).$

The solution is:

$T(t) = C e^{-3\left(\frac{n\pi}{a}\right)^2 t}.$

Step 4: General Solution

The general solution for $u(x,t)$ is the sum of the solutions for each $n$:

$u(x,t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi}{a} x\right) e^{-3\left(\frac{n\pi}{a}\right)^2 t}.$

Step 5: Applying the Initial Condition

Now, we apply the initial condition $u(x,0) = f(x)$, where $f(x)$ is given as: