To solve this boundary value problem using the collocation method, we will follow these steps:

1. Set up the problem:

We are given the second-order differential equation: $-\frac{d^2u}{dx^2} = \cos(\pi x), \quad 0 < x < 1$ with boundary conditions: $u(0) = 0, \quad u(1) = 0$

2. Trial solution using trigonometric basis functions:

The general solution is assumed to be of the form: $u(x) = \phi_0 + c_1 \phi_1 + c_2 \phi_2 + c_3 \phi_3$ where the basis functions are given as: $\phi_0 = 0, \quad \phi_1 = \sin(\pi x), \quad \phi_2 = \sin(2\pi x), \quad \phi_3 = \sin(3\pi x)$

Therefore, the trial solution becomes: $u(x) = c_1 \sin(\pi x) + c_2 \sin(2\pi x) + c_3 \sin(3\pi x)$

3. Collocation points:

The collocation method requires choosing specific points in the domain to enforce the differential equation. The problem specifies collocation points at: $x = \frac{1}{4}, \quad x = \frac{1}{2}, \quad x = \frac{3}{4}$

4. Apply the differential equation:

We substitute the trial solution into the given differential equation: $-\frac{d^2u}{dx^2} = \cos(\pi x)$ Taking the second derivative of the trial solution: