To solve the given problem, let's break down what is happening in the equations from the images. The first image contains a variational expression (Equation 2.23) involving integrals, while the second image contains a partial differential equation (PDE) system (Equation 1.5), likely describing a boundary value problem.

Step 1: PDE (Equation 1.5)

This equation describes a second-order PDE of the form:

$u_t - (a(x) u_x)_x + q(x) u = 0, \quad (x, t) \in Q$

with boundary conditions:

$u|_{x=0} = u|_{x=l} = 0,$ $u(x, 0) = \phi(x).$

The boundary and initial conditions imply that at $t = 0$, the solution is given by the function $\phi(x)$, and at $x = 0$ and $x = l$, the solution is zero. The additional condition $u(x, T) = g(x)$ provides the final time condition where the solution reaches $g(x)$ at $t = T$.

Step 2: Apply to Equation (2.23)

Equation (2.23) contains integrals that represent some form of energy or weak form of the original PDE (from 1.5). The integrals involve terms like $u(q_n) - g$, gradients $\nabla \psi$, and boundary terms.

This equation can be understood as deriving from multiplying the PDE (1.5) by a test function $\psi(x)$ and integrating over the domain, likely using integration by parts.

The form of Equation (2.23):