https://file.aiquickdraw.com/mathbot/file/1726881526656k8lk72ba.jpg

https://file.aiquickdraw.com/mathbot/file/1726881526656k9cyi71v.jpg

已知u(x,T)=g(x),再根据第二个图片的方程和边界条件计算第一个图片中的式子

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):

\[
\int_0^l (u(q_n) - g)_t \psi \, dx + \int_0^l a \nabla (u(q_n) - g) \cdot \nabla \psi \, dx + \int_0^l q_n (u(q_n) - g) \psi \, dx 
= - \int_0^l a \nabla g \cdot \nabla \psi \, dx - \int_0^l q_n g \psi \, dx.
\]

This variational formulation shows how the solution \( u \) interacts with the boundary and the given conditions.

### Step 3: Substitute the boundary conditions and solution

Since we know that \( u(x, T) = g(x) \), this suggests that at time \( t = T \), the solution \( u \) matches the function \( g(x) \). 

- We substitute \( u(q_n) \) with \( g(x) \) at the final time \( T \) because at \( t = T \), the solution equals \( g(x) \). 
- This simplifies some of the integrals where \( u(q_n) - g = 0 \).

### Step 4: Simplification

When \( u(q_n) = g \), several terms simplify. Specifically:

\[
(u(q_n) - g) = 0,
\]
so the first and third integrals vanish:

\[
\int_0^l (0)_t \psi \, dx = 0, \quad \int_0^l q_n (0) \psi \

Learn how to solve a complex boundary value problem involving PDEs and variational methods. The solution focuses on applying boundary conditions and using the weak formulation approach.

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