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The image you uploaded contains a partial differential equation (PDE) problem with boundary and initial conditions. Let's break it down:

### Problem Statement:
The PDE is given by:
\[ U_t = cU_{xx} - hU + hU_s \]

**Boundary and Initial Conditions:**
1. \( U(x, 0) = f(x) \) (Initial condition)
2. \( U(-\pi, t) = U(\pi, t) \) (Periodic boundary condition)
3. \( U_x(-\pi, t) = U_x(\pi, t) \) (Neumann boundary condition)

Here, \( c \), \( h \), and \( U_s \) are constants.

### Solution Approach:

#### 1. **Fourier Series Expansion:**
Since the boundary conditions are periodic, we can use Fourier series to represent the solution \( U(x, t) \):
\[ U(x, t) = \sum_{n=-\infty}^{\infty} A_n(t) e^{inx} \]

#### 2. **Substituting into the PDE:**
Substitute the Fourier series expansion into the PDE:
\[ U_t(x, t) = \sum_{n=-\infty}^{\infty} \frac{dA_n(t)}{dt} e^{inx} \]
\[ U_{xx}(x, t) = \sum_{n=-\infty}^{\infty} -n^2A_n(t) e^{inx} \]

This gives the PDE in terms of Fourier coefficients \( A_n(t) \):
\[ \frac{dA_n(t)}{dt} = -cn^2A_n(t) - hA_n(t) + hU_s \delta_{n0} \]

Where \( \delta_{n0} \) is the Kronecker delta, which is 1 when \( n = 0 \) and 0 otherwise.

#### 3. **Solving the ODEs for Fourier Coefficients:**
For \( n \neq 0 \):
\[ \frac{dA_n(t)}{dt} = -(cn^2 + h)A_n(t) \]
This is a simple first-order linear ODE, and its solution is:
\[ A_n(t) = A_n(0) e^{-(cn^2 + h)t} \]

For \( n = 0 \):
\[ \frac{dA_0(t)}{dt} = -hA_0(t) + hU_s \]
The solution is:
\[ A_0(t) = \left( A_0(0) - U_s \right)e^{-ht} + U_s \]

#### 4. **Initial Condition:**
Using the initial condition \( U(x, 0) = f(x) \), we can determine the initial Fourier coefficients \( A_n(0) \).

\[ A_n(0) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} dx \]

#### 5. **Constructing the Solution:**
Finally, the solution can be written as:
\[ U(x, t) = \sum_{n=-\infty}^{\infty} A_n(0) e^{-(cn^2 + h)t} e^{inx} + \left( A_0(0) - U_s \right)(1 - e^{-ht}) + U_s \]

### Final Answer:
The general solution is:
\[ U(x, t) = \sum_{n \neq 0} A_n(0) e^{-(cn^2 + h)t} e^{inx} + \left( A_0(0) - U_s \right)(1 - e^{-ht}) + U_s \]

Where \( A_n(0) \) is determined by the initial condition \( f(x) \).

Would you like any further details or have any other questions? Here are some related questions you might find useful:

1. How to determine \( A_n(0) \) for a specific function \( f(x) \)?
2. What is the physical interpretation of the constants \( h \) and \( U_s \)?
3. How does the solution behave for large \( t \)?
4. What happens if \( c \) or \( h \) is zero?
5. Can this method be used for non-periodic boundary conditions?
6. How would the solution change if the boundary conditions were Dirichlet instead?
7. What is the role of the Kronecker delta in the solution?
8. How does the Neumann boundary condition influence the solution?

**Tip:** When solving PDEs with periodic boundary conditions, Fourier series is a powerful tool for transforming the problem into a more manageable form, often reducing it to solving ordinary differential equations (ODEs).

Explore how to solve partial differential equations using Fourier series expansion. This detailed guide covers PDEs with periodic boundary conditions, including step-by-step solutions and key mathematical concepts like Fourier series and boundary conditions.

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