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The problem you're referring to is a **heat equation** initial-boundary value problem (IBVP). Let’s break down the components of the given problem.

### PDE (Partial Differential Equation):
The governing equation is:
\[
u_t = \alpha^2 u_{xx}, \quad 0 < x < 1, \quad 0 < t < \infty
\]
This is a typical diffusion (or heat) equation, where \( u(x,t) \) represents the temperature at position \( x \) and time \( t \). The parameter \( \alpha^2 \) represents the thermal diffusivity.

### Boundary Conditions (BCs):
The boundary conditions are mixed, and they are given by:
\[
\begin{aligned}
u(0,t) &= 1, \quad 0 < t < \infty \\
u_x(1,t) + h u(1,t) &= 1, \quad 0 < t < \infty
\end{aligned}
\]
- The first condition, \( u(0,t) = 1 \), is a Dirichlet boundary condition, fixing the value of \( u \) at \( x = 0 \).
- The second condition, \( u_x(1,t) + h u(1,t) = 1 \), is a Robin (or mixed) boundary condition at \( x = 1 \), which involves both \( u(1,t) \) and its derivative \( u_x(1,t) \).

### Initial Condition (IC):
The initial condition specifies the state at \( t = 0 \):
\[
u(x,0) = \sin(\pi x) + x, \quad 0 \leq x \leq 1
\]

### Solving the Problem:
To solve this, you would typically:

1. **Transform the boundary conditions to homogeneous form** by introducing a new function that satisfies the boundary conditions. Define a new function \( v(x,t) \) such that \( v(x,t) = u(x,t) - f(x) \), where \( f(x) \) satisfies the non-homogeneous boundary conditions.

2. **Solve the transformed problem** for the new function \( v(x,t) \), which will have homogeneous boundary conditions. This typically involves separation of variables or the use of eigenfunction expansions (Fourier series).

3. **Recover the original solution** by adding back the function \( f(x) \) to the solution of the transformed problem.

### Next Steps:
Would you like to proceed with the transformation to homogeneous boundary conditions, or do you need more clarification on any step?

Here are 5 related questions for deeper exploration:
1. How does the choice of \( f(x) \) help in simplifying the boundary conditions?
2. What is the significance of the Robin boundary condition at \( x = 1 \)?
3. How does separation of variables work in solving heat equations like this?
4. What is the role of the Fourier series in solving such PDEs?
5. How would we interpret the solution physically, especially with these boundary conditions?

**Tip:** Homogeneous boundary conditions often simplify the mathematical solution process and allow easier application of standard techniques like separation of variables or Fourier transforms.

Solve the initial-boundary value problem: PDE: u_t = α^2u_xx, 0 < x < 1, 0 < t < ∞; BCs: u(0,t) = 1, u_x(1,t) + h u(1,t) = 1; IC: u(x,0) = sin(πx) + x

Explore how to solve the heat equation u_t = α^2 u_xx with mixed boundary conditions (Dirichlet and Robin) and an initial condition u(x,0) = sin(πx) + x. Learn how to transform the problem and apply techniques such as separation of variables and Fourier series.

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