### Step 1: Interpretation of the Problem **Explanation:** - The given PDE is the heat (diffusion) equation: \( u_t = \alpha^2 u_{xx} \), describing how temperature (or another diffusing quantity) evolves over time on the interval \( 0 < x < 1 \). - Boundary conditions (BCs): - \( u(0, t) = 0 \): The value at the left end is always zero. - \( u_x(1, t) = 1 \): The spatial derivative (slope) at the right end is always 1, which is a Neumann boundary condition. - Initial condition (IC): - \( u(x, 0) = \sin(\pi x) \): The initial temperature profile is a sine wave. **Interpretation:** This problem models heat diffusion in a rod of length 1, where the left end is kept at zero temperature, and the right end has a fixed heat flux (slope). The rod starts with a sine-shaped temperature distribution. --- ### Step 2: Solution Behavior and Steady State **Explanation:** - **Early times (\( t \approx 0 \))**: The solution matches the initial sine wave, \( u(x, 0) = \sin(\pi x) \). - **As time increases**: The solution smooths out due to diffusion, and the boundary conditions start to dominate the shape. - **Long time (\( t \to \infty \)):** The system reaches a steady-state solution \( u_s(x) \) where \( u_t = 0 \), so \( u_{xx} = 0 \). Integrating twice and applying the BCs: - \( u_s(0) = 0 \) - \( u_s'(1) = 1 \) - Thus, the steady-state solution is a straight line: \( u_s(x) = x \). **Sketches:** - At \( t = 0 \): Sine curve from 0 to 0. - For small \( t \): The curve starts to flatten, especially near \( x = 1 \), to meet the boundary slope. - For large \( t \): The curve approaches a straight line from (0,0) to (1,1). **Steady State:** Yes, the solution will come to a steady state, and it is obvious because the heat equation dissipates initial shapes, and the boundary conditions dictate a unique steady state. --- ### **Final Answer:** **The initial-boundary-value problem describes heat diffusion in a rod with one fixed end and one end with a fixed heat flux. The solution starts as a sine wave and evolves over time to a straight line (\( u_s(x) = x \)), which is the steady state.**

### Step 1: Interpretation of the Problem **Explanation:** The PDE \( u_t = \alpha^2 u_{xx} \) models heat diffusion along a one-dimensional rod of length 1. - Boundary conditions: - \( u(0, t) = 0 \) indicates the left end of the rod is held at zero temperature. - \( u_x(1, t) = 1 \) specifies a constant heat flux (gradient) at the right end. - Initial condition: \( u(x, 0) = \sin(\pi x) \) means the initial temperature distribution is a sine wave along the rod. **Interpretation:** This setup describes how an initial sine-shaped temperature distribution evolves over time, with fixed temperature at one end and a constant heat flux at the other, illustrating a diffusion process influenced by these boundary conditions. --- ### Step 2: Solution Behavior and Steady State **Explanation:** - At early times, the solution resembles the initial sine wave. - As time progresses, diffusion causes the temperature profile to smooth out, influenced by boundary conditions. - The steady state occurs when \( u_t = 0 \), leading to \( u_{xx} = 0 \). Integrating twice: \[ u_s(x) = A x + B \] Applying BCs: - \( u_s(0) = 0 \Rightarrow B=0 \). - \( u_x(1) = A = 1 \). So, the steady state is \( u_s(x) = x \). **Sketches:** - Initially: sine wave \( \sin(\pi x) \). - With time: the profile becomes flatter near \( x=0 \) and adjusts near \( x=1 \) due to the boundary flux. - Long-term: the temperature profile approaches the straight line \( u(x) = x \). **Conclusion:** Yes, the solution will tend to the steady state \( u(x) = x \), as it is consistent with the boundary conditions and the nature of diffusion. --- ### **Final Answer:** The problem models heat diffusion in a rod with fixed temperature at one end and constant heat flux at the other. The solution starts as a sine wave and evolves over time toward a steady state linear profile \( u(x) = x \).

**Step 1: Understanding the Boundary Conditions and Steady State** **Explanation:** The boundary conditions are: - \( u(0,t) = 0 \): The temperature at \( x=0 \) is held fixed at zero for all times. - \( u_x(1,t) = 1 \): The heat flux (or spatial derivative) at \( x=1 \) is held constant at 1 for all times. In the long-term (steady state), the time derivative \( u_t \) becomes zero, so the PDE reduces to: \[ u_{xx} = 0 \] whose general solution is: \[ u_s(x) = A x + B \] Applying the boundary conditions for the steady state: - At \( x=0 \): \[ u_s(0) = B = 0 \] - At \( x=1 \): \[ u_x(1) = A = 1 \] Thus, the steady-state solution is: \[ \boxed{ u_s(x) = x } \] --- **Step 2: Interpretation of the Solution Behavior** **Explanation:** - The boundary condition \( u(0,t) = 0 \) fixes the temperature at the left end at zero. - The boundary condition \( u_x(1,t) = 1 \) imposes a constant heat flux at the right end, meaning heat flows into the domain at a steady rate. - Over time, the solution will evolve from any initial condition toward the steady state \( u_s(x) = x \), which satisfies both boundary conditions. **Will the solution come to a steady state?** Yes. The heat equation with these boundary conditions is well-posed and dissipative, so the solution will tend to the steady state \( u(x) = x \). **Sketching the solution:** - Initially, the solution might vary depending on the initial condition, but as \( t \to \infty \), it approaches the straight line \( u(x) = x \). - The profile will gradually smooth out and adjust to meet the boundary conditions, approaching the steady state from below or above depending on initial data. --- ### **Final Answer:** The boundary conditions specify a fixed temperature at \( x=0 \) and a constant heat flux at \( x=1 \). The solution of the PDE will evolve over time from initial conditions toward the steady state \( u(x) = x \), satisfying both boundary conditions, with the temperature profile becoming a straight line from 0 at the left end to a slope of 1 at the right end.