### **Step 1: Interpreting the Problem** - **Boundary Conditions (BCs):** - \( u(0, t) = 0 \): The solution is fixed at 0 at \( x = 0 \) for all time. - \( u_x(1, t) = 1 \): The spatial derivative (slope) at \( x = 1 \) is always 1 for all time. - **Initial Condition (IC):** - \( u(x, 0) = \sin(\pi x) \): The initial profile at \( t = 0 \) is a sine wave, starting at 0 (at \( x = 0 \)), peaking at \( x = 0.5 \), and returning to 0 at \( x = 1 \). - **Physical Interpretation:** - This is likely a diffusion or heat equation problem (though the PDE isn't stated, usually these are parabolic PDEs). - At \( t = 0 \), the system starts with the initial profile. - The left end is always at 0, while the right end's slope is always fixed at 1. **Explanation:** The system starts with a sine-shaped profile and is then influenced by the boundaries: one end is fixed, and the other end's slope is fixed. Over time, the profile will adjust to these constraints. --- ### **Step 2: Long-Term Behavior and Sketches** - **Sketch for \( t = 0 \):** - The curve is \( \sin(\pi x) \), peaking in the middle and zero at both ends. - **Sketch for \( t > 0 \):** - The solution will begin to change, especially near \( x = 1 \), where the slope must become 1. - Over time, the initial sine shape will fade, and the solution will transition to a shape that satisfies the boundary conditions for all \( t \). - **Steady-State Solution:** - At steady state, \( u(x, t) \to u_s(x) \) as \( t \to \infty \). - The steady-state must satisfy: - \( u_s(0) = 0 \) - \( u_s'(1) = 1 \) - For a typical diffusion equation (\( u_t = u_{xx} \)), the steady-state is a linear function with a slope of 1: - \( u_s(x) = x \) - This satisfies both BCs: \( u_s(0) = 0 \), \( u_s'(1) = 1 \). **Explanation:** The initial profile will decay (due to diffusion), and the solution will approach a steady-state linear profile \( u_s(x) = x \). This is because, at steady state, the system only needs to satisfy the boundary conditions, not the initial shape. --- ### **Final Answer** - The initial-boundary-value problem describes a process where an initial sine-shaped profile evolves over time, with one boundary fixed at zero and the other boundary's slope fixed at one. - **The solution will come to a steady state, which is the linear function \( u(x) = x \).** - This is obvious because the boundary conditions uniquely determine the steady-state solution, and the system will lose memory of its initial shape as time progresses.

### **Step 1: Interpretation of the Initial-Boundary-Value Problem** - **Given conditions:** - Initial condition: \( u(x, 0) = \sin(\pi x) \) for \( 0 \leq x \leq 1 \). - Boundary conditions: - At \( x=0 \): \( u(0, t) = 0 \). - At \( x=1 \): \( u_x(1, t) = 1 \) (the derivative with respect to \( x \) is fixed at 1). - **Physical/Mathematical context:** - Likely a diffusion or heat equation problem (e.g., \( u_t = u_{xx} \)), although the PDE isn't explicitly provided. - The initial profile is a sine wave, starting at zero, peaking in the middle, then decreasing back to zero at \( x=1 \). - Boundary conditions fix the value at \( x=0 \) and the slope at \( x=1 \), which influences how the initial wave evolves over time. - **Summary:** - The problem models how an initial sine-shaped profile diffuses or evolves under these boundary constraints. - The boundary conditions suggest the solution will adjust over time to satisfy both the fixed value at the left boundary and the fixed slope at the right boundary. **Explanation:** This initial-boundary-value problem describes a system starting with a sine wave, with fixed boundary conditions that will cause the solution to change over time toward a steady state that respects these constraints. --- ### **Step 2: Sketching the Solution and Its Long-Term Behavior** - **Initial profile at \( t=0 \):** \( u(x, 0) = \sin(\pi x) \), which starts at 0 at \( x=0 \), peaks at \( x=0.5 \), and returns to 0 at \( x=1 \). - **Evolution over time:** - The solution will smooth out and change shape, especially near the boundaries. - At \( x=0 \), \( u=0 \) remains fixed. - At \( x=1 \), the slope \( u_x \) is fixed at 1, implying the solution near \( x=1 \) will tend toward a linear function with slope 1. - **Steady-state solution:** - At long times, the solution approaches a steady state \( u_s(x) \). - The steady-state must satisfy: - \( u_s(0) = 0 \), - \( u_s'(1) = 1 \). - A linear function satisfying these conditions is: \[ u_s(x) = x, \] which has: - \( u_s(0) = 0 \), - \( u_s'(x) = 1 \), - \( u_s'(1) = 1 \). - **Sketches:** - At \( t=0 \): a sine wave starting at zero and peaking at \( x=0.5 \). - For \( t>0 \): the profile smooths out, gradually approaching the line \( u(x) = x \). - The solution will "flatten" the initial wave, especially near \( x=0 \), and develop a slope of 1 at \( x=1 \). **Will the solution come to a steady state?** Yes, because the boundary conditions are fixed and the PDE (assumed to be diffusive) leads to a smoothing process that stabilizes over time. The limiting shape is the linear function \( u(x) = x \). --- ### **Final Answer:** The initial-boundary-value problem models the evolution of a sine-shaped initial profile under fixed boundary conditions: zero at \( x=0 \) and slope 1 at \( x=1 \). Over time, the solution smooths out and approaches the steady state \( u(x) = x \). The solution sketches would show the initial sine wave gradually flattening into the linear function, confirming that the solution indeed approaches this steady state as \( t \to \infty \).