The problem involves determining the steady-state temperature for the given system governed by the Partial Differential Equation (PDE):

$u_t = u_{xx} + 1 \quad \text{for} \quad 0 < x < L$

with boundary conditions:

$u_x(0, t) = 1 \quad \text{and} \quad u_x(L, t) = \beta \quad \text{for} \quad 0 < t < \infty$

and initial condition:

$u(x, 0) = f(x) \quad \text{for} \quad 0 \leq x \leq L$

Part (a): Steady-State Solution

To find the steady-state solution, we assume the system reaches a state where the temperature no longer changes with time, i.e., $u_t = 0$. At steady-state, the PDE simplifies to:

$0 = u_{xx} + 1$

This can be rewritten as:

$u_{xx} = -1$

Integrating this equation twice will give us the steady-state temperature profile $u(x)$. First, integrate with respect to $x$:

$u_x(x) = -x + C_1$

where $C_1$ is an arbitrary constant of integration. Integrate again to find $u(x)$:

$u(x) = -\frac{x^2}{2} + C_1 x + C_2$

where $C_2$ is another arbitrary constant of integration.

Apply Boundary Conditions

We now use the boundary conditions to determine the constants $C_1$ and $C_2$.

1. From the condition $u_x(0, t) = 1$, we know that at $x = 0$:

$u_x(0) = -0 + C_1 = 1 \quad \Rightarrow \quad C_1 = 1$

2. From the condition $u_x(L, t) = \beta$, we have:

$u_x(L) = -L + C_1 = \beta$

Substitute $C_1 = 1$:

$- L + 1 = \beta \quad \Rightarrow \quad \beta = 1 - L$

Thus, the steady-state solution exists for $\beta = 1 - L$.

Steady-State Temperature Profile

Substituting $C_1 = 1$ into the equation for $u(x)$:

$u(x) = -\frac{x^2}{2} + x + C_2$

$C_2$ remains undetermined because it depends on the initial condition or additional information about the temperature distribution.

Thus, the steady-state temperature profile is:

$u(x) = -\frac{x^2}{2} + x + C_2$

where $C_2$ is an arbitrary constant, and the steady-state condition requires $\beta = 1 - L$.

Part (b): Value of $\beta$ for Steady-State Solution

From the boundary condition analysis, we determined that the steady-state solution exists when:

$\beta = 1 - L$

This is the value of $\beta$ that ensures a steady-state solution exists for the given system.

Summary:

• Steady-state temperature: $u(x) = -\frac{x^2}{2} + x + C_2$, where $C_2$ is arbitrary.
• Value of $\beta$ for steady-state: $\beta = 1 - L$.

Would you like more details on any step or specific clarification?

5 related questions:

1. How would the steady-state solution change if the source term in the PDE was different?
2. Can the initial condition $u(x, 0) = f(x)$ affect the steady-state solution?
3. What is the physical interpretation of the constants $C_1$ and $C_2$ in this system?
4. How does the length $L$ of the domain impact the steady-state solution and boundary condition $\beta$?
5. How would the solution differ if there was a time-dependent term in the PDE?

Tip:

To check if a steady-state solution exists, always start by assuming $u_t = 0$, which simplifies the time-dependent PDE into an easier ordinary differential equation (ODE).