Question

The steady state solution for the heat equation \[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad 0<x<2, \, t>0, \] with the initial condition \( u(x, 0) = 0, \, 0<x<2 \) and the boundary conditions \( u(0, t) = 1 \) and \( u(2, t) = 3, \, t>0 \) at \( x = 1 \) is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For steady-state solutions of the heat equation, the solution is a linear function of \( x \), which can be determined using the boundary conditions.

Answer 1

The Correct Option is B

We are given the heat equation with the boundary conditions and the initial condition. We need to find the steady-state solution at \( x = 1 \). Step 1: Steady-state condition
At steady state, the solution does not depend on time, so \( \frac{\partial u}{\partial t} = 0 \). Therefore, the heat equation becomes: \[ \frac{\partial^2 u}{\partial x^2} = 0. \] Step 2: General solution
The general solution to \( \frac{\partial^2 u}{\partial x^2} = 0 \) is a linear function of \( x \): \[ u(x) = Ax + B. \] Step 3: Apply boundary conditions
We apply the boundary conditions to determine the constants \( A \) and \( B \). From the boundary condition \( u(0) = 1 \): \[ A(0) + B = 1 \quad \Rightarrow \quad B = 1. \] From the boundary condition \( u(2) = 3 \): \[ A(2) + 1 = 3 \quad \Rightarrow \quad 2A = 2 \quad \Rightarrow \quad A = 1. \] Step 4: Steady-state solution
Thus, the steady-state solution is: \[ u(x) = x + 1. \] Step 5: Evaluate at \( x = 1 \)
At \( x = 1 \), the solution is: \[ u(1) = 1 + 1 = 2. \] Therefore, the steady-state solution at \( x = 1 \) is \( 2 \).