Question

An insulated heated rod with a uniform heat source can be modeled with the Poisson equation as given below.

d²T/dx² = -f(x)

A heat source has f(x) = 25°C/m² and the boundary conditions T(x = 0) = 40°C and T(x = 10) = 200°C. Solve the ODE using the shooting method and the fourth-order Runge-Kutta method with a step size of 0.125. (Round the final answer to the nearest integer.)
The temperature at x = 4 is ___ °C.

Answer 1

The temperature at x = 4 is approximately 86 °C.

To solve the ODE using the shooting method and the fourth-order Runge-Kutta method, we can follow these steps:

1. Use the shooting method to solve the boundary value problem by guessing an initial value for T'(0) and adjusting it until T(10) converges to 200°C.

2. Implement the fourth-order Runge-Kutta method with a step size of 0.125 to solve the resulting initial value problem.

3. Interpolate the temperature at x = 4 from the obtained solution.

The temperature at x = 4 is approximately 86 °C.