Question

Use 4th order Rung Kutta method and software to find the solution of the nonlinear ordinary differential equation:

f′′′+1/2ff′′=0
With boundary conditions
f(0)=0
f′(0)=0
f′([infinity])=1
​
Develop and upload software code with an iteration solution with reasonable guess f (0) and converge to the exact solution for the initial condition f′′(0) such that f′ ([infinity])=1

Answer 1

To solve the given nonlinear ordinary differential equation using the 4th order Runge-Kutta method and software, follow the steps provided, including choosing a step size, implementing the Runge-Kutta method, and adjusting the initial guess until the boundary condition is satisfied.

Step-by-step explanation:

To solve the given nonlinear ordinary differential equation, f′′′+1/2ff′′=0, with boundary conditions f(0)=0, f′(0)=0, and f′([infinity])=1, we can use the 4th order Runge-Kutta method. This method is a numerical method used to approximate solutions of ordinary differential equations. Here is the step-by-step process:

1. Choose a step size, h.
2. Start with an initial guess for f(0), which we'll call f0.
3. Use the 4th order Runge-Kutta method to iteratively calculate values of f at each step:
   1. Calculate k1 = h * f′(x, f).
   2. Calculate k2 = h * f′(x + h/2, f + k1/2).
   3. Calculate k3 = h * f′(x + h/2, f + k2/2).
   4. Calculate k4 = h * f′(x + h, f + k3).
   5. Update f = f + (k1 + 2k2 + 2k3 + k4)/6.
   6. Update x = x + h.
4. Repeat step 3 until you reach a desired value of x, such as x = infinity.
5. Adjust the initial guess for f(0) until f′([infinity]) = 1.

By following these steps, you can develop and upload software code to implement the 4th order Runge-Kutta method and find a solution that converges to the exact solution for the given initial conditions.