Numerical Solution of Ordinary Differential EquationA first order initial value problem of ODE may be written in the form

Example:

Numerical methods for ordinary differential equations calculate solution on the points,where h is the steps size

Numerical Methods for ODEEuler MethodsForward Euler MethodsBackward Euler MethodModified Euler Method

Runge-Kutta MethodsSecond OrderThird OrderFourth Order

Forward Euler MethodConsider the forward difference approximation for first derivative

Rewriting the above equation we have

So, is recursively calculated as

Example: solve

Solution:

etc

Graph the solution

Backward Euler MethodConsider the backward difference approximation for first derivative

Rewriting the above equation we have

So, is recursively calculated as

Example: solve

Solution:Solving the problem using backward Euler method for yields

So, we have

Graph the solution

Modified Euler MethodModified Euler method is derived by applying the trapezoidal rule to integrating ; So, we have

If f is linear in y, we can solved for similar as backward euler methodIf f is nonlinear in y, we necessary to used the method for solving nonlinear equations i.e. successive substitution method (fixed point)

Example: solve

Solution:f is linear in y. So, solving the problem using modified Euler method for yields

Graph the solution

Second Order Runge-Kutta MethodThe second order Runge-Kutta (RK-2) method is derived by applying the trapezoidal rule to integrating over the interval . So, we have

We estimate by the forward euler method.

So, we have

Or in a more standard form as

Third Order Runge-Kutta MethodThe third order Runge-Kutta (RK-3) method is derived by applying the Simpsons 1/3 rule to integrating over the interval . So, we have

We estimate by the forward euler method.

The estimate may be obtained by forward difference method, central difference method for h/2, or linear combination both forward and central difference method. One of RK-3 scheme is written as

Fourth Order Runge-Kutta MethodThe fourth order Runge-Kutta (RK-4) method is derived by applying the Simpsons 1/3 or Simpsons 3/8 rule to integrating over the interval . The formula of RK-4 based on the Simpsons 1/3 is written as

The fourth order Runge-Kutta (RK-4) method is derived based on Simpsons 3/8 rule is written as

Chapter 6 / Ordinary Differential Equation