Mathematical ODEs

1. Euler’s Method for
Solving ODEs: An
Approximate
Method
By: Ibrahim
Mohamed Diaaeldin

2. Ordinary Differential Equations
Our aims is to solve ODEs using:
Analytical methods
Approximate methods
a) Euler’s method
b) Improved Euler’s method
c) Runge-Kutta method (Third order)
d) Runge-Kutta method (Fourth order)

3. Analytical Solution of ODE
Solve the following ODE:
Aux. Equation:

4. Analytical Solution of ODE

5. Euler’s Method
2
Use Euler’s Method with a step size of to find approximate values of
the solution at = 0.5.

6. Euler’s Method
Step #1: Formulate the problem such that
Step #2: Get ,, and :
At any iteration (): We have , and
At iteration ():

7. Euler’s Method
Step #3: Formulate the Euler’s approximate function:
At any iteration ():
Inputs (R.H.S): Outputs (L.H.S):
, and
Given that:
, and

8. Euler’s Method
Algorithm of solution:
Our aim:
At iteration #1:
At iteration #2:
At iteration #3:
At iteration #4:
At iteration #5: (Our aim)

9. Euler’s Method
, and
Step #4: Solve ODE iteratively to get :
At iteration #1:
Inputs:
Outputs:

10. Euler’s Method
, and
At iteration #2:
Inputs:
Outputs:

11. Euler’s Method
, and
At iteration #3:
Inputs:
Outputs:

12. Euler’s Method
, and
At iteration #4:
Inputs:
Outputs:

13. Euler’s Method
, and
At iteration #5:
Inputs:
Outputs:

14. Comparison between Exact and Euler’s
method solutions
Iteration
()
Exact Error (%)
0 0 1 1 0
1 0.1 0.9 0.925794646 2.79
2 0.2 0.852967995 0.889504459 4.11
3 0.3 0.837441500 0.876191288 4.42
4 0.4 0.839833779 0.876283777 4.16
5 0.5 0.851677371 0.883727921 3.63