1. MATH 2140 Numerical Methods
Faculty of Engineering
Mechanical Engineering Department
MATH Numerical Methods
Instructor:
Dr. Mohamed El-Shazly
Associate Prof. of Mechanical Design and Tribology
Office: F072

2. Ordinary Differential Equations: Initial-Value Problems

3. BACKGROUND

4. BACKGROUND
Equation (10.6) is a first-order ODE, with (t) as the independent variable and (h) the dependent variable.
problems in which time t is the independent variable), the constraint is called an initial condition and the problem is called an initialvalue problem (IVP).

5. EULER'S METHODS
Euler's method is the simplest technique for solving a first-order ODE of the form of Eq. (10.7):

6. Euler's Explicit (صريح) (FORWARD) Method
It is a single-step, numerical technique for solving a first-order ODE.

7. Example : 1

10. Euler's Implicit (ضمني ) Method

13. RUNGE-KUTTA METHODS

14. Second-Order Runge-Kutta Methods

19. Fourth-Order Runge-Kutta Methods