Lecture Notes of Numerical Solution of Ordinary Differential Equations Using Euler s Method I

University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)







Numerical Solution of Ordinary Differential Equations

Undergraduate Level, 3th Stage



Mr. Waleed Ali Tameemi
College of Engineering/ Babylon University
M.Sc. Civil Engineering/ the University of Kansas/ USA



2016-2017
Lecture Outline
Introduction
Initial Value Problems
Runge-Kutta 4th Order Method
Ordinary Differential Equation
Simultaneous Ordinary Differential Equations
Euler’s Method
Improved Euler’s Method
Boundary Value Problems (Finite Differences Method)













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Euler’s Method
This method is utilized in solving ordinary deferential equation numerically. This method required an initial condition (y_0,x_0). The first derivative (dy/dx) provides a direct estimation of the slope that leads to a direct estimation of the ODE.
¬The following steps are required in solving ordinary differential equations using Euler’s Method:

The ordinary deferential equation (ODE) dy/dx=f(x,y)
Initial condition (x_0,y_0)
Required y value at a given x value ?(x_n,y_n)
Step size ?_x=x_(i+1)-x_i
Number of steps n=(x_n-x_0)/?_x
f(x_(i+1),y_(i+1) )=y_(i+1)=y_i+f(x_i,y_i )×?_x