Lecture Notes of Numerical Solution of Ordinary Differential Equations Using Runge-Kutta 4th Order Method II

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)













1.0 – Introduction
An ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives. The term "ordinary" is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
For example:
dy/dx=1.3e^(-x)-2y
Numerical solutions of ordinary differential equations will discuss in the following sections.

2.0 – Initial Value Problems
Runge-Kutta 4th Order method and Euler’s Method will be discussed in this section. Those methods required an initial condition (x_0,y_0).

2.1 – Runge-Kutta 4th Order Method
This method is utilized in solving ordinary deferential equation numerically. This method required an initial condition (x_0,y_0).

2.1.1 – Ordinary Differential Equation
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
¬The following steps are required:
f(x_(i+1),y_(i+1) )=y_(i+1)=y_i+1/6[k_1+?2k?_2+?2k?_3+k_4)
k_1=?_x×f(x_i,y_i )
k_2=?_x×f(x_i+?_x/2,y_i+k_1/2)
k_3=?_x×f(x_i+?_x/2,y_i+k_2/2)
k_4=?_x×f(x_i+?_x,y_i+k_3 )