Presentation of Boundary Value Problems (Finite Differences 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
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Boundary Value Problems (Finite Differences Method)
Finite Differences Method will be discussed in this section. This method required given conditions at the beginning and at the end (x_0,y_0) and (x_n,y_n). The derivatives of a function can be estimated using Finite Differences Method. This method requires the following steps:

Divided the range between the end boundary condition (x_n) and the initial boundary condition (x_n) by ?_x,
n=((x_n-x_0) )/?_x
x_0 x_1=x_0+?_x x_2=x_1+?_x … x_n
y_0 y_1 y_2 … y_n

Convert the derivatives (dy/dx) in the ordinary deferential equation (ODE) to finite deference equation and as follows:

The first derivative has to be changed to:
dy/dx=1/(2?_x ) (y_(i+1)-y_(i-1) )
The second derivative has to be changed to:
(d^2 y)/(dx^2 )=1/(?_x )^2 (y_(i+1)-??2y?_i+y?_(i-1) )
The third derivative has to be changed to:
(d^3 y)/(dx^3 )=1/?2(?_x )?^3 (?-y?_(i-2)+??2y?_(i-1)-2y?_(i+1)+y_(i+2) )
The fourth derivative has to be changed to:
(d^4 y)/(dx^4 )=1/(?_x )^4 (y_(i-2)-??4y?_(i-1)+6y_i-4y?_(i+1)+y_(i+2) )
Solve the systems of linear equations.

Ex1: If y(1)=1.1752 and y(3)=10.0179, solve the following ordinary differential equation using step size ?_x=0.5
(d^2 y)/(dx^2 )=y