Numerical Solution using Runge Kutta Method by the help of Programming in C++

1. Vijay Choudhary
vijaycttf@gmail.com
Department of Civil Engineering
Poornima University, Jaipur

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Introduction
Formulation of the problem
Solution of the problem
Approximations, errors
Graphical representation
Conclusion
References
Further Scope

3. A physical problem of finding how much concentration of
the pollutant would be there in a lake after certain time.
To find the concentration of the bacteria (pollutant), the
problem is modeled as an ordinary differential equation.
Runge Kutta 4th order method is used. Solutions obtained are
compared with exact solutions and are graphically
discussed and analyzed.

4. A polluted lake has an initial concentration of a bacteria
of 107 parts/m3 , while the acceptable level is only 5*106
parts/m3 . The concentration of the bacteria will reduce as
fresh water enters the lake. Find the concentration of the
pollutant after 7 weeks.
dC
+ 0.06C = 0
dt

5. The differential equation that governs the concentration C
of the pollutant as a function of time (in weeks) is given by
dC
+ 0.06 C = 0 ,C ( 0 ) =10 6
dt
We Use the Runge-Kutta 4th order method and take a
step size of 3.5 weeks.

12. #include<conio.h>
#include<iostream.h>
void main()
{
float c[10],f,t[10],h,n;
cout<<"enter the initial values of concentration of bacteria ";
int i;
cin>>c[0];
cout<<"enter the initial value of time ";cin>>t[0];
cout<<"enter the value of time in weeks at which we want to see the
concentration ";cin>>f;
cout<<"enter the difference ";cin>>h;
n= (f-c[0])/h;

13. for(i=1;i<=n;i++)
{
a=0
k1=-(.06*c[a]);
k2=-h*(.06*(c[a]+k1/2));
k3=-h*(.06*(c[a]+k2/2));
k4=-h*(.06*(c[a]+k3));
k=(k1+2*k2+2*k3+k4)/6;
y[i]=y[a]+1;
a++;
}
Cout<<“n Table";
for(i=0;i<n;i++)
{
cout<<"t x=%fty=%f",val[i][0],val[i][1]);
cout<<"n");
getch();
}

15. 1

16. 2

17. Figure 1 compare the exact solution with the numerical
solution using Runge kutte 4th order method using different
step size. It is observed that there is significant error when
the calculation is done using Runge Kutte 4th order method
with step size 7. This error can be minimized if we reduce
the step size from 7 to 3.5. Now the numerical solution is
close to the exact solution. Further reduction of step size
from 3.5 to 1.75 does not bring any major error reduction.
In Figure 2, we are comparing the exact results with Runge
Kutte 1st order method (Euler), Runge Kutte 2nd order
method( Heun) and the Runge Kutte 4th order method. It is
observed that 4th order method give close approximation
to exact solution than Heun’s method and Euler’s method

18. Book
Numerical Solutions using Programming in C++, Volume I, Mittal Publication
Chapter/Papers
AP Azai , ., Graph Behaviour, Sindh College Of Engineering, Sindh (*paper)
64 Pages
Internet
James Amtoel , Contouring and Its Applications[online] http://www.civilogyusa.gov
document
[21/02/2014].
http://civilsimplified.com
http://www.bournemouth.ac.uk/library/using/numerical-equations.html
[Accessed 4 Feb 2014].

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By the help of C++ program, it would be easy to analysis
results on different cases like step size, initial conditions,
boundary values, type of method etc.
Like application of 1st order differential equations in
Radioactive science, we can use 2nd order differential
equation in practical applications.
We can mould problems of Density, Population, Traffics
etc into 2nd order differential equations and can study the
behavior of result via graph on different inputs.

20. 
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Program 1
Program 2