Ordinary Differential Equations Part I

Ordinary equations finding its wide application in engineering, applied mathematics , finance and biological science has always been on the frontier of mathematics. At a higher level ODE can be divided into two classes

1. First order Differential equations
2. Higher Order Differential equations

First order ODE are the one where equation consist of first level derivative of equation. To solve these equations we need initial value of the Y to get final value at some defined value of X.Following are the basic methods used to solve first order ODEs -

1. Euler method
2. Predictor corrector method
3. Runge Kutta method

In all above methods we divide the path into n interval and then reach next value of Y in steps.Next value of Y is defined by Taylor series.

Taylor series -

Example -

Let us look at Euler method, in this we divide difference between intial value of X and final value of X into n intervals, calling it h.

Y' = Y+X Initial values Y(1)=2, Find value of Y(2)

Now x_initial = 1 and x_final = 2 , h = x_final-x_initial/n
at n =5 h= .2

Y(1.2) = Y(1)+ (0.2)*(Y(1)+X(1)) = 2.6
Y(1.4) = Y(1.2) + (0.2)*(Y(1.2) + X(1.2)) = 2.6+ .2* (2.6 + 1.2) = 3.36
and continue.

C++ Code for Euler method

I have developed code for the implementation of Euler method in c++, Please find it below (hope commenting is enough for understanding)

#include // for normal_distribution

using boost::math::normal; // typedef provides default type is double.

#include

using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint;

#include

using std::setw; using std::setprecision;

#include

using std::numeric_limits;

#include

#include

using namespace std;

int main()

{

float y_initial,y_final;

float x_initial,x_final;

float derv_y,derv_x,c_const;

float count,h;

int i;

//open file to store results

std::ofstream myfile;

myfile.open("plota.data");

// warning to enter 0 if variable has no value

cout<<"\n Enter 0 if variable has no valuee"<

// Get all the X and Y values

cout << "\n Enter intial value of X and Y respectively"<

cin>>x_initial>>y_initial;

//Taking final value of X where we need to calculate Y

cout << "\n Final value of X"<

cin>>x_final;

// Get Count of interval

cout<<"\n Number of Interval Required"<

cin>>count;

// Coeffciient of derivative equations

cout<<" \n Enter the coefficient of Y, X and constant of derivative"<

cin>>derv_y>>derv_x>>c_const;

//Calculating intervals

h= ( x_final - x_initial ) / count;

//Loop to find values

for(i =0;i

{

y_final = y_initial + h * (derv_y*y_initial+derv_x*x_initial+c_const);

y_initial = y_final;

x_initial = x_initial+h;

myfile<

}

return 0;

}

Output of the code-

Enter 0 if variable has no valuee

Enter intial value of X and Y respectively

1

2

Final value of X

2

Number of Interval Required

10

Enter the coefficient of Y, X and constant of derivative

1

1

0

The result is stored in "plota.data" file and can plotted using GNU plot, Run following commands

gnuplot> set terminal postscript

Terminal type set to 'postscript'

Options are 'landscape noenhanced defaultplex \

leveldefault monochrome colortext \

dashed dashlength 1.0 linewidth 1.0 butt noclip \

palfuncparam 2000,0.003 \

"Helvetica" 14 '

gnuplot> set output "ODE.ps"

gnuplot> plot "plota.data"

result looks like below