In a previous post, we compared the results from various 2nd order Runge-Kutta methods to solve a first order ordinary differential equation. In this post, I am posting the matlab program. It is better to **download** the program as single quotes in the pasted version do not translate properly when pasted into a mfile editor of MATLAB or see the **html version **for clarity and sample output .

**Do your own testing on a different ODE, a different value of step size, a different initial condition, etc. See the inputs section below that is colored in bold brick.**

% Simulation : Comparing the Runge Kutta 2nd order method of

% solving ODEs

% Language : Matlab 2007a

% Authors : Autar Kaw

% Last Revised : July 12, 2008

% Abstract: This program compares results from the

% exact solution to 2nd order Runge-Kutta methods

% of Heun’s method, Ralston’s method, Improved Polygon

% method, and directly using the three terms of Taylor series

clc

clear all

clc

clf

disp(‘This program compares results from the’)

disp(‘exact solution to 2nd order Runge-Kutta methods’)

disp(‘of Heuns method, Ralstons method, Improved Polygon’)

disp(‘ method, and directly using the three terms of Taylor series’)

**%INPUTS. If you want to experiment these are only things**

% you should and can change. Be sure that the ode has an exact

% solution

% Enter the rhs of the ode of form dy/dx=f(x,y)

fcnstr=’sin(5*x)-0.4*y’ ;

% Initial value of x

x0=0;

% Initial value of y

y0=5;

% Final value of y

xf=5.5;

% number of steps to go from x0 to xf.

% This determines step size h=(xf-x0)/n

n=10;

%REST OF PROGRAM

%Converting the input function to that can be used

f=inline(fcnstr) ;

% EXACT SOLUTION

syms x

eqn=[‘Dy=’ fcnstr]

% exact solution of the ode

exact_solution=dsolve(eqn,’y(0)=5′,’x’)

% geting points for plotting the exact solution

xx=x0:(xf-x0)/100:xf;

yy=subs(exact_solution,x,xx);

yexact=subs(exact_solution,x,xf);

plot(xx,yy,’.’)

hold on

% RUNGE-KUTTA METHODS

h=(xf-x0)/n;

% Heun’s method

a1=0.5;

a2=0.5;

p1=1;

q11=1;

xr=zeros(1,n+1);

yr=zeros(1,n+1);

%Initial values of x and y

xr(1)=x0;

yr(1)=y0;

for i=1:1:n

k1=f(xr(i),yr(i));

k2=f(xr(i)+p1*h,yr(i)+q11*k1*h);

yr(i+1)=yr(i)+(a1*k1+a2*k2)*h;

xr(i+1)=xr(i)+h;

end

%Value of y at x=xf

y_heun=yr(n+1);

% Absolute relative true error for value using Heun’s Method

et_heun=abs((y_heun-yexact)/yexact)*100;

hold on

xlabel(‘x’)

ylabel(‘y’)

title_name=[‘Comparing exact and Runge-Kutta methods with h=’ num2str(h)] ;

title(title_name)

plot(xr,yr, ‘color’,’magenta’,’LineWidth’,2)

% Midpoint Method (also called Improved Polygon Method)

a1=0;

a2=1;

p1=1/2;

q11=1/2;

%Initial values of x and y

xr(1)=x0;

yr(1)=y0;

for i=1:1:n

k1=f(xr(i),yr(i));

k2=f(xr(i)+p1*h,yr(i)+q11*k1*h);

yr(i+1)=yr(i)+(a1*k1+a2*k2)*h;

xr(i+1)=xr(i)+h;

end

%Value of y at x=xf

y_improved=yr(n+1);

% Absolute relative true error for value using Improved Polygon Method

et_improved=abs((y_improved-yexact)/yexact)*100;

hold on

plot(xr,yr,’color’,’red’,’LineWidth’,2)

% Ralston’s method

a1=1/3;

a2=2/3;

p1=3/4;

q11=3/4;

xr(1)=x0;

yr(1)=y0;

for i=1:1:n

k1=f(xr(i),yr(i));

k2=f(xr(i)+p1*h,yr(i)+q11*k1*h);

yr(i+1)=yr(i)+(a1*k1+a2*k2)*h;

xr(i+1)=xr(i)+h;

end

%Value of y at x=xf

y_ralston=yr(n+1);

% Absolute relative true error for value using Ralston’s Method

et_ralston=abs((y_ralston-yexact)/yexact)*100;

hold on

plot(xr,yr,’color’,’green’,’LineWidth’,2)

% Using first three terms of the Taylor series

syms x y;

fs=char(fcnstr);

% fsp=calculating f'(x,y) using chain rule

fsp=diff(fs,x)+diff(fs,y)*fs;

%Initial values of x and y

xr(1)=x0;

yr(1)=y0;

for i=1:1:n

k1=subs(fs,{x,y},{xr(i),yr(i)});

kk1=subs(fsp,{x,y},{xr(i),yr(i)});

yr(i+1)=yr(i)+k1*h+1/2*kk1*h^2;

xr(i+1)=xr(i)+h;

end

%Value of y at x=xf

y_taylor=yr(n+1);

% Absolute relative true error for value using Taylor series

et_taylor=abs((y_taylor-yexact)/yexact)*100;

hold on

plot(xr,yr,’color’,’black’,’LineWidth’,2)

hold off

legend(‘exact’,’heun’,’midpoint’,’ralston’,’taylor’,1)

% THE OUTPUT

fprintf(‘\nAt x = %g ‘,xf)

disp(‘ ‘)

disp(‘_________________________________________________________________’)

disp(‘Method Value Absolute Relative True Error’)

disp(‘_________________________________________________________________’)

fprintf(‘\nExact Solution %g’,yexact)

fprintf(‘\nHeuns Method %g %g ‘,y_heun,et_heun)

fprintf(‘\nImproved method %g %g ‘,y_improved,et_improved)

fprintf(‘\nRalston method %g %g ‘,y_ralston,et_ralston)

fprintf(‘\nTaylor method %g %g ‘,y_taylor,et_taylor)

disp( ‘ ‘)

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