want to solve (d^2 y)/(dx^2 )+dy/dx-6y=0 using 4th order Runge-Kutta method with y(0) = 3 and y’(0) = 1

My code is :

clc

clear all

h = 0.5;

x = 1:h:5;

y = zeros(2,length(x)); % y vector declaration

y(1,1) = 3; % y value

y(2,1) = 1; % y' value

f = @(x) (dy^2)/(dx^2 )+dy/dx-6*y;

for i=1:(length(x)-1) % calculating y values

z1 = f( x(i) , y(:,i));

z2 = f( x(i)+0.5*h, y(:,i)+0.5*h*z1);

z3 = f( x(i)+0.5*h, y(:,i)+0.5*h*z2);

z4 = f( x(i)+h, y(:,i)+h*z3);

y(:,i+1) = y(:,i) + (1/6)*(z1 + 2*z2 + 2*z3 + z4)*h;

end

%plotting results

Y = x.^2 - x - 6;

figure

hold on

plot(x,Y,'b'); % analytic solution plot--- blue

plot(x,y(1,:),'r'); % runga kutta solution --- Red

grid on

legend('Analytical solution','RK4');

xlabel('time (T)')

ylabel('y')

title('Runga gutta 4th order');

But My output is :

Too many input arguments.

Error in Untitled3 (line 11)

z1 = f( x(i) , y(:,i));

How can i fix this problem sir?

Answer 1

[James Tursa]

Change your function handle from this

f = @(x) (dy^2)/(dx^2 )+dy/dx-6*y;

to this

f = @(x,y) [y(2);-y(2)+6*y(1)];

The last part of that function handle is simply solving this

(d^2 y)/(dx^2 )+dy/dx-6y=0

for the 2nd derivative and then plugging in y(2) and y(1) for the other parts.

[John D'Errico]

Listen to @James Tursa.

Answer 2

[Riccardo Bonomelli]

Hi, the problem seems to be in the definition of the function f, you defined f as a single variable function or f(x) while, according to your code, f should be a function of two variables, like f(x,y). Furthermore, the dy and dx inside the function f in your definition must be defined in some way, say dx= something and dy= something. I guess you are trying to express a derivative, but you must define it in some way.

[PULAK Kumer]

Sir, i can not understand.. How can i do it?

[PULAK Kumer]

For this case how can i define derivative sir

[Riccardo Bonomelli]

The definition of the derivatives depends on the equation you are trying to solve, in your case you are solving a second order differential equation which requires a system of two first order differential equations to be solved using 4th order Runge Kutta. I'm no expert on the subject, so I am not comfortable explaining the details of the implementation.