Runge-Kutta 2

Hi, i have a homwork for school including the Runge-Kutta 2.

I have to do it in symbolic, i would be very greatfull if someone can help me in this taks.

intervalmin = 0;
intervalmax = 1;
h1 = 0.1;
g = 0.02; 
X0 = [0, 0];
[t1, x1] = fuggveny(intervalmin, X0(1), X0(2), h1, intervalmax);
[t2, x2] = fuggveny(intervalmin, X0(1), X0(2), g, intervalmax);

syms x(t);
dz = diff(x,t);
ode = diff(x,t,2) + 5.*diff(x,t,1) + 4.*x(t) == 3 - 2.*t - t.^2;
cond1 = x(0) == 1;
cond2 = dz(0) == 1;
RK2(t) = dsolve(ode,cond1,cond2);

plot(t1, x1, '-y');
hold on;
plot(t2, x2, '--r');
hold on;
fplot(RK2,'-*b');
hold on
legend('h=0.1','h=0.02','ode')
ax = gca;
ax.XAxisLocation = 'origin';
ax.YAxisLocation = 'origin';
grid on;

function dX = f(t, x1, x2)
    X1 = x2;
    X2 = -5.*x2 + 4.*x1 + 3 - 2.*t - t.^2;
    dX = [X1, X2];
end

function [t, x] = fuggveny(intervalmin, X0_1, X0_2, h, intervalmax)
    t = (intervalmin:h:intervalmax);
    X1 = zeros(size(t));
    X2 = zeros(size(t));
    X1(1) = X0_1;
    X2(1) = X0_2;
    for i = 1:1:length(t) - 1
        k1 = f(t(i), X1(i), X2(i));
        k2 = f(t(i) + h/2, X1(i) + h/2 * k1(1), X2(i) + h/2 * k1(2));
        k3 = f(t(i) + h, X1(i) - h*k1(1) + 2*h*k2(1), X2(i) - h*k1(2) + 2*h*k2(2));
        
        X1(i + 1) = X1(i) + h/6 * (k1(1) + 4*k2(1) + k3(1));
        X2(i + 1) = X2(i) + h/6 * (k1(2) + 4*k2(2) + k3(2));
    end
    x = X1;
end

Answer 1

[James Tursa]

The + 4.*x1 should be - 4.*x1 in your derivative function. Also, using both X1 and x1 and X2 and x2, even though MATLAB is case sensitive and these are different variables, is confusing to read. Suggest using different names such as x1dot and x2dot, or even just form the derivative vector directly. E.g.,

function dX = f(t, x1, x2)
    dX = [x2; -5.*x2 - 4.*x1 + 3 - 2.*t - t.^2];
end

Also, just to clarify, you are asked to solve this both symbolically and numerically?

Not sure why the assignment says Runge-Kutta II when it looks like you are using a 3rd order method.

[Nagy Csaba Norbert]

Thank you for your comment. In our task we have to use the Runge Kutta 2 order method. Also in Matlab it's enough to do it in symbolically. I'm not sure, about my code is working well, or i need to change something else.

[James Tursa]

Are you getting errors? The output is not what you expected? Or ...?

[Nagy Csaba Norbert]

I don't get any error messages. The problem is that, i don't know what the output should to be look like. My teacher noticed my work with that comment, "there are too many X s in my task.