I have a problem while programing in Mathematica 8, here is my code:
f[t_, y_] := {y, y};
RungeKutta3[a_, b_, Alpha_, n_, f_] :=
Module[{h, j, k1, k2, k3},
h = (b - a)/n;
Y = T = Table[0, {100 + 1}];
Y[[1]] = Alpha;
T[[1]] = a;
For[j = 1, j <= n, ++j,
k1 = f[T[[j]], Y[[j]]];
k2 = f[T[[j]] + h/2, Y[[j]] + k1*h/2];
k3 = f[T[[j]] + h, Y[[j]] + (-k1 + 2 k2)h];
Y[[j + 1]] = Y[[j]] + h/6(k1 + 4 k2 + k3);
(* Print[j, "----->", Y[[j]]];*)
T[[j + 1]] = T[[j]] + h;
];];
RungeKutta3[0., 1., {300., 500}, 2, f];
The thing is, I'm trying to implement a Runge-Kutta method. And I was successful actually, but only with a function f[x_] that had 1 dimension. This code is for 2 dimensions, but it simply doesn't work and I don't know why. Here is an example for a code with 1 dimension only (notice that I have to change the first line to define the function and the last line, when I call "RungeKutta3").
f[t_, y_] := y;
RungeKutta3[a_, b_, Alpha_, n_, f_] :=
Module[{h, j, k1, k2, k3},
h = (b - a)/n;
Y = T = Table[0, {100 + 1}];
Y[[1]] = Alpha;
T[[1]] = a;
For[j = 1, j <= n, ++j,
k1 = f[T[[j]], Y[[j]]];
k2 = f[T[[j]] + h/2, Y[[j]] + k1*h/2];
k3 = f[T[[j]] + h, Y[[j]] + (-k1 + 2 k2)*h];
Y[[j + 1]] = Y[[j]] + h/6*(k1 + 4 k2 + k3);
(* Print[j, "----->", Y[[j]]];*)
T[[j + 1]] = T[[j]] + h;
];];
RungeKutta3[0., 1., 300., 100, f];
To sum up, how do I implemented the Runge-Kutta method for a function with 2 dimensions??
If you could help me out I would be grateful.
Thanks in advance!
PS: the Runge-Kutta method is order 3
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Problem solved! Check the code, if anybody needs help with anything, just ask!
f[t_, y1_, y2_] := 3 t*y2 + Log[y1] + 4 y1 - 2 t^2 * y1 - Log[t^2 + 1] - t^2;
F[t_, {y1_, y2_}] := {y2, f[t, y1, y2]};
RungeKutta3[a_, b_, [Alpha]_, n_, f_] :=
Module[{h, j, k1, k2, k3, Y, T, R},
h = (b - a)/n;
Y = T = Table[0, {n + 1}];
Y[[1]] = [Alpha]; T[[1]] = a;
For[j = 1, j <= n, ++j,
k1 = f[T[[j]], Y[[j]]];
k2 = f[T[[j]] + h/2, Y[[j]] + k1*h/2];
k3 = f[T[[j]] + h, Y[[j]] + (-k1 + 2 k2)*h];
Y[[j + 1]] = Y[[j]] + h/6*(k1 + 4 k2 + k3);
T[[j + 1]] = T[[j]] + h;
];
R = Table[0, {n + 1}];
For[j = 1, j <= n + 1, j++, R[[j]] = Y[[j]][[1]]];
Print[ListPlot[Transpose[{T, R}]]]
];
RungeKutta3[0., 1, {1., 0.}, 1000, F];
I know basically have a mathematica program that can solve ANY 2nd order equation! Through Runge-Kutta method. just insert your function on
f[t_, y1_, y2_]:= [Insert your function here]
where t is the independent value, y1 is the function itself y(t), y2 is y'(t). Call the function through:
RungeKutta3[a, b, [Alpha], n, F];
where a is the initial "t" value, b the final "t" value, [Alpha] the initial value of your function and the first derivative (given in the form {y1(a),y2(a0)}), n the number of points equally spaced you want to represent. F is the function you have to insert despite of the function you give to f
Any questions feel free to ask!! PS: The Runge-Kutta problem solves differential equations with problems of initial values, i used this program as a base to solve a problem of boundary values, if you want it just text me!