solve a complex second order differential equation

the ode has a form: （）

and for, given , how could I use ode45 to solve it with plot? thx

Answer 1

[Bjorn Gustavsson]

The first step is to convert your second-order ODE to two coupled first-order ODEs:

Then you should write that ODE-system as a matlab-function:

function [dphidt_domegadt] = yourODEs(t,phi_w)

phi = phi_w(1);
w = phi_w(2);
dphidt = w;
if t == 0 % Here I assume that domegadt/t goes to zero as t -> 0+, perhaps there are solutions for other finite values of that ratio...
  domegadt = phi^3;
else
  domegadt = -2/t*dphidt - phi^3;
end
dphidt_domegadt = [dphidt;
                   domegadt];

This should be possible to integrate with ode45:

phi0w0 = [1 0];
t_span = [0 exp(2)]; % some limits of yours
[t,phi_w] = ode45(@(t,phi_w) yourODEs(t,phi_w),t_span,phi0w0);

HTH

[嘉杰 程]

I am afraid I can't run this code?

[Bjorn Gustavsson]

My bad, forgot to combine the derivatives. Done now.

However, you seriously need to consider the division-by-zero issue, as hinted at in the code and outlined by Walter in his answer.

[嘉杰 程]

I have a further question, if I have a parameter t, the eqs goes like: ,

I want to solve it when t varies from certain scale, when , I want to plot relation, given too, could you help me switch the code for it? thx

(or can we just use integral to solve this problem?)

[Walter Roberson]

That only has a solution when t = 1.

[嘉杰 程]

What I want to plot is lane-emden eqs, refer to Lane-Emden Differential Equation -- from Wolfram MathWorld

but I can't do like that when n=3.

[Walter Roberson]

implies that is a contant independent of η . further tells us that which tells us that . When then because is indepdent of . This carries over to as we know the must be 0, and tells us that y must be 0 except perhaps when is infinite.

But your talk about which is irrelevant, as you have no other places with

I have a suspicion that you intended to imply that ϕ is a family of functions in variable t but your notation is wrong for that.

[Walter Roberson]

I do not see any resemblence between the forms posted in https://www.mathworks.com/matlabcentral/answers/1567938-solve-a-complex-second-order-differential-equation#comment_1793988 and the link to tbe Lane-Emden Differential Equation

[嘉杰 程]

every t determine certain curvature of phi and eta, when eta=0, phi(eta)=1, phi'(eta)=0, that's one side, another side is determined by t

[嘉杰 程]

actually this is only a specific function when n=3

Answer 2

[Walter Roberson]

You cannot use any numeric solver for that. You have initial conditions at η = 0, but at 0 you have a division by 0 which gets you a numeric infinity. That numeric infinity is multiplied by the boundary condition of 0, but numeric infinity times numeric 0 gives you NaN, not 0.

If you work symbolically you might think that the infinity and the 0 cancel out, but that only works if the φ' approaches 0 faster than 1/η approaches infinity, which is something that we do not immediately know to be true.

[嘉杰 程]

So is there any suggestion?

[嘉杰 程]

If I let to be a quite small number?

[Bjorn Gustavsson]

That is not enough. The ratio of 1/t*dphi/dt has to behave well for t = 0.