what I should do-----second order variable coefficients ODE??

Hi, matlab person,I have some problems about second order variable coefficients ODE.

Thank you very much for any suggestion or help.

equation: x^2 y''+x y'-sin(y)cos(y)+4*k/pi*x sin(y)^2-x^2 sin(y)cos(y)=0

boundary condition: y(0)=0;y'(0)=-0.126

The following is my code, but there are several problem:

first defining function *********************************************

function yprime = skyrmion(x,y)

k=0.95;

yprime = [y(2); -1/x*y(2)+1/x^2*sin(y(1))*cos(y(1))-4*k/pi* (1/x)*sin(y(1))^2+sin(y(1))*cos(y(1))];

*********************************************************

Then transfer function

*****************************************************

xspan=[0.01,10];

y0=[-0.126;pi];

[x,y]=ode45('skyrmion',xspan,y0);

plot(x(:),y(:,2))

****************************************************

problem one:

Because The ODE was divided by x^2, so the x span can't use x==0. whether the code can be modified to not divide x^2.

problem two:

the results get by using my code are not consist with some papers.

so I want to know my code is right or wrong.

[Walter Roberson]

To confirm: x is independent and there is y(x) ? In particular, it is not x(t) and y(t) ?

And y'' is diff(diff(y(x),x),x) ?

[petter]

Yes,y is y(x) and y''is diff(diff(y(x),x),x) .

can you give some advise for solving this ODE.

Thank you again.

Answer 1

[Star Strider]

Your initial equation:

x^2 y''+x y'-sin(y)cos(y)+x sin(y)^2-x^2 sin(y)cos(y)=0

does not contain the ‘-4*k*pi’ term that appears in your second one:

yprime = [y(2); -1/x*y(2)+1/x^2*sin(y(1))*cos(y(1)) -4*k/pi * (1/x)*sin(y(1))^2+sin(y(1))*cos(y(1))];

So should

-4*k*pi

be

-4*k*pi/x^2

instead?

I found no other problems with your conversion of the first equation to your ‘yprime’ matrix expression. You could simplify it a bit by combining the ‘sin(y)*cos(y)’ terms to ‘(1+1/x^2)*sin(y)*cos(y)’ but I doubt that would make much of a performance difference.

I believe ‘ode45’ will avoid the singularity at x=0 on its own, and search for solutions elsewhere. If you want to be certain about this, contact TMW Tech Support and ask them.

[petter]

Thank you for your reply and careful examine.Indeed,I lost 4*k*pi in my initial ODE. I have change it. I know the problem of the singularity at x=0 is trouble and I can't solve it.

now except the problem of the singularity at x=0,other problem is if I want to carry out other boundary condition such as y(infinite)=0 what I should do.

you know solve second order ODE just need two initial condition. here I want to consider y(infinite)=0.

can you give me some advice for solving this problem?

[Star Strider]

Thank you for clarifying the ‘-4*k/(pi*x)’ in the system ode45 uses. I am running my own integration of your ode to test the validity of my answers, and incorporated your correction.

With respect to the asymptotic behavior of your system near x=0, you mentioned you were comparing your results to those in other papers. How did they approach that problem?

Considering that your equation contains trigonometric functions, I doubt y(x->infinity) is defined. With the constants and initial conditions you specified, it appears to converge to zero, but that is my empirical observation only.

Thank you for accepting my answer.

[petter]

In other paper, they solved this ODE by Runge-Kutta method. the different of asymptotic behavior near x=0 is not the largest, My results are different from their results in the region of large x

In deed, their results show y appears to converge to 0 with x -> infinity. My results show trigonometric performance for large x. I feel I should consider the consideration y(infinite)=0 But I don't know how I should add the boundary condition y(infinite)=0 to ODE45.

Thank you for your help.

[Star Strider]

My pleasure! The ‘ode45’ function is also a Runge-Kutta solver, according to the documentation. You and the authors you quote are both correct as I understand it. The behavior of the integrated differential equation is similar to that of ‘exp(-x)*cos(x)’ so while it approaches zero asymptotically for large x, it also remains periodic.

I doubt it is possible to add the boundary condition of y(x->infinity) -> 0 to ‘ode45’, although the function seems to do this itself. You might be able to do this with ‘bvp4c’. I have used ‘bvp4c’ once, so I do not feel qualified to comment on how you would apply it to your differential equation. I will experiment with it later.

Answer 2

[Walter Roberson]

Please check my transcription of your equations:

dsolve( [x^2*((D@@2)(y))(x)+x*(D(y))(x)-sin(y(x))*cos(y(x))+x*sin(y(x))^2-x^2*sin(y(x))*cos(y(x)), (D(y))(0) = 0, ((D@@2)(y))(0) = -.126])

When I use that, Maple believes there is no solution. Indeed it thinks that even when I do not provide the boundary conditions.

[petter]

Thank you for your reply. I have no idea about maple.Indeed,I believe my ODE has not analytical solutions. So I want to use ODE45 to get numerical solutions.

now except the problem of the singularity at x=0,other question is if I want to carry out other boundary condition such as y(infinite)=0

can you give me some advice for solving this problem?