Solving 3rd order ODE with B.C. of Numan and Dirichlet Type

[oby7842]

Hiii,

In my research i encountared with following 3rd order ODE:

d^3y/dx^3 = (1-y)/y^3.

my B.C. are:

at x = 50 y=1(dirichlet B.C.) ; dy/dx=0 ( numan B.C.) and d^2y/dx^2=0.

i need to integrate from x=0:50. I tried ode45 but it gives same result as B.C. throughout the range. If anyone have any idea how can i approach to solve this problem please let me know. thanks.

Omar Bin Yusuf

 

[gato_]

the command is
[x,y]=ode45(@(x,y)[y(2);y(3);(1-y(1))/y(1)^3],[50 0],[1;0;0]);

I've tried, and obtained the same result as you. This is an equilibrium point of the system. if you try y(50)=1+1e-15, it blows up. If you try y(0)=1-1e-15, it blows up too (when reaching y=0).

 

[oby7842]

Thanks gato. i tried with y(50)=1+1e-15. and it works. I have 3 more question i hope you will be able to help me.

1. In my problem the upper end ( x = 50 ) is not fixed. In original problem it was said that Y( as x goes to infinity) =1. at first I sovled the problem by assuming upper limit as 50. What should be the approach of fixing this upper limit.

2. Again although if I use y(50)=1+1e-15 and use ode45, we can see that y remains equal 1 far below x= 50. Why is this happening?

3. Inliterature i found for this problem authors described another BC by dy/dx(x=0)=alpha. where 'alpha' is unknown. So now can i solve this problem by bvp4c and solve the problem along with that unknown parameter? I tried but my results are heavily influenced by initial guess.

Thank you again for your reply. your ans really help me alot.