Problem with finite diffence method

[zeynepkisa]

I have been trying to solve this problem for hours and hours but derivative boundary condition makes the it very hard. can anybody help me about nonlineer eq. solution with finite difference??

question is:

u'' + 3*u^2 *( 1 / (sin^2(x)) =2.5
BCs:

u'(1)=0.95
u(2)=0.83
h=0.25
(radian for sin function)

 

[HallsofIvy]

Exactly how are you applying "finite difference"? Are you assuming a linear solution overe each interval?

 

[arildno]

As for the boundary condition, since it involves a derivative, you should regard it the actual left boundary as an interior point.
Thus, introduce a fictitious boundary point [itex]x_{0}=1-h[/tex]
and you have n+1 points [itex]x_{0}, x_{1}=1,\cdots{x}_{n}=2[/tex]

The differential equation should be satisfied at all interior points, so that you have n+1 non-linear equations like this:
$$\frac{u_{2}-u_{0}}{2h}=0.95, u_{n}=0.83$$
and:
$$\frac{u_{i+1}-2u_{i}+u_{i-1}}{h^{2}}+3u_{i}^{2}\frac{1}{\sin^{2}(x_{i})}=2.5, i=1,\cdots{n-1}$$

Now, you are ready to see how to implement a loop structure to actually solve this system approximately..

 

[zeynepkisa]

thanks for your attention

 

[arildno]

Have you decided upon a strategy to take care of the non-linearity?

 

[zeynepkisa]

yes Newton method