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February 19th, 2016, 10:54 PM  #1 
Newbie Joined: Feb 2016 From: universe Posts: 7 Thanks: 0  Question about solving ODE
hi guys, Hope it's okay, but I typed out my question to make the math equations more clear. I had a general question about solving ODEs. So as we know, if you have an equation of the form: thanks in advance for your help 
February 20th, 2016, 02:00 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,828 Thanks: 2160 
I would suggest solving for each set of conditions separately. It may be possible to spot whether the solutions obtained are compatible.

February 20th, 2016, 09:14 PM  #3  
Newbie Joined: Feb 2016 From: universe Posts: 7 Thanks: 0  Quote:
Yes, the solutions do intersect. However, the solution point is a singularity which is something I don't want. I would like to solve for both simultaneously, if possible, so that I get a smooth continuous function. I was sure that there would be some kind of approximation method to do this, but I've searched and can't find one. Maybe I'm missing something? Any other ideas?  
February 21st, 2016, 03:18 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,828 Thanks: 2160 
If there is a "solution" that "has a singularity", it's usually the case that every solution has a singularity somewhere. I don't understand your remark "the solution point is a singularity" as a function and its derivatives aren't defined at a singularity.

February 23rd, 2016, 09:10 PM  #5  
Newbie Joined: Feb 2016 From: universe Posts: 7 Thanks: 0  Quote:
What I'm looking for is a method for numerical integration that SIMULTANEOUSLY uses the information at BOTH points x0, v0, a0 and x1, v1, a1 to create a smoother curve, shown below. There has to be some way to do multipoint numerical integration. The method would need to somehow use the information for both points and at the same time or in some intelligent way to create the overall curve. Are there advanced methods that do this?  
February 23rd, 2016, 11:13 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,828 Thanks: 2160 
The two curves in your first illustration show the common situation that they can't be joined where they intersect because the original equation wouldn't be satisfied at that point. You can't change the curves to produce a smooth join without producing a function that doesn't satisfy the equation you solved. In particular cases, it sometimes happens that the two curves intersect at a point where they have the same slope (and the same second derivative), in which case, more than one set of conditions can be satisfied by a single "smooth" solution.

February 24th, 2016, 01:06 AM  #7  
Newbie Joined: Feb 2016 From: universe Posts: 7 Thanks: 0  Quote:
For example, if I start at x0 and iterate forward with +e. At the same time I start at x1 and iterate backward with e. Then using some intelligent way, perhaps using weighted distance or some other approach, I could form the overall green curve shown above. 1) Are there any methods at all that attempt to tackle/investigate this issue? 2) What, in your opinion, is the best way to create a smooth curve using both x0(blue curve) and x1(red curve)? thanks again for all your help. Really appreciate it.  
February 24th, 2016, 03:46 AM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
Given the picture you give, there must be a stationary point somewhere near the intersection of the two approximations. Can you use that fact? Is there any benefit in writing $y=\dot x$ and studying the phase space (x,y) of the system? Last edited by skipjack; February 24th, 2016 at 04:39 AM. 
February 24th, 2016, 04:33 AM  #9 
Global Moderator Joined: Dec 2006 Posts: 20,828 Thanks: 2160 
In general, no "green curve" exists that satisfies the equation. If, for example, your differential equation has an arbitrary straight line as its general solution, there's little point in trying to join a "red" line that satisfies one set of conditions to a separate "blue" line that satisfies another set of conditions. 
February 24th, 2016, 08:19 PM  #10  
Newbie Joined: Feb 2016 From: universe Posts: 7 Thanks: 0  Quote:
Quote:
I'm just surprised that there aren't other approximate methods besides the simple numerical/perturbation approach I stated in my first post. I have found things like the Newmarkbeta method, but unless I'm mistaken, that is simply a variation. I would need some kind of multipoint technique.  

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