boundary value problem - constrained paramter


by Deadstar
Tags: boundary, constrained, paramter
Deadstar
Deadstar is offline
#1
Nov25-12, 03:41 PM
P: 106
Let's say I have a set of nonlinear differential equations of the form.

[tex]x' = f(x,y) \\
y' = g(x,y)[/tex]

Where f and g contain some parameter 'a' that is constrained to within certain values.

Let's say I know x(0), y(0) and x(T), y(T) where T isn't a set value. What methods can I use to solve/integrate this to match the boundary conditions with the parameter 'a' free to change. I suppose if T could be minimized that would be nice but it's not essential, just looking for general methods used to solve these sort of problems.
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
pasmith
pasmith is online now
#2
Nov26-12, 09:50 AM
HW Helper
P: 772
Quote Quote by Deadstar View Post
Let's say I have a set of nonlinear differential equations of the form.

[tex]x' = f(x,y) \\
y' = g(x,y)[/tex]

Where f and g contain some parameter 'a' that is constrained to within certain values.

Let's say I know x(0), y(0) and x(T), y(T) where T isn't a set value. What methods can I use to solve/integrate this to match the boundary conditions with the parameter 'a' free to change. I suppose if T could be minimized that would be nice but it's not essential, just looking for general methods used to solve these sort of problems.
If I understand correctly, the system you have is this:
[tex]
x' = f(x,y,a) \\
y' = g(x,y,a)
[/tex]
where a is an unknown constant lying between certain values. You are given values of x(t) and y(t) at t=0 and t=T, and you need to determine a.

This is essentially a root-finding problem: given x(0), y(0) and T, the value x(T) is then a function of a. Let's call it X(a). Similarly for Y(a). Thus your problem is to find a such that
[tex]
X(a) - x_1 = 0, \\
Y(a) - y_1 = 0
[/tex]
where [itex]x_1[/itex] and [itex]y_1[/itex] are the given values for x(T) and y(T).

Let's concentrate on the first of those. If we can find a such that [itex]X(a) = x_1[/itex], we can check whether the second equation is satisfied; if it isn't then there is no solution.

There is no general analytic method for this; it must be done numerically.

There are a number of root-finding algorithms which may be suitable, but the first step is always to obtain a graph of X(a) for suitable a, and see whether a solution is likely to exist. In general, there is no guarantee that a solution exists, and no guarantee that a solution is unique if it does.

Given a, you find X(a) (and Y(a)) by solving the ODEs numerically subject to the given conditions at t = 0 and with the given value of a to determine x(T) (and y(T)).
Deadstar
Deadstar is offline
#3
Nov27-12, 06:12 AM
P: 106
Hi thanks for the reply.

That's sort of what I'm after but it seems like that solution will give a fixed choice of a such that the boundary conditions will be met. But what if a can change?

Let's use a discrete dynamical system as an example to show what I mean.

Let [tex]x_{n+1} = a x_n[/tex] where a = {2,3}

We have x(0) = 1, x(T) = 12.

Clearly having a fixed at 2 or 3 throughout will not give match the boundary conditions (a = 2: 1 -> 2 -> 4 -> 8 -> 16 ..., a = 3: 1 -> 3 -> 9 -> 27 ...)

But if we change a...

a = 2

1 -> 2 -> 4

a = 3

4 -> 12.


So this might be a bad example since I'm literally thinking it out as a type this but going back to the original problem. If a is fixed, we may not be able to find a solution, but the selection of a may still be valid at x(0), it will just change in time.

Perhaps this should be solved by some sort of multiple shooting method? Such that a can change in each time interval.


Register to reply

Related Discussions
Eliminate paramter Calculus & Beyond Homework 2
constrained minimum problem Calculus 1
Constrained extreme problem Calculus & Beyond Homework 6
Constrained Minimization Problem(HELP ASAP) Calculus & Beyond Homework 2
Lagraingian constrained optimization problem General Math 3