How do you solve a nonlinear recurrence relation?

In summary: The steady-state solution to many charge and current flow problems can be formulated as solution to Laplace's equation. That in turn is a differential operator that can be interpreted as an averaging operation. This is essentially a differencing operation between the function at a point and the average of it's neighboring values. Once you cast your problem in this form, iteration leads to the correct solution.
  • #1
lugita15
1,554
15
While solving a problem involving equilibrium positions of charges on a line, I came up with a recurrence relation which is nonlinear, and moreover implicitly defined. Here it is: [itex]x_{0}=0[/itex] and [itex]\sum^{n-1}_{i=0} \frac{1}{(x_{n}-x_{i})^{2}} = 1[/itex]. I should also mention that [itex]0 \leq x_{n}< x_{n+1}[/itex] for all [itex]n[/itex].

I can't even find an explicit expression for [itex]x_{n}[/itex] as a function of the previous terms, let alone as a function of [itex]n[/itex]. I've dealt with linear recurrences before, but how would I go about solving a nonlinear recurrence like this? Is it even possible to find a closed-form expression using elementary functions? If an exact solution is impossible, is there some way to get a numerical approximation?

Any help would be greatly appreciated.

Thank You in Advance.
 
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  • #2
I think the only way forward is numerical methods.
 
  • #3
Numerical and graphical solutions. "nonlinear dynamics and chaos" is an excellent book by Strogatz that outlines such methods.
 
  • #4
What you have is a statement about the field in integral (sum) form. Write it as a differential equation which will become a difference equation. Then you're home free linear or not.
 
  • #5
Antiphon said:
What you have is a statement about the field in integral (sum) form. Write it as a differential equation which will become a difference equation. Then you're home free linear or not.
How exactly would I "write it as a differential equation which will become a difference equation"?
 
  • #6
OK, I have two additional facts: [itex]x_{n}\geq n[/itex] for all [itex]n[/itex] and [itex]lim_{n\rightarrow\infty} (x_{n}-x_{n-1}) = \infty[/itex]

I don't know how much they'll help.
 
  • #7
lugita15 said:
How exactly would I "write it as a differential equation which will become a difference equation"?

The steady-state solution to many charge and current flow problems can be formulated as solution to Laplace's equation. That in turn is a differential operator that can be interpreted as an averaging operation. This is essentially a differencing operation between the function at a point and the average of it's neighboring values. Once you cast your problem in this form, iteration leads to the correct solution.
 

1. How do you determine the general solution to a nonlinear recurrence relation?

The general solution to a nonlinear recurrence relation can be determined by using various methods such as substitution, iteration, and generating functions. These methods involve finding a pattern in the recurrence relation and using it to create a formula that can be used to find any term in the sequence.

2. Can a nonlinear recurrence relation have multiple solutions?

Yes, a nonlinear recurrence relation can have multiple solutions. This is because there can be multiple patterns or formulas that can be used to find the terms in the sequence. It is important to check the initial conditions of the recurrence relation to ensure that the correct solution is being used.

3. What is the difference between a linear and a nonlinear recurrence relation?

A linear recurrence relation is one where the next term in the sequence can be determined by a linear combination of the previous terms. This means that the coefficients of the previous terms are constant. On the other hand, a nonlinear recurrence relation does not have constant coefficients and the next term cannot be determined by a linear combination of the previous terms.

4. Can a nonlinear recurrence relation be solved using a recursive algorithm?

Yes, a nonlinear recurrence relation can be solved using a recursive algorithm. This involves breaking down the recurrence relation into smaller sub-problems and using the solutions to these sub-problems to find the solution to the entire recurrence relation. However, this method may not be efficient for large or complex recurrence relations.

5. How do you know if a recurrence relation is nonlinear?

A recurrence relation is nonlinear if the coefficients of the previous terms are not constant, or if the next term cannot be determined by a linear combination of the previous terms. This can also be identified by the presence of variables with exponents or other nonlinear terms in the relation.

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