Approximate Solutions to Non-Linear Differential Equations

In summary: Perturbations can be a useful technique for solving nonlinear equations. In quantum mechanics, the "WKB approximation" is a method of perturbing the system to get a more accurate solution.
  • #1
nassboy
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I'm interested in coming up with an function that approximates the solution to a non-linear differential equation. (There is no known closed form solution)

The equation is Y''=(1/Y)(Y')^2-Y*A+Y^2*A where "A" is a constant, and Y' and Y'' are the first and second derivatives with respect to x.

How would I look for approximations to the answer in the literature? Is this equation a member of a certain class of differential equations?

Is there a technique known to work well?

Any help would be appreciated!
 
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  • #2
Since the independent variable does not appear explicitely in this equation, it looks like a candidate for "quadrature". Taking x to be the independent variable, let z= dy/dx. Then d2y/dx2= dz/dx= (dz/dy)(dy/dx)= zdz/dx. Now the equation becomes
[tex]z\frac{dz}{dy}= \frac{z^2}{y}- Ay+ Ay^2[/tex]
a separable first order equation.

Another typical method for non-linear equations is "perturbations". The "WKB approximation" in quantum mechanics is a type of perburbation method.
 
  • #3
quadrature as in numerical integration?
 
  • #4
If I work the problem using this form [tex] z\frac{dz}{dy}= \frac{z^2}{y}- Ay+ Ay^2[/tex]

, I'm still left with an integral that has no solution when I try to integrate a second time!
 
  • #5
Well, set [tex]u=\frac{1}{2}z^{2}[/tex], rewriting the equation as:
[tex]\frac{du}{dy}-\frac{2}{y}u=-Ay+Ay^{2}[/tex]
Multiplying this by the integrating factor y^-2, we get:
[tex]\frac{d}{dy}(\frac{1}{y^{2}}u)=-\frac{A}{y}+A[/tex],
or, by integrating:
[tex]u=Cy^{2}+Ay^{3}-Ay^{2}\ln(y)[/tex]
Thus, we get:
[tex]\frac{dy}{dx}=\pm\sqrt{2(Cy^{2}+Ay^{3}-Ay^{2}\ln(y))}[/tex]

which is separable, if not easy to solve..
 
  • #6
Thanks for the help!

[TEX]\int \frac{1}{\sqrt{2\left(C Y^2+A Y^3-A Y^2 \text{Log}[Y]\right)}} [/TEX]

(I hope that is right, the latex preview isn't working for me)

After separating the variables, one is left with the integral above which has no closed form solution. What is the best way to approximate the integral, keeping in mind that I want to solve algebraically for y in the end.
 
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Related to Approximate Solutions to Non-Linear Differential Equations

1. What is the difference between linear and non-linear differential equations?

Linear differential equations involve terms that are only directly proportional to the dependent variable and its derivatives. Non-linear differential equations involve terms that are not directly proportional, making them more complex and difficult to solve.

2. Why are approximate solutions used for non-linear differential equations?

Non-linear differential equations often do not have exact solutions that can be found using traditional methods. Therefore, approximate solutions are used to estimate the behavior of the system and provide a close enough solution.

3. How are approximate solutions to non-linear differential equations obtained?

Approximate solutions can be obtained using numerical methods such as Euler's method, Runge-Kutta methods, or finite difference methods. These methods involve breaking the differential equation into smaller, simpler equations and solving them iteratively.

4. What are the limitations of using approximate solutions for non-linear differential equations?

Approximate solutions may not be accurate for all values of the independent variable, and they may also be sensitive to small changes in the initial conditions. Additionally, the complexity of the non-linear equation may require a large number of iterations to obtain an accurate solution.

5. Are there any real-world applications of using approximate solutions for non-linear differential equations?

Approximate solutions for non-linear differential equations are commonly used in fields such as physics, engineering, and economics to model real-world systems. For example, they can be used to predict the behavior of a chemical reaction or the movement of a satellite in space.

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