What Are the Best Algebraic Methods for Approximating Non-Linear ODE Solutions?

[nassboy]

I'm interested in knowing if there are any techniques besides taylor series and picards method to find approximate general solutions to non-linear ordinary differential equations.

I'm not interested in numerical techniques only algebraic approximations.

 

[matematikawan]

I saw at seminar people use Adomian Decomposition Method to solve nonlinear DE or pde. The method doesn't look that difficult to understand. It seem that convergence is faster than the Taylor series method. But I haven't try it yet. May be in the future when need arises.

 

[nassboy]

This looks really cool. Do you know if anybody has developed a mathematica function to produce the series?

 

[lurflurf]

What do you consider algebraics? All the standard numerical methods like Runge-Kutta have variants that return functions such as polynomials. There are perturbation methods including the Lanczos tau method. There are other series methods like Fourier series.

 

[matematikawan]

What do you consider algebraics? All the standard numerical methods like Runge-Kutta have variants that return functions such as polynomials.

I think what nassboy meant is analytical solution. You know something like series solution where we can integrate or differentiate.

I never know that Runge-Kutta method can return polynomial as solution. This will be great because I always though that the method only give us points which we can display graphically. Not equation.

 

[kosovtsov]

I believe that the systematical method to obtaine approximate analytical general solutions to non-linear ODEs or PDEs is the operator method

1. http://arxiv.org/abs/0910.3923
2. http://arxiv.org/abs/math-ph/0409035

You can find in [1] the references to Maple procedures to produce the series.

 

[alejandrito29]

i too will to solve non-linear ordinary differential equations of second order. Using dsolve comand on Maple, the solutions delivered by maple are very poor and extensive.
someone recommend me some method by using maple?
my level of awareness is low
there is a some package for to solve diferential equations non lineal second order?

 

[nassboy]

I just implemented the Adomian Decomposition method for my problem in mathematica. It didn't produce answers too much better than the regular old taylor series.

Lots of people have modified the adomian decomposition method to get better results for their specific problem, but being an engineer I don't really have time for that stuff.

 

[matematikawan]

I won't recommend the method next time. Sorry that you had taken all the trouble coding the method but didn't see much improvement in the result.

 

[nassboy]

Don't be sorry...it is cool. It might work better for some differential equations than others.