Numerical method to solve high order ODEs.

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Discussion Overview

The discussion centers on numerical methods for solving high order ordinary differential equations (ODEs), specifically a simplified version of a third-order ODE and its associated boundary conditions. Participants explore various approaches to tackle the problem, including both numerical and analytical perspectives.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents a simplified third-order ODE and seeks hints for numerical solutions.
  • Another participant suggests that the problem might not require numerical solutions, although they admit to not having tested this idea.
  • A participant emphasizes the complexity of the original equations, indicating that an exact analytical solution is not feasible and expresses a desire to learn general numerical methods.
  • One participant proposes transforming the third-order ODE into three interconnected first-order equations using substitutions for derivatives, suggesting the use of a fourth-order Runge-Kutta method for numerical integration.
  • Another participant reiterates the suggestion of using the fourth-order Runge-Kutta method and mentions the Adams-Moulton-Bashforth method as an alternative that may offer greater accuracy, while noting the necessity of using RK4 for initial steps.

Areas of Agreement / Disagreement

Participants express differing views on whether numerical methods are necessary, with some suggesting alternative approaches while others focus on numerical techniques. The discussion remains unresolved regarding the best method to apply.

Contextual Notes

Participants acknowledge the complexity of the original equations and the limitations of finding exact solutions, which may affect the applicability of proposed methods.

Who May Find This Useful

This discussion may be useful for individuals interested in numerical methods for differential equations, particularly those dealing with high order ODEs and seeking various approaches to solve them.

meridian
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here is a simplified version of my working equtions
[tex] y''' = \frac{(y'' y+y' y) y + y'y''}{y' + y''}[/tex]
and 3 related boundary conditions, is there some hints to solve such equation numerically?

ThX
 
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This might sound crazy, but note that everything here is a function of y. You may not have to resort to numerical solutions. But then again I haven't tried it out yet. Looks a little intimidating.
 
in the equation,
[tex]y = y(x)[/tex]
The original equation are much more complex, it is not possible to get a exact analytical solution for that. What I want to learn is the general numerical method to solve such equation.
 
Let u= y'(x), v= y"(x). Then your equation is becomes
[tex]y'= \frac{(v y+u y) y + uv}{y + v}[/tex]
That together with y'= u and u'= v gives you three interconnected first order equations. Do, say, a 4th order Runge-Kutta, advancing the step in all three equations at the same time.
 
Do, say, a 4th order Runge-Kutta, advancing the step in all three equations at the same time.

Thats would I would suggest. Although the Adams-Moulton-Bashforth method would work as well and probably give you a little bit more accuracy. But you will need the RK4 for the first couple steps anyway.
 

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