Solving nonlinear singular differential equations

In summary, when tackling singularities in nonlinear and singular differential equations like the Lane-Emden equation, the method used to solve the equations is crucial. One approach is to choose initial values that avoid singularities, while another is to classify and handle different types of singularities depending on the situation and goal. When using numerical methods, the robustness of the solution must also be considered. Viewing solutions as a vector field can be helpful in understanding and approaching these types of equations.
  • #1
wasi-uz-zaman
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TL;DR Summary
how to tackle singularity in the differential equations
hi, i am going through differential equations which are nonlinear and singular - like Lane-Emden equation etc.
my question is how to tackle singularity - while coding.
regards
 
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  • #2
What method are you using to solve these equations?

If you solve [tex]
\frac{1}{r^{n-1}} \frac{d}{dr}\left(r^{n-1}\frac{d\theta}{dr}\right) = -\theta^m[/tex] for [itex]n = 2[/itex] or [itex]3[/itex] subject to [itex]\theta(0) =1[/itex] and [itex]\theta'(0) = 0[/itex] then there is no difficulty: The singular point is at the origin, but you aren't enforcing the ODE there because [itex]\theta[/itex] and [itex]\theta'[/itex] are fixed by the boundary conditions.
 
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  • #3
I like to think of the solutions of an ODE as a vector field. @pasmith's answer means to avoid singularities by choosing initial values, i.e. trajectories which lead you through the field without encountering singularities.

If you still consider the entire field, then there are certain types of singularities: attractors, repellers, or intersecting trajectories, maybe even isolated points. Such a classification allows handling different types of singularities differently. It's always the same: what is the situation and what is the goal?

If you approach it numerically, then you have to consider how robust a solution is. This means if you vary the initial values, how much do the trajectories vary? Again a matter of the given situation and goal.
 
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  • #4
Yes, I remember the idea of solutions as a vector field being a revelation for me. I was fortunate that this notion was introduced in my initial ODE course.
 
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Related to Solving nonlinear singular differential equations

1. What is a nonlinear singular differential equation?

A nonlinear singular differential equation is a type of mathematical equation that involves a function and its derivatives. It is called "nonlinear" because the function and/or its derivatives are raised to powers other than 1, and it is called "singular" because the equation becomes undefined at certain points, known as singularities.

2. Why are nonlinear singular differential equations difficult to solve?

Nonlinear singular differential equations are difficult to solve because they do not have a general, analytical solution like linear equations do. This means that there is no one method that can be used to solve all nonlinear singular differential equations, and each one must be approached individually.

3. What techniques can be used to solve nonlinear singular differential equations?

There are several techniques that can be used to solve nonlinear singular differential equations, including separation of variables, substitution, and numerical methods such as Euler's method or Runge-Kutta methods. The specific technique used will depend on the form of the equation and the availability of initial conditions.

4. How are nonlinear singular differential equations used in science?

Nonlinear singular differential equations are used in many scientific fields, including physics, engineering, and biology, to model complex systems and phenomena. They can be used to describe the behavior of physical systems, such as the motion of objects under the influence of forces, or the growth and development of biological populations.

5. What are some real-world applications of solving nonlinear singular differential equations?

Some real-world applications of solving nonlinear singular differential equations include predicting the trajectory of a rocket, modeling the spread of diseases, and understanding the behavior of weather patterns. They are also used in designing control systems for machines and processes, such as in robotics and chemical engineering.

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