Second ODE, initial conditions are zeros at infinity

In summary, the conversation discusses finding the temperature profile of a phase transition layer in the interstellar medium. The stationary solution involves a second-order differential equation with initial conditions of T(x=-∞)=1 and dT/dx(x=-∞)=0, and the need for numerical solving and choosing an appropriate integration starting point. One approach suggested is to multiply by dT/dx and use an iterating scheme of recursion equations.
  • #1
omyojj
37
0
second ODE, initial conditions are zeros at infinity!

I want to know the temperature profile of phase transition layer in the interstellar medium.
For stationary solution, the dimensionless differential equation I ended up with is

[tex]\frac{d^2T}{dx^2} = \frac{f(T)}{T^2} - \frac{1}{T} [/tex]
where [itex]f(T)[/itex] is some complicated but well-behaved function.
Boundary conditions are
[tex]T(x=-\infty) = 1 ,[/tex]
[tex]\frac{dT}{dx}(x=-\infty) = 0,[/tex]

However, [itex]f(T=1) = 1[/itex], one obtains [itex]\frac{d^2T}{dx^2}(x=-\infty) = 0[/itex]

How do I solve this numerically? where should I start the integration? and what should be the initial condition?
Do I need to Taylor expand the differential equation?

Thank you for your attention.
 
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  • #2


One approach is to multiply by dT/dx this eqaution to get:

[itex]1/2 d/dx (dT/dx)^2 = (f(T)/T^2-1/T)dT/dx = (f(T)-T)/T^2 dT/dx[/itex]

Now perhaps use an iterating scheme of recursion equations, and integrate from minus infinity to some arbitrary x.

Hope I helped somehow.
 
  • #3


hmm..can you elaborate on the iterating scheme of recursion eq. or share some links I can refer to?
 

What is a "Second ODE"?

A Second ODE (ordinary differential equation) is a mathematical equation that describes the relationship between a function and its derivatives. It contains a second derivative, making it a second-order differential equation.

What does it mean for initial conditions to be zeros at infinity?

Initial conditions refer to the values of the function and its derivatives at a certain point. When these initial conditions are considered at infinity, it means that the function and its derivatives approach zero as the independent variable approaches infinity.

Why is it important to specify initial conditions at infinity?

Specifying initial conditions at infinity is important because it allows us to fully define the solution to the Second ODE. Without these conditions, the solution may not be unique and may not accurately represent the behavior of the function at infinity.

Can a Second ODE have initial conditions that are not zeros at infinity?

Yes, a Second ODE can have initial conditions that are not zeros at infinity. In fact, many real-world problems do not involve functions that approach zero at infinity. In these cases, the initial conditions at infinity are simply not considered.

What are some applications of Second ODEs with initial conditions at infinity?

Second ODEs with initial conditions at infinity are commonly used in physics, engineering, and other scientific fields to model various phenomena. For example, they can be used to describe the motion of a pendulum, the growth of a population, or the flow of electricity in a circuit.

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