Confusion about series solutions to differential equations

In summary, the Frobenius method uses a function to approximate the solution at a singular point. The regular singular point is a point at which the function is singular and the approximation is good. The Frobenius approximation is valid for points within a certain distance from the regular singular point.
  • #1
timetraveller123
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i have used series solutions to differential equations many times but i never really stopped to think why it works i understand that the series solution approximates the solution at a local provided there is no singularity in which frobenius is used but i am not understanding how exactly it works because the series is only valid for local value of x then what is the use of using such a solution and more specifically how is this justified when using it to solve Legendre differential equation is it because we are only interested in the behavior of solution form -1 to 1
thanks for the help
 
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  • #2
The series solution is only valid up to the nearest singularity, but that is often enough. The local solution may be all you care about. If you care about the solution around other points beyond the validity of a local series solution, you can expand the series at that point and get a local solution there.
 
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  • #3
ok considering one of the Legendre differential equations
media%2Ffff%2Ffff06006-3e23-4790-9012-d24ef2838893%2FphpQ5slwD.png

so for this the normal first kind solution is only valid between -1 and 1 ?(where the singularities occur)

also speaking of locality, so in the case where there is no singularity the series solution becomes the global solution
say for something like
##
\frac{d y }{d t} = t
##
series solution gives us ## y = e^t## which also happens to be our global solution
or is this because the radius of convergence of e^t happens to be infinity that's why the local series soltuion become the global solution also which one is it? thanks you have already helped me clear up a lot
 

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  • #4
vishnu 73 said:
ok considering one of the Legendre differential equations
View attachment 224426
so for this the normal first kind solution is only valid between -1 and 1 ?(where the singularities occur)
Yes. Exactly correct.
also speaking of locality, so in the case where there is no singularity the series solution becomes the global solution
say for something like
##
\frac{d y }{d t} = t
##
series solution gives us ## y = e^t## which also happens to be our global solution
or is this because the radius of convergence of e^t happens to be infinity that's why the local series soltuion become the global solution also which one is it? thanks you have already helped me clear up a lot
The convergence of the series is up to the nearest singularity in the complex plane. So if there are no singularities in the complex plane, the series solution is good everywhere. A factor of ##(1+x^2)## in your first example would probably be a problem in the complex plane because of ##x= \pm i ##.
 
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  • #5
oh wow i see thanks are most applications of legendre de only concerned between -1 to 1 like the angular solution of hydrogen

edit:
also what would happen
##
(1 + x^2) \frac{d y }{d x} + y =0
## if i blindly apply series solution to this without considering the complex singularity would the solution be valid for just real numbers? thanks once again
 
  • #6
vishnu 73 said:
also what would happen
##
(1 + x^2) \frac{d y }{d x} + y =0
## if i blindly apply series solution to this without considering the complex singularity would the solution be valid for just real numbers? thanks once again
A power series will converge for numbers in a disk of the complex plane up to the closest singularity and diverge outside of the disk. So the singularities at ##z = \pm i## would mean that the expansion at ##z=0## would also not converge for real numbers outside of [-1, +1]. The reason for it not converging for the real numbers would not be apparent unless you consider the complex singularities.
 
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  • #7
oh now i get it is it like two dimensional circle whose radius is the distance to the nearest singularity?
 
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  • #8
vishnu 73 said:
oh now i get it is it like two dimensional circle whose radius is the distance to the nearest singularity?
Yes. It's the part of the real line that is in the disk of convergence in the complex plane.
 
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  • #9
thanks sir now there is one last thing i don't really know how the legendre polynomial of second kind is derived it would be of help if you can enlighten me on that
i doubt it would not be the series method how is it derived? thanks
 
  • #10
They can be very easily derived from the power series. It's a very powerful technique. Also, the derivative values of a function at a point b are closely related to the coefficients of the power series expanded at point b, so there are multiple ways to reach the same result. The n'th coefficient, ##a_n##, is ##a_n = \frac{f^{(n)}(b)}{n!}##
 
  • #11
sorry for the late reply i am understanding about that but now something has been bugging why and how does frobenius method work and where does the definition of regular singular point come from
thanks
 

1. What is a series solution to a differential equation?

A series solution to a differential equation is a method of finding an approximate solution by representing the unknown function as a power series and using this representation to solve the differential equation. This is useful when an exact solution cannot be found analytically.

2. How is a series solution different from other methods of solving differential equations?

A series solution is different from other methods, such as separation of variables or substitution, because it involves expanding the unknown function into a power series and solving for the coefficients. This allows for an approximate solution to be found, rather than an exact solution.

3. When is it appropriate to use a series solution to a differential equation?

Series solutions are often used when the differential equation cannot be solved through other methods, or when an exact solution is not necessary. They are also useful when studying the behavior of a system near a certain point, such as in a Taylor series expansion.

4. What are the advantages and disadvantages of using series solutions?

The advantages of using series solutions include their ability to provide approximate solutions for a wide range of differential equations, as well as their usefulness in studying the behavior of a system. However, they can be time-consuming to calculate and may not always provide a solution that is accurate enough for practical purposes.

5. Are there any limitations to series solutions for differential equations?

Series solutions may not always converge, meaning that the calculated solution may not be accurate. They also may not provide a solution for all types of differential equations, particularly those with singularities or discontinuities. Additionally, finding the coefficients of a power series can be difficult for more complex equations.

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