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A Malik
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can we use only frobenius method to solve bessel equation?
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Why do you want to use series developments to solve Bessel ODEs ?A Malik said:can we use only frobenius method to solve bessel equation?
JJacquelin said:Why do you want to use series developments to solve Bessel ODEs ?
On the contrary, the Bessel functions are closed forms which avoid the series developments.
Sorry, I cannot understand what is the confusion.jackmell said:That's confussing. I mean I look in my DE book, even gotta' section on Bessel DE, and bam! Series solutions.
JJacquelin said:Sorry, I cannot understand what is the confusion.
For example, in attachment, two ODEs are compared.
Both solutions can be expressed on closed form thanks to functions which are also infinite series. Of course, the name given to the functions are not the same : J0(x) and cos(x) for example. But I cannot see any other difference, except that cos(x) is more popular than J0(x). But popularity has nothing to do with maths.
jackmell said:Ain't there suppose to be a factor of [itex]2^{2k}[/itex] in there?
May I also ask how is that first equation solved directly if one does not know before-hand, it is a Bessel DE? I think we'd have to resort to power series. However, the OP I think is asking, is there another way other than power series to solve it?
By the way, thanks for helping us in here as you solve in an elegant manner, equations I can't solve and maybe others too. :)
The Bessel equation is a second-order ordinary differential equation that arises in many areas of physics and engineering, particularly in problems involving circular or cylindrical symmetry.
The solution to the Bessel equation is a class of special functions known as Bessel functions. They are denoted by the letter J and have various orders and types.
Bessel functions have many important properties, including orthogonality, recursion relations, and asymptotic behaviors. They are also closely related to other special functions, such as the hypergeometric function and the gamma function.
Bessel functions have a wide range of applications in areas such as electromagnetics, acoustics, fluid mechanics, and signal processing. They are also commonly used in the solutions of partial differential equations that arise in these fields.
There are several methods for solving the Bessel equation, including power series, Frobenius series, and integral representations. Other techniques, such as the WKB approximation and numerical methods, can also be used in certain cases.