Frobenius Method (1)

Frobenius Method (1)

– Let b(x) and c(x) be any analytic functions at x = 0. Then, ODE

has at least one sol. represented in the form

where r may be any (real or complex) number.

– The ODE has a second sol. (which is linearly independent) similar to above series form with a different r and different coeff. or may

contain a logarithm term.

   

2

0

b x c x

y y y

x x

    

  



0

r m r

m m

y x x a x x a a x a x a x





       ^{ a}

^{ 0 }

Frobenius Method (2)

– Application example 1 of Frobenius method: Bessel’s eqn.

– Application example 2: hyergeometric DE

– Regular point --- a point x₀ at which p(x) and q(x) are analytic. Then, power series method can be applied.

Or, a point x₀ at which are analytic, and

– Singular point --- not regular

2 2

2

1 x 0

y y y

x x

  

      

 

 ^{b x}   ^{ 1, c x}   ^{ x}

^{ }

    ⁰

y   p x y   q x y 

      ⁰

h x y   p x y   q x y  , ,

h p q h x  

 0

Indicial Equation (1)

– Multiplying x² on the given DE

– Expand b(x) and c(x) in power series

– Term by term differentiation of the assumed sol.

   

2

0

x y   xb x y   c x y 

 

,  

b x   b b x b x   c x   c c x c x  

   

 

0

m r r

1

m m

y x m r a x x ra r a x

   



           

   

2

0 1

1 1

x

r r a r ra x

        

    

0

1

m

y x m r m r a x

  



     

Indicial Equation (2)

– Substituting into the given DE

– Sum of the coeff. corresponding to x^r

– Two roots of the indicial eqn.: r₁ and r₂

   1 

 

0 r r b r c

     

   

0 0 1 0

0 1 0 1

1

0

     

 

 

     

r r

r

x r r a b b x x ra

c c x x a a x

Indicial equation of ODE

Indicial Equation (3)

– Case I: Distinct roots not differing by an integer

– Case II: Double root r₁ = r₂ = r

– Case III: Roots differing by an integer

  



1









y x x a a x a x

    



2



ln 





y x y x x x A x A x

 





1









y x x a a x a x

   





2



ln 







y x ky x x x A A x A x

 





1









y x x a a x a x

 





2









y x x A A x A x

Example of Case III (1)

– DE

– Applying Frobenius method

– Collect all the terms with power x^m+r and simplify

 ^x

^{ x y}  ^{  xy    y 0}



 ^{  }

^{ }

0 0

0

1

0

 

   

 

 



      



 



m r m r

m m

m m

m r m

m

x x m r m r a x x m r a x

a x

 

  

0 0

1 1 0

 

  

 

      





m m

m r a x m r m r a x

Example of Case III (2)

– Set m = s in the first series, and m = s + 1 in the second

– Lowest power is x^r-1, and the resulting indicial eqn.

– , differ by an integer, Case III – First sol. y₁ corresponding to

– Recurrence formula

 

  

0 1

1 1 0

 

 



 

      





s s

s r a x s r s r a x

 ^{  1}  ⁰

r r

1

 1,

 0

r r

1

 1 r

  

2 1

1 0

2 1 0

 





     

 



s

s a s s a x

  

2

1

2 1





 

s s

a s a

s s

Example of Case III (3)

– , Taking

– Second sol. y₂ --- can be obtained by the reduction of order

– Substituting

– Dividing by x and simplifying

2



 ,

    ,

    2  y y u xu y xu u y xu u

1

 0,

 0,

a a

1

1



 y x a x

0

 1 a

 ^x

^{ x}  ^{ xu   2 u     x xu   u   xu  0}

 ^x

^{ x u}  ^{   x  2  u   0}

Example of Case III (4)

– Partial fractions and integrating

– Taking exponents and integrating

– y₁, y₂: linearly independent

2

2 2 1

1 ,

      

  

u x

u x x x x

ln   ln x  1

u x

2 2

2

1 1 1 1

, ln ,

ln 1

      

  

u x u x

x x x x

y xu x x

Bessel’s Eqn. and Functions (1)

– Bessel’s DE

– Electrical field, vibrations, heat conduction with cylindrical symmetry ν: real and nonnegative

– Can apply Frobenius method

– Substituting

 

2 2 2

0 x y ^  xy ^  x   y 

 

0

r m r

m m

y x x a x

 



  ^{ a}

^{ 0 }

    

0 0

2 2

0 0

1

0

m r m r

m m

m m

m r m r

m m

m m

m r m r a x m r a x

a x  a x

 

 

 

 

  

 

    

  

 

 

Bessel’s Eqn. and Functions (2)

– Collecting the coeff. of x^s+r

– Indicial eqn.

– Two roots:

– Coeff. recurrence for the first root r = r₁ = ν

 ^{s  2, 3,} 

 1 

0

r r  a  ra   a   ^{s  0} 

 r  1  ra

   r 1  a

 

a

 0  ^{s  1} 

 s  r  s   r 1  a

   s r a 

 a

 

a

 0

 ^{r  }  ^{r  }  ^{ 0}

 

1

0 ,

r    r   

 s   r   s  

r   a

 a

 0 0

a 

Bessel’s Eqn. and Functions (3)

– Resultantly,

– We only have the coeff. of even numbers s = 2m

– Even numbered coeff.

 s  2   sa

 a

 0

3

0,

0,

a  a 

 ^{2 m  2 }  ^{2 ma}

^{ a}

^{ 0}

 

2 2 2 2

1 ,

m

2

a a

m  m

   m  1, 2,



 

 

2 2

,

2 1 2 2! 1 2

a a

a a

  

  

  

   

  

2 2

1 ,

2 ! 1 2

m

m m

a a

m    m

 

   m  1, 2,

Bessel Function of the 1

Kind J

(x) (1)

– Integer ν now denoted by n

– Arbitrary a₀ determined as follows:

– Bessel function of the 1^st kind of order n --- converges for all x

   

  

2 2

1 ,

2 ! 1 2

m

m m

a a

m n n n m

 

   m  1, 2,

0

1 , 2

!

a  n  

 

2 2

1

2 ! !

m

m m n

a

m n m

 

 m  1, 2,

   

 

2 2

0

1

2 ! !

m m

n

n m n

m

J x x x

m n m



 

 

 

– Bessel function of order 0

   

       

2 2 4 6

0 2 2 2 2 4 2 6 2

0

1 1

2 ! 2 1! 2 2! 2 3!

m m

m m

x x x x

J x

m





        

Bessel Function of the 1

Kind J

(x) (2)

Bessel Function of the 1

Kind J

(x) (1)

– Extend from integer ν = n to any ν ≥ 0 – Gamma function Γ(ν)

– Integrating by parts

 

0

e t

dt

  

 ^  

 

0

0 0

1 e t dt

e t

e t

dt

 

    

   ^ 



 ^ 

 ¹   

      

   

0 0

1 e dt

e

0 1 1

   

        

  ²   ^{1 1!,}   ^{3 2}   ^{2 2!,}

       

 ^{n 1}  ^{n !}

    ^{n  0,1,} 

– Now for any ν

– The even numbered coeff.

– Bessel function of the 1^st kind of order ν --- converges for all x

 

0

1 1

2 ! 2 1

a  





  

   

2 2

1

2 ! 1

m

m m

a

m m

 

  





   

 

2 2

0

1

2 ! 1

m m

m m

J x x x

m m



 

 

  



  



Bessel Function of the 1

Kind J

(x) (2)

General sol. Of Bessel’s DE

– Replacing ν by - ν

– General sol. of Bessel’s DE for non-integer ν

– If ν is an integer, above form is not a general sol. because they are linearly dependent.

   

 

2 2

0

1

2 ! 1

m m

m m

J x x x

m m

 

 



 

  



  



 

 

 

y x  c J

x  c J

x

    ¹

 

n n

J

x   J x  ^{n  1, 2,} 

   

   

 

2 2

2 2

0

1 1

2 ! ! 2 ! !

m m n n s s n

n m n s n

m n s

x x

J x

m m n n s s

  

 

  

 

 

 

 

 

Properties of Bessel function

– Derivatives

– Recurrence relation

– For half-integer order ν

 

 

x J x  x J

x

  





 

 

x J

x  x J

x

   





     

1 1

J x J x 2 J x





 x

  



     

1 1

2

J

x  J

x  J

 x

   

1 2 1 2

2 2

sin , cos

J x x J x x

x

x

 

 

– When ν is an integer, need a second linearly independent sol.

--- Bessel function of the 2^nd kind Y_n(x) – n = 0: Bessel function of the 2^nd kind Y₀(x)

– Indicial eqn. has double root, , Case II – Second desired sol. of the form

0 xy    y  xy 

0 r 

   

2 0

1

ln

m

y x J x x A x





  

0 1

2 0

1

ln

m

y J x J mA x

x

 



     

 

0 0

2 0 2

1

ln 2 1

m

J J

y J x m m A x

x x

 



        

Bessel Function of the 2

Kind Y

(x) (1)

– Substituting into the DE and resulting eqn.

– Expression for J’₀(x)

– Resulting eqn.

– Collecting the power of x⁰, x^2s

 

0

1 1 1

2 1

0

m m m

J m m A x mA x A x

  

 

  

        

   

 

   

2 1 2 1

0 2 2 2 1

1 1

1 2 1

2 ! 1 !

2 !

m m m m

m m

m m

mx x

J x

m m m

 

 

  

 

  

  

   

2 1

2 1 1

2 2

1 1 1

1 0

2 ! 1 !

m m

m m

m m

m

m m m

x m A x A x

m m

   

 

   

   

   

1

0,

0,

0,

A  A  A 

Bessel Function of the 2

Kind Y

(x) (2)

– Even numbered coeff.

– Expression for the second linearly independent sol. y₂(x)

– Another basis if y₂ replaced by an independent particular sol. With (γ: Euler const.)

 

 

1

2 2 2

1 1 1 1

1 ,

2 3

2 !

m

m m

A m m



 

         m  1, 2,

     

 

1

2

2 0 2 2

1

ln 1

2 !

m

m m

m m

y x J x x h x

m

 



   

1

1 1

1, 1

m

2

h h

     m m  2, 3,





, 2 , ln 2

a y  bJ a  b   



Bessel Function of the 2

Kind Y

(x) (3)

– Bessel function of the 2^nd kind of order 0 --- Neumann’s function of order 0

     

 

1

2

0 0 2 2

1

2 1

ln 2 2 !

m

m m

m m

x h

Y x J x x

m

 



    

 _       ^       

Bessel Function of the 2

Kind Y

(x) (4)

Bessel Function of the 2

Kind Y

(x) (1)

– When ν = n = 1,2,…, y₂ can be obtained as in Case III – Standard second sol. for all ν

--- Bessel function of the 2^nd kind of order ν, Neumann’s function of order ν

– n = 1,2,…: Bessel function of the 2^nd kind of order n Y_n(x)

  ¹   ^cos  

Y x sin J x J x

x

     

 





  ^lim  

n n

Y x Y x





       

 

1

2 2

0

2 1

ln 2 2 ! !

n m

m m n m

n n m n

m

h h

x x

Y x J x x

m m n

 



 

 

 

 _    ^    _  

Bessel Function of the 2

Kind Y

(x) (2)

 

1

2 2

0

1 !

2 !

n n

m m n

m

x n m

m x

 

 

 _   

– Furthermore

– General sol. of Bessel’s DE

– Bessel function of the 3^rd kind of order ν, 1^st and 2^nd Henkel function of order ν

      ¹

n n

Y

x   Y x

 

 

 

y x  c J

x  c Y

x

 

     

 

     

1 2

H x J x iY x

H x J x iY x

 

 

  

  