Page 9 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION
P. 9

Differential and Integral Equation




                    Notes          2.  Recurrence relations are quite useful as they help in finding whole class of J (x) in terms of
                                                                                                   n
                                       two or three J (x) of lower values of n i.e., n = 0, 1, 2.
                                                  n
                                   3.  Generating function  for J (x) is introduced  so that certain formulas involving  Bessel
                                                             n
                                       functions can be deduced. With the help of generating functions we can deduce recurrence
                                       relations or certain other formulas straight away.
                                   4.  Finally we also discuss the zeros of Bessel functions as they will lead us to the completeness
                                                                           ,
                                       as well as orthogonality properties of Bessel s Functions.
                                              ,
                                   1.1 Bessel s Differential Equations from Laplace Equations

                                   In dealing with the theory of potential problems in electrostatics or in gravitational field we
                                   commonly use Laplace equations

                                                              2    2    2
                                                               V   V    V
                                                                           = 0                             ...(1)
                                                              x 2  y 2  z 2
                                   Here V is a function of the Cartesian Co-ordinates. Any solution  V  of this equation, which is a
                                                                                        n
                                   homogeneous polynomial of degree n in x, y, z is called the solid spherical Harmonies.
                                   Depending upon the symmetry of the problem we can express Laplace equation in cylindrical
                                   co-ordinates (r,  ,  z) or spherical polar co-ordinates (r,  ,  ). You must be knowing that the
                                   relations between x, y, z and r,  , z are
                                                                         x = r cos

                                                                         y = r sin                         ...(2)
                                                                         z = z
                                   Also the relation between x, y, z and r,  ,   are

                                                                         x = r sin   cos
                                                                         y = r sin   sin                   ...(3)
                                                                         z = r cos
                                              ,
                                   1.2 Bessel s Differential Equations

                                                                               ,                              ,
                                   To define Bessel functions we first of all obtain Bessel s Differential equation from Laplace s
                                                                 ,
                                   equation. To do that we write Laplace s equations (1) in cylindrical co-ordinates as
                                                      2            2    2
                                                       V  1 V   1  V    V  = 0                             ...(4)
                                                      r 2  r  y  r 2  2  z 2
                                   We assume that V as a function of r,   and z can be written as
                                                             ,   ,
                                                   V = R(r)   ( ) z (z)                                    ...(5)
                                            ,  ,
                                   Where R,   , Z  are functions of r,  , z alone respectively. Substituting in (4) we get
                                             2
                                                                        2
                                            d R  1    dR   RZ d 2      d Z
                                          Z   2     Z       2    2  R    2  = 0
                                            dr   r     dr  r  d        dz
                                               2
                                                                       2
                                             1 d R  1 dR  1  1 d 2  1 d Z
                                   Or           2          2     2       2  = 0                            ...(6)
                                             R dr  rR dr  r    d    Z dZ

          2                                 LOVELY PROFESSIONAL UNIVERSITY
   4   5   6   7   8   9   10   11   12   13   14