Lecture 10a: Introduction to Bessel Functions

IntroductionIn many branches of engineering and applied mathematics the following differential equation appears:

t2d2 y

dt2+ t

dy

dt+ It2 - 2M y = 0

This differential equation is known as Bessel's equation. The parameter is a given number. It may take on integer ornon-integer values. Our goal in this tutorial is to study the solution properties of Bessel's equation and to illustratehow Bessel's equation arises in the solution of heat conduction problems. Before we begin it is important to note thatBessel's equation can also be expressed as

r2d2 y

dr2+ r

dy

dr+ I2 r2 - 2M y = 0

This form arises from a simple change of variables t = r. To show the steps leading to the above form we use thechain rule of differentiation. Recall if we are given y = y HtL, then by the chain rule of differentiation we have

dy

dt=dy

dr

dr

dt=dy

dr

1

,

d

dt

dy

dt=

d

dr

dy

dr

1

dr

dt=d2 y

dr21

2

Thus the derivatives can be rewritten as follows:

t2d2 y

dt2= I2 r2M d

2 y

dr21

2= r2

d2 y

dr2

tdy

dt= H rL dy

dr

1

= rdy

drSubstituting these forms into Bessel's equation gives the desired result. In the next section we discuss the generalsolution of Bessel's equation and the properties of Bessel functions of the First and Second kind.

General Solution to Bessel's EquationWe can use Mathematica's DSolve function to determine the general solution to Bessel's equation for an arbitrary

DSolveAt2 y''@tD + t y'@tD + It2 - 2M y@tD == 0, y@tD, tE88y@tD BesselJ@, tD C@1D + BesselY@, tD C@2D

Consider the case when is a non-integer:genSol1@.75D

0 5 10 15 20

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

tClearly, the two functions are linearly independent. Now consider the case when is an integer:genSol1@3D

0 5 10 15 20

-0.4

-0.2

0.0

0.2

0.4

tThe two plots can be made coincident with each other by multiplying one of the plots by -1, indicating thatJ2 HtL and J 2 HtL are linearly dependent. Indeed, one can show that when = n is an integer then

Jn HtL = H-1Ln J-n HtLimplying that Jn HtL and J-n HtL are linearly dependent.Because of the symmetry property of Bessel functions of the first kind when is an integer, it is customary to expressthe general solution of Bessel's equation in terms of Bessel functions of the first and second kind, denoted as J@tDand Y@tD . In Mathematica these expressions are denoted by BesselJ@, tD and BesselY@ , tD. Let usexplore these functions with the following plot function:genSol2@n_D := Plot@8BesselY@n, tD, BesselJ@n, tD

genSol2@.75D

0 5 10 15 20

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

tWhen we compare the graphs in this plot with the previous generated set, we see that J-0.75 HtL -Y.75 HtL,though the functions are qualitatively similar. Indeed, one can show that Y HtL is related to J HtL by the followingformula

Y HtL = J HtL Cos H L - J- HtLSin H L

We show this equivalence in the plot below. For convenience we definemyBesselY@_, t_D := HBesselJ@, tD Cos@ D - BesselJ@-, tDL Sin@ DPlot@8myBesselY@.75, tD, BesselY@.75, tD

When we apply FullSimplify to the above expression we see indeed that Y2 HtLis a solution0

Asymptotic Behavior of Bessel functionsOne way to obtain insight into the properties of the Bessel functions is to examine the asymptotic behavior of thefunction about t = 0 and t = . These asymptotic forms can be readily obtained via a Taylor series expansion. InMathematica the function Series will determine the appropriate series expansion about a specified point and for aspecified number of terms. The output from series is shown in this simple exampleSeries@Sin@xD, 8x, 0, 4

AsymJ0@, 2D

2

1

tCosB

4- tF - I

1tM32 SinA

4- tE

4 2

This result show that BesselJ@0, tD behaves asJ0 HtL ~ 1

tCos I

4- tM at t

Plot@Evaluate@8AsymJ0@, 2D, BesselJ@0, tD

BC1 : u Hr, 0, zL < BC2 : u H1, zL = 0

BC3 :u

zHr, LL = 0

BC4 : u Hr, 0L = f HrL

We seek a solution of the form

u Hr, zL = X HrL Z HzLSubstituting this expression into the PDE and regrouping terms we obtain

1

X

d2 X

d r2+1

r

dX

dr= -

1

Z

d2 Z

dz2=

-2

02

The solution to our PDE will be of the form

u Hr, zL = n=1

An Xn HrL Z HzL

such that it satisfies

u Hr, 0L = f HrL = n=1

An Xn HrL Z H0LThis means we must have an eigenvalue problem for a set of eigenfunctions Xn HrLwhich will allow us to expandf HrL in terms of these eigenfunctions as shown above. The eigenvalue problem is thus

r2d2 Xd r2

+ rdXdr+ 2 r2 X = 0

BC1: X H0L < BC2: X H1L = 0

As can be seen from the previous section, the ODE is a Bessel equation of order zero. Thus the general solution is

X HrL = C1 J0 H rL + C2 Y0 H rLBC1 implies that C2 = 0 as Y0 H0L = -. Applying BC2 we obtain

C1J0 HL = 0Thus for a non-trivial solution, we require that

J0 HL = 0From our knowledge of the properties of Bessel functions of the first kind, we must find the zeros of J0 HL. Thesezeros (or roots) are infinite in number and can be ordered as

1 < 2 < 3

6 ECH140bIntroBesselFunc.nb

The eigenvalue problem is thus

r2d2 Xd r2

+ rdXdr+ 2 r2 X = 0

BC1: X H0L < BC2: X H1L = 0

As can be seen from the previous section, the ODE is a Bessel equation of order zero. Thus the general solution is

X HrL = C1 J0 H rL + C2 Y0 H rLBC1 implies that C2 = 0 as Y0 H0L = -. Applying BC2 we obtain

C1J0 HL = 0Thus for a non-trivial solution, we require that

J0 HL = 0From our knowledge of the properties of Bessel functions of the first kind, we must find the zeros of J0 HL. Thesezeros (or roots) are infinite in number and can be ordered as

1 < 2 < 3

Next we solve for the functions Z HzL which must satisfy

d2 Z

d z2-

2Z = 0

BC3 :dZ

dzHLL = 0

The general solution is given by

Z HzL = C3 Cosh H zL + C4 Sinh H zLApplying BC3 gives

C4 = -C3 Tanh H LLThus we can express the solution as

Z HzL = C3 8Cosh H zL Cosh H LL-Sinh H zL Sinh H LL

The coefficients An are thus found by applying the orthogonality condition to BC4

0

1J0 Hm rL f HrL rr =

n=1

An Cosh@n LD 0

1Jo Hn rL J0 Hm rL rr

Thus

An =1

Cosh@n LD0

1f HrL J0 Hn rL rr

0

1J0 Hn rL2 rr

Mathematica SolutionIn this section the Mathematica code is given to compute various aspects of the solution. Evaluate the input cells to seethe results. You may also be interested in evaluating the solution with different parameter values and exploringdifferent possibilities for boundary condition BC4. First, we need to calculate the roots of J0 HL = 0. We will make use of the asymptotic results that we found earlier.ApproxRoots = TableB3.

4+ n , 8n, 0, 20

We can assess how well our eigenfunctions approximate the function f HrL in BC4 by comparing the partial sum(shown in blue) with the original function (shown in red) Plot@Evaluate@8f@rD, fpar@r, 5D

ContourPlot@Evaluate@upar@r, z, 10DD, 8r, 0, 1