Fourier-bessel Expansions with Arbitrary Radial Boundaries

Series expansion of single variable functions is represented in Fourier-Bessel form with unknown coefficients. The proposed series expansions are derived for arbitrary radial boundaries in problems of circular domain. Zeros of the generated transcendental equation and the relationship of orthogonality are employed to find the unknown coefficients. Several numerical and graphical examples are explained and discussed.


Introduction
Several boundary value problems in the applied sciences are frequently solved by expansions in cylindrical harmonics with infinite terms.Problems of circular domain with rounded surfaces often generate infinite series of Bessel functions of the first and second types with unknown coefficients.In this case, the intention is to find the series coefficients which should satisfy the boundary conditions.
The subject of Fourier-Bessel series expansions was investigated and examined in many texts [1][2][3][4][5][6][7][8][9][10].Nearly all of them has derived cylindrical harmonics expansions in J 0 (r) for the interval [0, a] only, where J 0 (r) is the Bessel function of the first kind with order zero and argument r [8].The existence of the origin point excludes Y 0 (r), Bessel function of the second kind with order zero and argument r, because it goes to negative infinity as r approaches zero [9].Both J 0 (r) and Y 0 (r) are shown plotted in Figure 1.
In many other problems in the applied sciences, the interval of expansion is found to be [a, b] such that a, b ∈ R.An example of this could be a hollow cylinder in heat conduction problems or a circular band in vibrations analysis solved in the cylindrical coordinate system.In this case, cylindrical harmonics expansions in both J 0 (r) and Y 0 (r) are necessary.
In this paper, the derivation of cylindrical harmonics expansion of a single variable function in [a, b] in both J 0 (r) and Y 0 (r) is solved.In accordance with the boundaries at r = a and r = b, zeros of the obtained transcendental equation are first calculated.As shown in Figure 2, the solution region is for a ≤ r ≤ b where the desired series expansions are forced to be zero at r = a and r = b respectively.Unknown coefficients are then found and the complete series expansion can be achieved.

Formulation and Solution
The Bessel differential equation of order zero is well known as [1,4]: ∀ α and r ∈ R and a ≤ r ≤ b.
The general solution to Equation ( 1) for real values of  is known to be [2,3]: As in Equation ( 1), the assumed boundary conditions at r = a and r = b are of Dirichlet type as f(a) = 0 and f(b) = 0 respectively.Both A n and B n are then related as: Going after the elimination method, the transcendental equation can be obtained as: In order for Equation ( 5) to be satisfied, there exist many zeros or values of  to be calculated.Thus, in all former and coming equations  can be replaced by  n which are the zeros obtained from the transcendental equation ∀ n ∈ I.That is: The orthogonality feature of Bessel functions can be applied to Equation ( 2 The terms under the summation in the left side of Equation (7) are zeros for all values of m ≠ n [5,6,7].Hence, Equation ( 7) can be simplified to: Either Equation ( 3) or (4) can help.Using Equation (3) we can obtain the B n coefficients as: where, S 0 ( n r) is given by: By Equation ( 3) or ( 4), the A n coefficients can also be found.Once the coefficients A n and B n are calculated, the function f(r) can be expanded as in Equation (2).

Numerical Examples
The transcendental expression in Equation ( 6) shows a gradual decay as  increases which mean small magnitudes between high zeros.This leads to the convergence of the series in Equation ( 2) above as n increases.As a consequence, a finite number of terms in Equation ( 2) can be sufficient for numerical approximations.
The zeros are first evaluated using the transcendental cross product Bessel functions equation for the interval [a, b].A graph of Equation ( 6) is shown in Figure 3 for the solution regions [0.65, 2.5] and [0.65, 5].Table 1 shows the first 50 zeros of Equation ( 6) for a = 0.65 and b = 2.5.Zeros obtained from the transcendental equation changes according to the values of a and b assumed for the solution region.The data presented in Table 1 indicates that the calculated zeros are not periodic and should be calculated using a proper numerical technique.
Let's assume that the function f(r) to be expanded as in Equation ( 2  Many variations can be noticed for the numerical values of A n and B n with a general absolute scale of < 1 except for B 0 = 2.328.Some coefficients are in the order of ×10 -3 meaning that their associated terms are very small such as B 4 and A 31 in Tables 2 and 3 respectively. The function sin(r) and its approximate expansions are plotted in Figure 4. Summation over the first 10 terms produced an acceptable estimation in the interval [0.65, 2.5] with some apparent oscillations around the exact function.An improved approximate expansion is also plotted for n = 0 to 49 with less fluctuations in the same radial domain.In addition, f(r) = cos(r) is expanded as in Equation ( 2) and the first fifty coefficients are listed in Tables 4 and 5 for the B n and A n respectively.Similar to the sin(r), the cos(r) coefficients go through several variations with a general absolute scale of < 1 except A 1 = −1.550.Also, only four coefficients are in the order of ×10 -3 implying that their related terms in the series are extremely small such as B 4 and A 41 in Tables 4 and 5 respectively.The calculated coefficients for the function e r are also shown in Tables 6 and 7 for B n and A n respectively.Apparently, the coefficients swing around the exact values with an absolute level of > 1 or < 1.
The greatest values in Tables 6 and 7 are found as B 0 = 13.852 and A 1 = 11.499.In addition, no coefficients are calculated in the order of ×10 -3 implying that all coefficients are to be included in the series expansion.
The function exp(r) and its estimated expansions are shown plotted in Figure 6 in [0.65, 2.5].A satisfactory estimation of a finite summation over the first 10 terms are generated with several oscillations close to the exact function.A good approximated expansion is also plotted for n = 0 to 49 with fewer variations in the same solution region.
The last numerical example to be discussed is the square function expressed as: The calculated B n and A n coefficients for this function are shown in Tables 8 and 9   in the order of ×10 -4 such as B 41 indicating that their associated terms in the series are very small.
The function expressed by Equation (13) and its approximate expansions are plotted in Figure 7. Summation over the first 10 terms produced an acceptable estimation in the interval [0.65, 2.5] with some noticeable oscillations around the exact function.A better approximate expansion is also plotted for n = 0 to 49 with less fluctuations in the same radial domain.
In all graphical plots previously shown, the curves re-turn to zero at the assumed boundaries a = 0.65 and b = 2.5.In addition, accuracy of the expanded curves may appear better as n increases due to larger number of terms involved in the series and less fluctuations seen around the exact values.

Conclusions
Functions were expanded as a Fourier-Bessel series summation in both J 0 (r) and Y 0 (r).A finite series expan- Copyright © 2010 SciRes.AM

Figure 2 .
Figure 2. The solution region in radial boundaries.
) by multiplying both sides by [ ] over all possible values of r from a to b as: ) is sin(r) with a radial solution region in [0.65, 2.5].The coefficients B n can be evaluated from Equation (11) and the A n coefficients are then obtained by Equation (3).Both coefficients are shown in Tables 2 and 3 respectively for n = 0 to 49.

Figure 5 .
Figure 5. ▬▬ cos(r), ••• Equation (2) with n = 0 to 10, ▬ ▬ Equation (2) with n = 0 to 49.The function cos(r) and its estimated expansions are shown plotted in Figure 5. Finite summation over the first 10 terms generated a satisfactory estimation in the interval [0.65, 2.5] with several obvious oscillations close to the exact function.A better approximate expansion is also plotted for n = 0 to 49 with less fluctuations in the same solution region.The calculated coefficients for the function e r are also shown in Tables6 and 7for B n and A n respectively.Apparently, the coefficients swing around the exact values

Figure 7 .
Figure 7. ▬▬ Equation (13), ••• Equation (2) with n = 0 to 10, ▬ ▬ Equation (2) with n = 0 to 49. sion was obtained for arbitrary radial boundaries in [a, b].Coefficients were found by calculating the zeros of the transcendental equation and by employing the relationship of orthogonality.A number of examples were numerically and graphically discussed.