BESSEL FUNCTIONS. 1. Introduction The standard form for any second order homogeneous differential

Transcription

1 BESSEL FUNCTIONS FAYEZ KAROJI, CASEY TSAI, AND RACHEL WEYRENS Abstract. We briefly aress how to solve Bessel s ifferential equation an escribe its solutions, Bessel functions. Aitionally, we iscuss two real-life scenarios to motivate an emonstrate the importance of Bessel functions. Finally, we iscuss an prove orthogonality for Bessel functions of the first kin. 1. Introuction The stanar form for any secon orer homogeneous ifferential equation is (1.1) y + P (x)y + Q(x)y =. If both P (x) an Q(x) are analytic at x, meaning there is a power series, n= a n(x x ) n converging to the require function on some neighborhoo of x, then the point x is sai to be orinary, otherwise it is singular. To solve equation (1.1), we first etermine whether x is an orinary or singular point. When a singular point is regular, that is, both (x x )P (x) an (x x ) 2 Q(x) are analytic at x, we can use the the metho of Frobenius to solve the ifferential equation. Bessel functions of orer n are solutions to the secon orer ifferential equation (1.2) x 2 y + xy + (x 2 n 2 )y =, where n is an arbitrary, constant value. We will limit our focus to values where n. Equation 1.2 has two linearly inepenent solutions J n (x) an Y n (x) an thus a general solution y = c 1 J n (x) + c 2 Y n (x). We enote the Bessel function of the first kin with orer n by J n (x) an the secon kin by Y n (x). Similarly, we notate the zeros of these functions as j n an y n, respectively. Zeroth orer Bessel functions of the first an secon kin can be graphically represente by Figures 1 an 2, respectively. 1

2 2 FAYEZ KAROJI, CASEY TSAI, AND RACHEL WEYRENS Figure 1. Bessel Function of the first kin, J Figure 2. Bessel Function of the secon kin, Y For ifferential equations of the form 1.2, x is regular singular. Solving the inicial equation r(r 1) + a r + b r = for r gives the values we use in the Frobenius metho. In this equation, a an b are the analytic values of P (x) an Q(x), in equation 1.1, respectively. The solutions to the inicial equation of 1.2 are n an n for all n. The general form for J n (x) is given by J n (x) = m= where the Gamma function is efine by Γ(p) = ( 1) m ( x 2 )n+2m Γ(m + 1)Γ(n + m + 1), e x x p 1 x

3 BESSEL FUNCTIONS 3 for p, 1, 2,... an has the following properties: (1) Γ(1) = 1 (2) Γ(p + 1) = pγ(p) (3) Γ(p + 1) = p! for positive integral p. As a result of the properties of the gamma function, J n (x) has certain properties uner specific conitions as seen in the following chart. J n (x) Even function O function Complex values Conition p even p o x < or x / Z Since J n (x) has complex values for x < an non-integer values of p, we consier only values of x > when p is not an integer. 2. Application 2.1. Heat Diffusion. One common application that results in a Bessel function solution is steay-state temperature in a cyliner. This particular example is a cyliner with a raius of r = 3 an a height of z = 5. The temperature is then expresse as an equation of three variables: r, θ, an z. The lateral surface an bottom face of this raially symmetric cyliner are hel at temperature zero. The top face is hel at an arbitrary temperature g(r). Since the cyliner is raially symmetric, the temperature will never epen on θ, an all functions of θ can thus be omitte from the equation. The conitions of the aforementione scenario can be mathematically moele by the following bounary value problem where r < 3 an < z < 5: u rr + 1 u r r + u zz = u(3, z) = u(r, ) = u(r, 5) = g(r) Lemma 2.1 (Separation of Variables). Suppose (2.1) u(r, z) = R(r)Z(z) ; then the preceeing bounary value problem results in the following two orinary ifferential equations { Z λz = rr + R + λrr =.

4 4 FAYEZ KAROJI, CASEY TSAI, AND RACHEL WEYRENS Proof. Let u(r, z) = R(r)Z(z), an observe u rr + 1 r u r + u zz = R Z + 1 r R Z + RZ =. Diviing through by u(r, z) we obtain R R + 1 r R R + Z Z =. Since R(r) an Z(z) are inepenent from each other, they must both be some arbitrary constant, so we can equate them by R 1 R + r R R = Z Z = λ. This is then ecompose into two separate secon orer orinary ifferential equations R + 1 r R + λr = an Z λz =. The general solution for Z is (2.2) Z = c 1 cosh(αz) + c 2 sinh(αz) when we set λ = α 2 >. To fin c 1 =, we substitute the bounary value u(r, ) = into equation 2.2. Likewise, the general solution for R is R = c 3 J (αr) + c 4 Y (αr). In orer to keep R boune, we require c 4 = because Y approaches negative infinity as x approaches zero. The solution α n = jn is foun 3 using the bounary conition u(3, z) =. Plugging the solution back into equation 2.1 an applying the metho of linearity, we have the general solution (2.3) u(r, z) = A n J ( j nr 3 ) sinh(j nz 3 ). We fin A n to be A n = n=1 3 rg(r)j ( jnr )r 3 sinh( 5jn ) 3 rj 3 ( jnr )J 3 ( jnr )r, 3

5 BESSEL FUNCTIONS 5 for n = 1, 2, 3..., using knowlege of orthogonality an substituting the bounary conition u(r, 5) = g(r) into equation 2.3. This is further simplifie to 3 A n = rg(r)j ( jnr )r 3 sinh( 5jn )( 9)(J 3 2 1(j n )) 2 for n = 1, 2, Wave Propagation. The next physical application we iscuss is a raially symmetrical circular rumhea of raius r = 1 with fixe eges that is isplace irectly in the center with an arbitrary force. The velocity of the rumhea at time t = is zero. The initial position of the center s isplacement is represente by f(r). We have the bounary value problem where r < 1, t > : u tt = u rr + 1 u r r u(r, ) = f(r) u t (r, ) = u(1, t) = Using separation of variables where (2.4) u(r, t) = R(r)T (t) the following results: { T + µt = R + 1 r R + µr =. We isregar ampening forces an assume the rumhea will vibrate forever. The general solution (2.5) T (t) = c 1 cos(αt) + c 2 sin(αt) results from the eigenvalue µ = α 2 >. We substitute the initial conition u t (r, ) = into this ifferential equation as T () =. Since α 2 >, we know c 2 =. The general solution for equation 2.5 becomes T (t) = c 1 cos(αt). We note that r = is a regular singular point of the orinary ifferential equation of R(r). A slight change in variables leas to the following Bessel function (2.6) R(r) = c 3 J (αr) + c 4 Y (αr). We know R(r) must be boune at. Observing the graphs of J an Y in Figure 1 an Figure 2, we have c 4, in equation 2.6, must be

6 6 FAYEZ KAROJI, CASEY TSAI, AND RACHEL WEYRENS equal to zero. The bounary conition R(1) = yiels α n = j n. Using this information we can substitute back into equation 2.4 to obtain u n (r, t) = A n J (j n r) cos(j n t). The general solution u(r, t) is the Fourier-Bessel series (2.7) u(r, t) = A n J (j n r) cos(j n t). n=1 The initial conition u(r, ) = f(r) an knowlege of orthogonality can be use to fin the amplitue of the initial isplacement of the rumhea. The amplitue of the isplacement, or Fourier coefficient A n, is foun by evaluating the orthogonal relationship A n = rj (j n r)f(r)r rj (j n r)j (j n r)r which epens on the force. From equation 2.7, the frequency, funamental pitch of the rum, an all overtones can be foun. The perio is foun using the parameter of cos(j n t), an is 2π j n, so the frequency is jn. The funamental pitch 2π is the frequency j 1. All overtones are foun as the frequency evaluate 2π at each zero j n with n = 2, 3, 4... where j 2 efines the first overtone, j 3 the secon, an so forth. When the ratios between overtones an the funamental pitch are integer values, the soun is harmonious. When the relation is not an integer value, the soun is cacophonous. 3. Orthogonality We now explore the orthogonality property of J n (λx) an J n (ρx), the Bessel functions of the first kin of orer n where λ an ρ are istinct positive roots of J n (x) =. We o this by proving xj n (λx)j n (ρx) x =. A few remarks are in orer. Recall the ot prouct or inner prouct efine for R n. Two vectors x, y R n are sai to be orthogonal if n x, y = x i y i =. i=1 As a generalization we efine the inner prouct for the space of Riemann integrable functions Ra, b] on a close an boune interval by f, g = b a f(x)g(x) x.

7 BESSEL FUNCTIONS 7 We say f, g are orthogonal if f, g =. Sometimes, as in the case of Bessel functions, inner proucts are efine with a weight function w(x). In this case, we say f, g Ra, b] are orthogonal if f, g = b a w(x)f(x)g(x) x =. In the case of Bessel functions, we have w(x) = x. We are now reay to prove the following theorem. Theorem 3.1. If λ an ρ are positive roots of J n (x), then { 1, if λ ρ xj n (λx)j n (ρx) x = 1 J 2 2 n+1(λ), if λ = ρ Proof. (i) Suppose λ ρ. Then λ an ρ are istinct positive roots of J n (x). Since J n (λx) an J n (ρx) are solutions of the Bessel equation in parametric form, we can write (3.1) an x 2 J n(λx) + xj n(λx) + (λ 2 x 2 n 2 )J n (λx) = (3.2) x 2 J n(ρx) + xj n(ρx) + (ρ 2 x 2 n 2 )J n (ρx) =. Equations (3.1) an (3.2) may be written in the form x (3.3) x x J n(λx) + (λ 2 x 2 n 2 )J n (λx) = an (3.4) x x x J n(ρx) + (ρ 2 x 2 n 2 )J n (ρx) =. Multiplying equation (3.3) by Jn(ρx) x obtain (3.5) an equation (3.4) by Jn(λx), we x J n (ρx) x x J n(λx) + 1 x (λ2 x 2 n 2 )J n (λx)j n (ρx) = an (3.6) J n (λx) x x J n(ρx) + 1 x (ρ2 x 2 n 2 )J n (ρx)j n (λx) =.

8 8 FAYEZ KAROJI, CASEY TSAI, AND RACHEL WEYRENS Then subtracting (3.6) from (3.5), we obtain J n (ρx) x x J n(λx) (3.7) We rewrite (3.7) as J n (ρx)x x (3.8) x an simplify to obtain (3.9) x ] x J n(λx) ] J n (λx)x x J n(ρx) J n (ρx)x x J n(λx) ] J n (λx) x x J n(ρx) + (λ 2 ρ 2 )xj n (λx)j n (ρx) =. ] x J n(ρx) ] + x J n(λx) x x J n(λx) x x J n(ρx) +(λ 2 ρ 2 )xj n (λx)j n (ρx) = J n (λx) x x J n(ρx) + (λ 2 ρ 2 )xj n (λx)j n (ρx) =. Now integrating equation (3.9) with respect to x from to 1, noting that J n (λ) = J n (ρ) =, we get (3.1) Since λ ρ, we have (λ 2 ρ 2 ) xj n (λx)j n (ρx) x =. (3.11) as esire completing the first part of the proof. (ii) To prove the secon part, let u = J n (λx), then Multiplying by 2u, we get We can rewrite this equation as xj n (λx)j n (ρx) x = x 2 u + xu + (λ 2 x 2 n 2 )u =. 2x 2 u u + xu 2 + (λ 2 x 2 n 2 )uu =. x 2 u 2 n 2 u 2 + λ 2 x 2 u 2] 2λ 2 xu 2 =. x

9 BESSEL FUNCTIONS 9 Integrating with respect to x from to 1, we get or x 2 ( x J n(λx) x 2 u 2 n 2 u 2 + λ 2 x 2 u 2] 1 2λ2 xu 2 x = ) 2 n 2 Jn(λx) 2 + λ 2 x 2 Jn(λx) 2 But J n (λ) = an nj n () =, so ( ) ] 2 x J n(λx) 2λ 2 x=1 Hence (3.12) xj 2 n(λx) x = 1 2λ 2 ] 1 2λ 2 xjn(λx) 2 x =. ( ) ] 2 x J n(λx), x=1 an from the recurrence relations of Bessel Functions, we have or (λx) J n(λx) = n λx J n(λx) J n+1 (λx) Then equation (3.12) becomes x J n(λx) = n x J n(λx) λj n+1 (λx). xj 2 n(λx) x = 1 2λ 2 (n = 1 2 J 2 n+1(λ) xj 2 n(λx) x =. x J n(λx) λj n+1 (λx)) 2] x=1 as esire, completing the secon part of the proof. Remark 3.2. The proof of the secon part of Theorem 3.1 justifies ivision by rj (j n r)j (j n r) r in the computation of the Fourier Coefficient A n = rj (j n r)f(r) r rj (j n r)j (j n r) r for the wave equation in polar coorinates.

10 1 FAYEZ KAROJI, CASEY TSAI, AND RACHEL WEYRENS 4. Acknowlegments We woul like to thank Dr. Davison an the grauate stuents, Joel Geiger an Lokenra Singh, for their motivation an guiance with this project. References 1] Benson, Dave, The Drum. Music: A Mathematical Offering N.p., n.. Web. 9 July bensonj/html/music.pf. 2] Boyce, W.E. an DiPrima, R.C., Elementary Differential Equations an Bounary Value Problems, John Wiley & Sons, New York, ] Coington, E.A., An Introuction to Orinary Differential Equations, Dover Publications, New York, ] El Attar, R., Bessel an Relate Functions, Lulu Press, North Carolina, 27. 5] Jackson, D., Fourier Series an Orthogonal Polynomials, Dover Publications, New York, 24. 6] Tolstov, G. P., Fourier Series, Dover Publications, New York, Mathematics Department, Louisiana State University, Baton Rouge, Louisiana Mathematics Department, University of Arkansas, Fayetteville, Arkansas