FOURIER-BESSEL SERIES AND BOUNDARY VALUE PROBLEMS IN CYLINDRICAL COORDINATES

Transcription

1 FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES The paametic Bessel s equation appeas in connection with the aplace opeato in pola coodinates. The method of sepaation of vaiables fo poblem with cylindical geomety leads a singula Stum-iouville with the paametic Bessel s equation which in tun allows solutions to be epesented as seies involving Bessel functions.. The Paametic Bessel Equation The paametic Bessel s equation of ode α with α is the the second ode ODE x y + xy + λx α y =, with λ > a paamete, o equivalently, x y + xy + ν x α y =, whee ν = λ. We have the following lemma. emma. The geneal solution of equation in x > is yx = AJ α νx + BY α νx, whee J α and Y α ae, espectively, the Bessel functions of the fist and second kind of ode α and A, B ae constants. Moeove, lim x + yx is finite if and only if B = and so yx = AJ α νx. Poof. We tansfom equation into the standad Bessel equation of ode α by using the substitution t = νx. Indeed, we have dy dx = ν dy dt, and d y dx = ν d y dt. In tems of the vaiable t, equation becomes x ν d y dy + xν dt dt + x ν α y = t d y dt + tdy dt + t α y = which is the Bessel equation of ode α in the vaiable t. The geneal solution is theefoe, yt = AJ α t + BY α t. This is pecisely, yx = AJ α νx + BY α νx. Date: Apil 4, 6.

2 FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES Note that J α = if α > and J =, while the second solution Y α satisfies lim x + Y α x =. Hence, if the solution yx is bounded in the inteval, ϵ with ϵ >, then necessaily B =. We can ewite equation in a self-adjoint fom by dividing by x and noticing that xy = xy + y to obtain xy + λx α y =. x We have hee with the notation of Note 9 that px = x, x = x, and qx = α /x. The point x = is a singula point. Any Stum-iouville poblem associated with this equation on the inteval [, R] is a singula S. As mentioned in Note 9, we can use use only the bounday condition at the endpoint x = R.. A Singula Stum-iouville Poblem We associate to the paametic Bessel equation the following S-poblem xy 3 + λx α y = x b yr + b y R = It is undestood in this poblem that yx is a bounded solution, o equivalently that lim x + yx exists and is finite. In the space Cp[, R], of piecewise continuous function on [, R], we conside the following inne poduct < f, g > x = R fxgxxdx. The poof given in Theoem of Note 9, can be caied out almost vebatim to establish the othogonality of the eigenfunctions of poblem 3. Thee is an exceptional case which occus when b = b α/r with b. In this case λ = is an eigenvalue of Poblem 3 with an eigenfunction x α. To simplify the discussion we will assume that b b α/r. By emma, we know that the bounded solutions of the ODE ae yx = J α νx with ν = λ. Note that y x = νj ανx In ode fo such a solution to satisfy the bounday condition, the paamete ν must satisfy 4 b J α νr + b νj ανr =. This means that λ = ν is an eigenvalue of Poblem 3 with eigenfunction J α νx. It can also be poved that these solutions fom a complete set basis in the space of piecewise continuous functions. This imply that any function f C p[, R] has a epesentation as a seies in these eigenfunctions. Denote by ν < ν < ν 3 < < ν j <, be the inceasing sequence of solutions of equation 4. The seies expansion is then fx C j J α ν j x, whee the coefficients ae given by, / C j = < f, J αν j x > R R x J α ν j x = fxj α ν j xxdx J α ν j x xdx x

3 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES3 It is of impotance to find the squae noms J α ν j x x. We have the lemma. emma. et ν j be a positive solution of 4, then we have the following: If b =, then J α ν j x x = R If b, let h = Rb/b, then we have J α ν j x x = R J α ν j x xdx = R J α+ν j R J α ν j x xdx = R νj α + h J α ν j R Poof. Since J α ν j x satisfies Poblem 3 with λ = ν j, then We multiply by J α ν j x to get ν j xxj α ν j x = α ν j x J α ν j x. xj α ν j x [xj α ν j x ] = α ν j x J α ν j xj α ν j x. By using ff = f, we obtain [ xj α ν j x ] = α ν j x J α ν j x Now we integate fom to R. The left side is R [xj α ν j x ] dx = [xj α ν j x ] x=r = RJ α ν j R. Fo the ight side we use integation by pats, R x= α ν j x J α ν j x dx = [ α ν j x J α ν j x ] R + ν j Afte equating the two sides, we get. R = ν j J αν j x x + α ν j R J α ν j R 5 ν j J α ν j x x = R J α ν j R + ν j R α J α ν j R We distinguish the two cases. If b =, then the condition is J α ν j R =, and elation 5 becomes J α ν j x x = R J α ν j R ν j. J α ν j x xdx We simplify this expession by using Popety 4 of Bessel junctions x α J α x = x α J α+ x in the fom J α x α x J αx = J α+ x. Fo x = ν j R, we have J α ν j R = and it follows that J α ν j x x = R J α+ ν j R ν j If b, then the bounday condition can be witten as ν j J αν j R = b b J α ν j R = h R J αν j R..

4 4FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES Relation 5 becomes ν j J α ν j x x = h J α ν j R +ν j R α J α ν j R = R ν j +h α J α ν j R. Fom this we deduce that J α ν j x x = R νj + h α J α ν j R. ν j 3. Bessel Seies Expansions We summaize the above discussion in the following theoems. Theoem. Conside the S-poblem x y + xy + λx α y =, < x < R, yr = with bounded solutions yx. The eigenvalues values consist of the sequence λ j = ν j, with j Z + and whee ν j R is the j-th positive zeo of J α x. That is, ν < ν < ν 3 < < ν j < satisfy J α ν j R =. The coesponding j-th eigenfunction is y j x = J α ν j x. Futhemoe, the family of eigenfunctions J α ν j x is complete in Cp[, R]. The squae nom of the j-th eigenfunction is J α ν j x x = R Theoem. Conside the S-poblem J α ν j x xdx = R J α+ν j R x y + xy + λx α y =, < x < R, b yr + b y R =, b b αb /R with bounded solutions yx. The eigenvalues consist of the sequence λ j = ν j, with j Z +, such that, ν < ν < ν 3 < < ν j < satisfy the equation hj α ν j R + ν j RJ αν j R =, h = Rb b The coesponding j-th eigenfunction is y j x = J α ν j x. Futhemoe, the family of eigenfunctions J α ν j x is complete in Cp[, R]. The squae nom of the j-th eigenfunction is J α ν j x x = R J α ν j x xdx = R νj α + h J α ν j R Consequence of Theoem and. If f Cp[, R], then f has the expansion, f av x = fx+ + fx = C j J α ν j x, ν j

5 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES5 with C j = < fx, J αν j x > x J α ν j x x and whee the functions J α ν j x ae as in Theoem o as in Theoem. Remak. If we set z j = ν j R, then the above seies takes the fom fx + + fx x = C j J α z j R whee in the case of Theoem, the z j s ae the positive oots of J α, i.e., J α z j =, j =,, 3,, in the case of Theoem, the z j s ae the positive oots of the equation hj α z j + z j J αz j =, j =,, 3,. Remak. Since J α = fo α > and J =, then λ = is an eigenvalue of the S-poblem 3 only in the case of bounday condition y R = and the eigenfunction is J z j x/r, whee the z j s fom the sequence of positive zeos of J. We illustate this theoems with some examples. Example. Expand the function fx = on [, 3] into a J -Bessel seies with condition J z j =. This is the case of Theoem. We have zj x = C j J < x < 3, 3 Whee 3 C j = J z j x/3 J z j x/3xdx. x The squae noms ae J z j x/3 x = 9J z j / and to find the integal we use Popety 9 t α J α tdt = t α J α t + C and a substitution. 3 Hence, J z j x/3xdx = 9 z j and the seies expansion is = zj tj tdt = 9 z j C j = z j J z j J z j J z j zj x 3 z j J z j J = 9J z j z j. < x < 3. Now we give appoximations of the fist five tems of the seies j z j J z j /z j J z j

6 6FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES The beginning of the seies expansion is.8j.8x.53j.84x +.46J.88x.365J 3.93x +.34J 4.98x + Example. An analogous calculation gives the expansion of fx = ove [, R] with espect J α with end point condition J α z j = as Whee C j = = R J α+ z j R zj x C j J α R J α z j x/rxdx = < x < R, zj zj J α+z j tj α tdt. Example 3. Expand fx = on [, ] in J α -Bessel seies with endpoint condition J αz j =. This is the case of Theoem with R = and h =. The squae noms ae J α z j x/ x = z j α zj J α z j and with C j = z j z j α J α z j = zj x C j J α, xj α z j x/dx = When α =, we get a moe compact fom of C j C j = z j J z j zj tj tdt = zj zj α J α z j tj α tdt z j J z j [tj t] z j = J z j z j J z j The expansion of fx = ove [, ] in a Bessel seies with endpoint J z j = is = J z j z j J z j J zj x. Example 4. Find the J -Bessel seies of fx = x ove [, ] with endpoint J z j =. We have fo < x <, x = C j J z j x, C j = < x, J z j x > x J z j x x We know that J z j x x = J z j see pevious calculations. To compute < x, J z j x > x, we use the substitution t = z j x in the integal and the.

7 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES7 popeties t α J α t = t α J α t and t α J α t = t α J α+ t: zj < x, J z j x > x = zj 4 zj t tj tdt = zj 4 zj t tj t dt [ z j t tj t ] z j = zj 4 + zj zj 4 t J tdt Hence the coefficient C j is = z 4 j zj = [ t J t ] z j zj 4 t J tdt = z 4 j zj zj + 4 zj 4 = 4 zj 4 [tj t] zj = 4J z j zj 3 C j = 4J z j /z 3 j J z j / = 8 z 3 j J z j. zj t J t dt tj tdt = 4 z 4 j The seies epesentation of x is x J z j x = 9 zj 3J, x,. z j zj tj tdt Example 5. The expansion of x m ove [, ] in J m -Bessel seies with endpoint condition J m z j = can be obtained by using simila aguments as befoe. x m = A calculation gives and the seies is C j J m z j x, x m = C j = C j = < xm, J m z j x > x J m z j x x z j J m+ z j J m z j x, x,. z j J m+ z j 4. Vibations of a Cicula Membane The following wave popagation poblem models the small vetical vibations of a cicula membane of adius whose bounday is held fixed. The initial position and velocity ae given by the functions f and g. whee u tt x, y, t = c ux, y, t x + y <, t >, ux, y, t = x + y =, t >, ux, y, = fx, y x + y <, u t x, y, = gx, y x + y <. = x + y

8 8FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES is the -dimensional aplace opeato. Fo the pupose of using the method of sepaation of vaiables, this poblem is bette suited fo pola coodinates x = cos θ, y = sin θ. Recall the expession of the aplace opeato in pola coodinates is = + The expession of the BVP becomes u tt = c u + u + u θθ 6 + θ. <, θ π, t >, u, θ, t = θ π, t >, u, θ, = f, θ <, θ π, u t, θ, = g, θ <, θ π,. It is undestood that the sought solution u, θ, t and its deivatives need to be peiodic in θ: i.e., u, θ, t = u, θ + π, t and so on. Since in this fom the PDE has a singulaity at =, we impose that the solution u be continuous at the oigin. Befoe we conside the geneal case, we ae going to solve poblem 6 in the paticula case when f and g ae independent on θ invaiant unde otations. 4.. Rotationally invaiant case. Since f and g depend only on, we seek solution u that is also independent on θ. That is, at each time t the displacement, u = u, t depends only on the adius and not on the angle θ. In this case poblem 6 becomes a poblem in vaiables: u tt = c u + u <, t >, 7 u, t = t >, u, = f <, u t, = g <. We seek nontivial solutions with sepaated vaiables u = RT t of the homogeneous pat of the poblem: u tt = c u + u, u, t =. The sepaation of vaiables of the PDE leads to R R + R R = T t c = λ constant. T t This togethe with the bounday condition give the ODE poblems fo R and T t { R + R + λ R =, < < R, and R = T + c λt =. The R-poblem is the eigenvalue poblem. Its ODE is the paametic Bessel equation of ode. Since we ae seeking bounded solutions, then the eigenvalues and eigenfunctions ae λ j = z j, R zj j = J, j =,, 3, whee z j is the j-th positive oot of J x =.

9 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES9 Fo each j Z +, the T -equations becomes T +cz j / T = with independent solutions coscz j t/ and sincz j t/. Fo each j Z +, the homogeneous pat has two independent solutions with sepaated vaiables u czj t zj j, t = cos J and u czj t zj j, t = sin J. The geneal seies solution of the homogeneous pat is the linea combination of all the u j s and u j s: u, t = [ A j cos czj t + B j sin ] czj t zj J Now we use the nonhomogeneous conditions to find the constants A j and B j so that the seies solution satisfies the whole poblem. Fist we compute u t : [ ] cz j czj t czj t zj u t, t = A j sin + B j cos J. We have zj u, = f = A j J, cz j u t, = g = B zj jj. These ae, espectively, the J -Bessel seies of the functions f and g ove [, ] with endpoint condition J z j =. We have then, A j = < f, J z j / > J z j / = B j = cz j < g, J z j / > J z j / J z j = cz j J z j fj z j /d, gj z j /d. Example. The stuck dum head. Suppose that the above cicula membane, with = which was oiginally at equilibium is stuck at time t = at its cente in such a way that each point located at distance < fom the cente is given a velocity. The BVP in this case is we take c = u tt = u + u <, t >, u, t = t >, u, = u t, = g <. whee g = In this case the coefficients ae A j = and B j = { if < <, if < <. 5z j J z j gj z j /d.

10 FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES Fo the last integal, we have We have theefoe and gj z j /d u, t = = = z j J z j /d = z j zj / [tj t] z j/ = z j J z j / B j = J z j / z j J z j J z j / z j J z j sin The following table list the appoximate values zj t j z j J z j J z j / B j We have then u, t zj J. tj tdt.53 sin.4tj.4.5 sin.55tj sin.86tj.86.3 sin.8tj.8.7 sin4.9tj Geneal case. Conside again BVP 6. This time we assume that the initial position f and initial velocity g depend effectively on θ and and also u = u, θ, t. We emphasize the π-peiodicity of u and its deivative u θ, with espect to θ, by adding it to the poblem. The poblem is theefoe u tt = c u + u + u θθ <, θ π, t >, u, θ, t = θ π, t >, 8 u, π, t = u, π, t < <, t >, u θ, π, t = u θ, π, t < <, t >, u, θ, = f, θ <, θ π, u t, θ, = g, θ <, θ π. We apply the method of sepaation of vaiables to the homogeneous pat. et u, θ, t = RΘθT t be a nontivial solution of the homogeneous pat. Fo such a function the wave equation can be witten as T t c T t = R R + R R + Θ θ = λ constant. Θθ Thus T t + c λt t = and R R + R R λ Θ θ Θθ =.

11 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES The last equation can be sepaated: R R + R R + λ = Θ θ Θθ = α constant. The ODEs fo R and Θ ae R + R + λ α R = and Θ + α Θ =. The homogeneous bounday conditions imply that R =, Θπ = Θ π and Θ π = Θ π. The thee ODE poblems ae Θ + α Θ = Θπ = Θ π Θ π = Θ π { R + R + λ α R = R = T t + c λt t = The Θ-poblem is the egula peiodic S-poblem and the R-poblem is the singula Bessel poblem of ode α with endpoint condition R =. The eigenvalues and eigenfunctions of the Θ-poblem ae. fo m Z +, α =, Θ θ = ; α m = m, Θ mθ = cosmθ, Θ mθ = sinmθ. We move to the R-poblem. Fo each α = m with m Z + {}, the eigenvalues and eigenfunctions of the R-poblem ae λ mj = z mj, R zmj mj = J m, whee {z mj } j is the inceasing sequence of zeos of J m. That is, < z m < z m < < z mj < satisfy J m z mj =. Now that we have the eigenvalues and eigenfunctions of the Θ- and R-poblems, we solve the T -poblem. Fo each m Z + {} and each j Z +, the coesponding T -equation is T czmj + T = with two independent solutions Tmj czmj t = cos and Tmj czmj t = sin. The collection of solutions with sepaated vaiables of the homogeneous pat of Poblem 8 consists of the following functions whee m =,,, and j =,, u zmj mj, θ, t = J m u mj, θ, t = J m zmj u 3 mj, θ, t = J m zmj u 4 mj, θ, t = J m zmj czmj t cosmθ cos, czmj t sinmθ cos, czmj t cosmθ sin, czmj t sinmθ sin.

12 FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES These ae the m, j-modes of vibations of the membane. Up to a otation, the pofile of each mode is given by the gaph of the function zmj v mj, θ = J m cosmθ each m, j-mode is just v mj times a time-amplitude. Fo each m, j, the membane can be divided into egions whee v mj > and egions whee v mj <. The cuves whee v mj = ae called the nodal lines see figue. m,j=, m,j=, m,j=, m,j=, m,j=, m,j=, Some of the m, j-modes ae gaphed in the figue. m=, m=, m=, m=, j= m=, j= m=, j=

13 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES3 The geneal seies solution of the homogeneous pat is a linea combination of the solutions u k mj : zmj czmj t u, θ, t = J m cos A mj cosmθ + B mj sinmθ m= zmj czmj t +J m sin C mj cosmθ + D mj sinmθ and In ode fo such a solution to satisfy the nonhomogeneous condition, we need zmj u, θ, = f, θ = J m [A mj cosmθ + B mj sinmθ] u t, θ, = g, θ = m= m= cz mj t J m zmj [C mj cosmθ + D mj sinmθ] These ae the double Fouie-Bessel seies of f and g. To find the coefficients, we can poceed as follows. Expand f, θ into its Fouie seies view as a paamete, f, θ = A + A m cosmθ + B m sinmθ whee A m = π π π m= f, θ cosmθdθ, m =,,, B m = f, θ sinmθdθ, m =,, 3, π Now expand A m and B m into a J m -Bessel seies with end point condition J m z =. We get, zmj zmj A m = A mj J m, A mj = J m A m J m d and B m = zmj B mj J m, B mj = J m zmj B m J m d Afte substituting the expessions of A m and B m in tem of the integals of f, we obtain the coefficients A mj and B mj of the seies solution as: π zmj A mj = π J m z mj / f, θj m cosmθddθ, π zmj B mj = π J m z mj / f, θj m sinmθddθ. The othe coefficients C mj and D mj ae obtained in a simila way by using the second initial condition. Example Take =, c =, g, θ = and f, θ = + sin θ. In this situation C mj = D mj = fo m and j. Futhemoe, A mj = fo m

14 4 FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES and B mj = fo m Hence, the solution has the fom u, θ, t = A j J z j cosz j t + B j J z j cosz j t sin θ, j j whee and B j = π J z j The squae noms ae A j = J z j π J z j = J z j J z j d J z j sin θ ddθ = J z j and J z j = J z j. J z j d. The integal involved in A j can be evaluated. Fist, notice that by using the popety t α J α t = t α J α t and integation by pats, we get t 3 J tdt = t tj t ttj tdt = t 3 J t t J t + C. Hence, by using the substitution t = z j, we get and so The solution u is u, θ, t = 4 J z j d = z 4 j = z 4 j zj = J z j z j A j = 4J z j z j J z j. J z j J z j z j J z j cosz j t + [ z j t t 3] J tdt [ z j t t 3 J t + t J t ] z j B j J z j cosz j t sin θ. 5. Heat Conduction in a Cicula Plate The following bounday value poblem models heat popagation in a thin cicula plate with insulated faces and with heat tansfe on the cicula bounday. In pola coodinates, the poblem is: u t = k u + u + u θθ 9 <, θ π, t >, hu, θ, t + u, θ, t = θ π, t >, u, θ, = f, θ <, θ π,

15 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES5 whee h is a constant. et us assume fo simplicity that f is otation invaiant f = f is independent on θ. We seek then a solution that is also independent on θ and the poblem becomes. u t = k u + u <, t >, hu, t + u, t = t >, u = f <. The solutions with sepaated vaiables: u, t = RT t of the homogeneous pat leads to the ODE poblems { R + R + λ R = hr + R and T + kλt =. = The R-equation is the paametic Bessel equation of ode. The values λ < cannot be eigenvalues. The case λ = is a paticula case. Remak about the case λ =. In this case, the R-equation is also a Cauchy- Eule equation with geneal solution A + B ln and the bounded solutions ae R = A. Such a solutions satisfies the bounday condition if and only if h =. This coespond to the situation whee the bounday is insulated The positive eigenvalues and eigenfunctions ae: λ j = z j, R zj j = J, j Z +, whee z j is the j-th positive oot of the equation A coesponding T -solution is hj z + zj z =. T j t = exp kz j t The seies solution of the homogeneous pat, when h >, is u, t = C j e kz j /t zj J and when h = is u, t = C + C j e kz j /t zj J In ode fo such a seies to satisfy the nonhomogeneous condition, we need zj u, = f = C + C j J with C = when h >. This is the J -Bessel seies of f with endpoint condition hj z + zj z =. We have then see Theoem, C j = J z j / fj zj d =. z j z j + h J z j fj zj d.

16 6 FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES Remak. If instead of being independent on θ, the initial condition is of the fom f cosmθ, the solution u of poblem 9 can be put in the fom u, θ, t = v, t cosmθ, whee v solves the two vaiables poblem v t = k v + v v m <, t >, hv, t + v, t = t >, v = f <. This time the sepaation of vaiables of the v-poblem leads to the solution v of the fom v, t = C j e kz j /t zj J m 6. Helmholtz Equation in the Disk The Helmholtz equation in the disk is the following two-dimensional eigenvalue poblem: Find λ and nontivial functions u, θ defined in the disk of adius such that: { u, θ + λu, θ = < <, θ π, u, θ = θ π. The solutions with sepaated vaiables u, θ = RΘθ leads to the following one-dimensional eigenvalue poblems { Θ t + α Θt = Θ and Θ π peiodic and { R + R + λ α R = R =. The eigenvalues and eigenfunctions of the Θ-poblem ae α m = m, Θ mθ = cosmθ and Θ mθ = sinmθ m =,,, Fo each m, the R-poblem is an ode m Bessel type eigenvalues poblem with eigenvalues and eigenfunctions zj,m zj,m λ j,m =, Rj,m = J m. whee z j,m is the j-th positive oots of the equation J m z =. The eigenvalues and eigenfunctions of the Helmholtz equation ae λ j,m = z j,m, u zj,m j,m, θ = cosmθj m, u zj,m j,m, θ = sinmθj m, Eigenfunctions expansion fo a nonhomogeneous poblem. The eigenfunctions u j,m, u j,m can can be used to solve nonhomogeneous poblems such as the following { u, θ + au, θ = F, θ < <, θ π, 3 u, θ = θ π.

17 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES7 whee a is a constant. We expand F and u into these eigenfunctions Fouie-Bessel seies. We seek then a solution u of the fom u, θ = m= By using u, m,j = λ m,ju, m,j, we deduce that 4 u + au = m= zj,m [A m,j cosmθ + B m,j sinmθ] J m zj,m a λ m,j [A m,j cosmθ + B m,j sinmθ] J m The Fouie-Bessel seies of F is [ 5 F, θ = F m,j cosmθ + Fm,j sinmθ ] zj,m J m m= whee the coefficients F, m,j ae given by 6 F m,j F m,j } = π J m z m,j / π F, θj m zm,j By identifying the coefficients in the seies 4 and 5, we get { cosmθ d sinmθ dθ. A m,j = F m,j a λ m,j and B m,j = F m,j a λ m,j. povided that a λ m,j fo evey m, j nonesonant case. Example. Conside the poblem { u, θ u, θ = + sinθ < < 3, θ π, u3, θ = θ π. In this poblem, we have F, m,j =, fo evey m except when m = and m =. Futhemoe, F,j = fo evey j. Fo the emainde of the coefficients, we have F,j = J z,j /3 = z,j J z,j = z,j J z,j 3 z,j J z,j /3d tj tdt and F,j = J z,j /3 9 = z,j 4 J 3z,j 6 = z,j J 3 z,j 3 z,j 3 J z,j /3d t 3 J tdt

18 8 FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES The coesponding coefficients of the solutions ae The solution of the poblem is u = 8 F,j A,j = z,j /9 = 8 z,j 9 + z,j J z,j B,j F,j = z,j /9 = 458 z,j 9 + z,j J 3z,j J z,j /3 z,j 9 + z,j J z,j Execises J z,j /3 sinθ z,j 9 + z,j J 3z,j In Execises to 9, find the J α -Bessel seies of the given function fx ove the given inteval and with espect to the given endpoint condition. Execise. fx = ove [, 5], and J z =. Execise. fx = x ove [, 7], and J z =. Execise 3. fx = 5 ove [, ], and J z =. { if x Execise 4. fx = if < x ove [, ], and J z =. Execise 5. fx = x ove [, 3], and J z = Execise 6. fx = ove [, 3], and J z + zj z =. Execise 7. fx = x ove [, 3], and J z = leave the coefficients in an integal fom. Execise 8. fx = x ove [, 3], and J 3 z = leave the coefficients in an integal fom. Execise 9. fx = x ove [, π], and J / z = use the explicit expession of J / and elate to Fouie seies. In execises to 4, solve the indicated bounday value poblem that deal with heat flow in a cicula domain. Execise. u t = u + u < <, t >, u, t = t >, u, = 5 < <. Execise. Execise 3. u t = u + u < <, t >, u, t = t >, u, = 5 < <. u t = u + u < <, t >, u, t u, t = t >, u, = 5 < <.

19 FOURIER-BESSE SERIES ANDBOUNDARY VAUE PROBEMS INCYINDRICA COORDINATES9 Execise 3. Find a solution u of the fom u, θ, t = v, t sinθ of the poblem. u t = u + u + u θθ < <, θ π, t >, u, θ, t = θ π, t >, u, θ, = 5 sinθ < <, θ π,. Execise 4. u t = u + u 9u < <, t >, u, t = t >, u, = 3 < <. In execises 5 to 8, solve the indicated bounday value poblem that deal with wave popagation in a cicula domain. Execise 5. u tt = u + u < < 3, t >, u3, t = t >, u, = 9 < < 3, u t, = < < 3. Execise 6. whee u tt = u + u < <, t >, u, t = t >, u, = < <, u t, = g < <. g = { if < < /, if / < <. Execise 7. Find a solution u of the fom u, θ, t = v, t cos θ u tt = u + u + u θθ < <, t >, u, t = t >, u, = < <, u t, = g cos θ < <. whee g = { if < < /, if / < <. Execise 8. Find a solution u of the fom u, θ, t = v, t cos θ u tt = u + u + u θθ < <, t >, u, t = t >, u, = J z, sin θ < <, u t, = g cos θ < <. whee z, is the fist positive zeo of J and whee { if < < /, g = if / < <.

20 FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES In execises 9 to, solve the Helmholtz equation in the disk with adius Execise 9. = u u =, u, θ = Execise. = u = sin θ, u, θ = Execise. = 3 u + u = cos3θ, u3, θ = Execise. Solve the following Diichlet poblem in the cylinde with adius and height u + u + u zz = < <, < z <, u, z = < z <, u, = < <, u, = < <.