Boundary Value Problems in Cylindrical Coordinates
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1 Boundary Value Problems in Cylindrical Coordinates 29
2 Outline Differential Operators in Various Coordinate Systems Laplace Equation in Cylindrical Coordinates Systems Bessel Functions Wave Equation the Vibrating Drumhead Heat Flow in the Infinite Cylinder Heat Flow in the Finite Cylinder
3 Differential Operators in Various Coordinates Differential Operators in Various Coordinates
4 Differential Operators in Polar Coordinates Polar Coordinates x = r cos θ, y = r sin θ r 2 = x 2 + y 2, tan θ = y x Differential relations r x = x r, θ x = y, and 2 r r 2 x 2 r y = y r, θ y = x, and 2 r r 2 y 2 2 θ x θ θ =, and y2 x = y2 r 3, 2 θ x 2 = x2, 2 θ r 3 y 2 r x + θ y Differential operators u u x = cos θ r sin θ r 2 u = 2 u x u y 2 = 2 u r r = 2xy r 4 = 2xy r 4 r y = u u u θ, and y = sin θ r + cos θ u r θ u r u r 2 θ 2
5 Laplacian in Various Coordinates Polar Coordinates (r, θ) : x = r cos θ, y = r sin θ 2 u = 2 u r u r r u r 2 θ 2 Cylindrical Coordinates (ρ, φ, z) : x = ρ cos φ, y = ρ sin φ, z = z 2 u = 1 ( ρ u ) u ρ ρ ρ ρ 2 φ u z 2 Spherical Coordinates (r, θ, φ) : x = r cos φ sin θ, y = r sin φ sin θ, z = r cos θ 2 u = ( 1 r 2 r 2 u ) ( 1 + r r r 2 sin θ u ) 1 2 u + sin θ θ θ r 2 sin 2 θ φ 2
6 Laplace Equation on Cylindrical Coordinates Laplace Equation on Cylindrical Coordinates
7 Dirichlet Problems on a Disk or inside Inf Cylinder A Dirichlet s problem inside a Disk or Infinite Cylinder Variables u(ρ, φ, z) = u(r, φ). PDE 2 u = 2 u r + 1 u 2 r r u =, < r < L, φ < 2π. r 2 φ2 Periodic Boundary Conditions u(r, φ) = u(r, φ + 2π), < r < L, φ < 2π; u(l, φ) = f(φ), φ < 2π. Regularity Conditions: u is finite in < r < L.
8 Dirichlet Problems on a Disk or inside Inf Cylinder Separation of variables u(r, φ) = R(r)Φ(φ) = R R + 1 r R = Φ Φ = λ. Sturm-Liouville Eq. associated to Φ is Harmonic Eq. Φ + λφ =, φ < 2π, Φ() = Φ(2π), Φ () = Φ (2π). Eigenvalue and eigenfunction for the Sturm-Liouville Eq. λ n = n 2, n =, 1, 2,... { A, n = ; Φ n (φ) = A n cos nφ + B n sin nφ, n = 1, 2,.... R
9 Dirichlet Problems on a Disk or inside Inf Cylinder ODE associated to R is a Euler Equation r 2 R + rr λr =, < r < L. { C + D ln r for λ = ; R(r) = C n r n + D n r n, for λ n, n = 1, 2,.... Regularity condition: u is finite in < r < L requires that D = D n =, because ln r and r n as r. A general solution < r < L and φ < 2π is u(r, φ) = a 2 + (a n r n cos nφ+b n r n sin nφ), n = 1, 2,... n=1
10 Dirichlet Problems on a Disk or inside Inf Cylinder Coefficients a = 1 2π f(φ) dφ 2π an L n = 1 2π f(φ) cos nφ dφ 2π bn L n = 1 2π f(φ) sin nφ dφ 2π A complex form of the general solution is u(r, φ) = c n r n e inφ, < r < L, φ < 2π. c L n = 1 2π n= f(φ)e inφ dφ, n =, ±1, ±2,...
11 Neumann Problems on a Disk or inside Inf Cylinder A Neumann s problem on a Disk or inside Infinite Cylinder Variables u(ρ, φ, z) = u(r, φ). PDE 2 u = 2 u r + 1 u 2 r r u =, < r < L, φ < 2π. r 2 φ2 Periodic Boundary Conditions u(r, φ) = u(r, φ + 2π), < r > L, φ < 2π; u r (L, φ) = f(φ), φ < 2π. Regularity Conditions: u is finite in < r < L.
12 Neumann Problems on a Disk or inside Inf Cylinder General solution u(r, φ) = a 2 + (a n r n cos nφ + b n r n sin nφ) n= Boundary condition u ( r = an nl n 1 cos nφ + b n nl n 1 sin nφ ) = f(θ) r=l n=1 Coefficients an nl n 1 = 1 2π f(φ) cos nφ dφ 2π bn nl n 1 = 1 2π f(φ) sin nφ dφ 2π Note that the B.C. must satisfies 2π f(φ) dφ =
13 Dirichlet Problems outside a Disk or Inf. Cylinder A Dirichlet s problem outside a Disk or Infinite Cylinder Variables u(ρ, φ, z) = u(r, φ). PDE 2 u = 2 u r + 1 u 2 r r u =, r > L, φ < 2π. r 2 φ2 Periodic Boundary Conditions u(r, φ) = u(r, φ + 2π), r > L, φ < 2π; u(l, φ) = f(φ), φ < 2π. Regularity Conditions u and u dr is finite for r > L L and u as r.
14 Dirichlet Problems outside a Disk or Inf. Cylinder Similar to the Laplace equation inside a disk: Eigenvalue: λ = n 2, n =, 1, 2,... Eigenfunction: { A, n = ; Φ n (φ) = A n cos nφ + B n sin nφ, n = 1, 2,.... Solution { for Euler Eq: C + D ln r for λ = ; R(r) = C n r n + D n r n, for λ n, n = 1, 2,..., and D = C n = as R(r) must be finite when r. General solution u(r, φ) = a 2 + (a n r n cos nφ + b n r n sin nφ) n=
15 Neumann Problems outside a Disk or Inf Cylinder A Neumann s problem outside a Disk or Infinite Cylinder Variables u(ρ, φ, z) = u(r, φ). PDE 2 u = 2 u r + 1 u 2 r r u =, r > L, φ < 2π. r 2 φ2 Periodic Boundary Conditions u(r, φ) = u(r, φ + 2π), r > L, φ < 2π; u r (L, φ) = f(φ), φ < 2π. Regularity Conditions u and u dr is finite for r > L L and u as r.
16 Neumann Problem outside a Disk or Inf Cylinder General solution u(r, φ) = a 2 + (a n r n cos nφ + b n r n sin nφ) n= Boundary condition u ( r = an nl n 1 cos nφ b n nl n 1 sin nφ ) = f(φ) r=l n=1 Coefficients an nl n 1 = 1 2π f(φ) cos nφ dφ 2π bn nl n 1 = 1 2π f(φ) sin nφ dφ 2π Note that the B.C. must satisfies 2π f(φ) dφ =
17 Dirichlet Problem on a Ring A Dirichlet s problem on a ring Variables u(ρ, φ, z) = u(r, φ). PDE + B.C. 2 u r u r r u r 2 =, A < r < B, φ < 2π. φ2 u(r, φ) = u(r, φ + 2π), A < r < B, φ < 2π; u(a, φ) = fa (φ), u(b, φ) = f B (φ), φ < 2π. General solution u(r, φ) = a + b ln r + { (an r n + b n r n ) cos nφ + (a n r n + b n r n ) sin nφ } n=1
18 Robin s Problem on a Wedge A Robin s problem on a wedge Variables u(ρ, φ, z) = u(r, φ). PDE + B.C.. 2 u = 2 u r u r r u r 2 =, < r < L, φ < α. φ2 u(r, ) = u(r, α) =, < r < L; r u(l, φ) = u(l, φ) φ, φ < α. Solution u(r, φ) = b n r nπ nπ α sin α φ where b n = 2α2 ( 1) n nπl(α + nπ) n=1
19 Bessel Functions Bessel s functions have wide application in mathematics, especially when dealing with cylindrical symmetry or cylindrical coordinate systems. Definition (Bessel s functions & Bessel s equation) Bessel s functions J ν or Y ν are solutions to the Bessel s equation of order ν x 2 y + xy + (x 2 ν 2 )y =. (1)
20 Bessel Functions If we seek a series solution about x = for the Bessel s equation, then y(x) = n= a n x n+r, a n = for n <, as x = is a regular singular point. Substitute into the Bessel s equation, we obtain Indicial equation r 2 ν 2 =, = r = ±ν r1 r 2 = 2ν, and if 2ν is integer, we need to find another linearly independent series solution Recurrence relation: a1 =, and n(n ± 2ν)a n = a n 2, n 2.
21 Bessel Functions If 2ν is not an integer, two linearly independent solutions for Eq.(1)are [ ] y1 (x) = x ν 1 x2 2(2+2ν) + x 4 2(4)(2+2ν)(4+2ν) [ +... ] y2 (x) = x ν 1 x2 2(2 2ν) + x 4 2(4)(2 2ν)(4 2ν) +... After normalizing the solutions, we obtain the general solution to the Bessel s equation y(x) = c 1 J ν (x) + c 2 J ν (x), where Jν (x) y 1 (x); and J ν (x) y 2 (x); and J ν (x) is called the Bessel s functions of the first kind.
22 Bessel Functions Definition (Bessel s functions of the first kind) The Bessel s function of the first kind of order ν is defined as ( 1) ( ) n x ν+2n 2 J ν (x) = n!(n + ν)! n= = 1 [ ν! (1 2 x)ν 1 ( x 2 )2 ν ( x ] 2 )4 2(ν + 1)(ν + 2) Definition (Gamma/factorial function) For ν is not an integer, ν! is defined as ν! Γ(ν + 1) t ν e t dt.
23 Bessel Functions If 2ν is an integer, and ν = N + 1, for some integer N, 2 the resulting functions are called spherical Bessel s functions jn (x) = (π/2x) 1/2 J N+1/2 (x) (We will come back to speherical Bessel s function later) ν = N, for some integer N, JN (x) = ( 1) N J N (x) is linearly dependent to J N (x); a linearly independent second solution is YN (x) and it is called the Bessel s function of the second kind; now the general solution of Eq.(1) is y(x) = c 1 J N (x) + c 2 Y N (x).
24 Bessel Functions Definition (Bessel s functions of the second kind) The Bessel s function of the second kind of order ν is defined as Y ν (x) = J ν(x) cos(πν) J ν (x). sin(πν) If ν = N, for some integer N, then Y N (x) = lim ν N Y ν (x).
25 Bessel Functions A variation of Bessel s equation of order ν is of the form, x 2 y + xy + (λ 2 x 2 ν 2 )y = (2) Eq.(2) can be converted to the standard Bessel s Equation by defining new variable t = λx. The general solution is y(x) = c 1 J ν (λx) + c 2 Y ν (λx).
26 Bessel Functions A modified Bessel s Equation is of the form x 2 y + xy (x 2 + ν 2 )y =. (3) The general solution to modified Bessel s equation is y(x) = c 1 J ν (ix) + c 2 Y ν (ix); or y(x) = c 1 I ν (x) + c 2 K ν (x). Definition (Modified Bessel s functions) First kind I ν (x) = i ν J ν (ix) = n= ( x 2 )ν+2n n!(n + ν)! ; I r (x) cos rπ I r (x) Second kind K ν (x) = lim. r ν sin rπ
27 Bessel Functions A variation of modified Bessel s Equation is of the form x 2 y + xy (λ 2 x 2 + ν 2 )y =, (4) with the general solution y(x) = c 1 I ν (λx) + c 2 K ν (λx).
28 Bessel Functions Figure: Bessel s functions of the first kind, J n (x). Figure: Bessel s functions of the second kind, Y n (x).
29 Bessel Functions Figure: Modified Bessel s functions of the first kind, I n (x). Figure: Modified Bessel s functions of the second kind, K n (x).
30 Bessel Functions Another representation of the Bessel s functions for integer values of ν = N is in the integral form J N (x) = 1 π π π cos(nτ x sin τ) dτ Y N (x) = 1 sin(x sin τ Nτ) dτ π 1 [ e Nτ + ( 1) N e Nτ] e x sinh τ dτ π
31 Bessel Functions Also the Bessel s functions could be generated from the generating function e (x/2)(t 1/t) = J n (x)t n e iz cos φ = n= n= i n J n (z)e inφ
32 Bessel Functions Recursion relations of Bessel s functions d dx [xν J ν (x)] = x ν J ν 1 (x) d dx [x ν J ν (x)] = x ν J ν+1 (x) J ν 1 (x) + J ν+1 (x) = 2ν x J ν(x) J ν 1 (x) J ν+1 (x) = 2J ν(x) J ν(x) = ν x J ν(x) + J ν 1 (x) = ν x J ν(x) J ν+1 (x)
33 Bessel Functions Orthogonality properties of Bessel s functions: If α and β are two roots of J ν, then 1 xj ν (αx)j ν (βx) dx = δ α,β 2 J 2 ν+1(α) = δ α,β 2 J 2 ν (α) Thus, the generalized Fourier series in terms of J ν is f(x) = c n J ν (λ n x), and where the coefficient is given by c n = n=1 R 1 xf(x)jν(λnx) dx R 1 xj ν 2 (λ nx) dx = 2 J 2 ν+1 (λn) 1 xf(x)j ν(λ n x) dx.
34 The Vibrating Drumhead (Radial Symmetric I.C.) The vibrating of a thin circular membrane with uniform density with radial symmetric initial conditions is given by ( PDE: 2 u 2 t = u 2 c2 r + 1 ) u, < r < l, t >. 2 r r B.C.: u(l, t) =, t ; I.C.: u(r, ) = f(r), u (r, ) = g(r), r < r < l. The initial conditions are said to be radial symmetric as f and g are depend only on r but not φ.
35 The Vibrating Drumhead (Radial Symmetric I.C.) Separation of variables: u(r, t) = R(r)T (t) = T c 2 T = 1 R (R + 1 r R ) = λ 2. The ODEs and corresponding B.C. rr + R + λ 2 rr =, R(l) = (5) T + c 2 λ 2 T = (6) Solutions to ODEs R(r) = c 1 J (λr) + Y (λr) Since u is bounded near r =, this gives c2 = ; B.C. R(l) = = J (λl) = ; Let αn, n = 1, 2,... be the roots of J, then λ = λ n = αn l, n = 1, 2,....
36 The Vibrating Drumhead (Radial Symmetric I.C.) Non-trivial solutions to the ODEs ( αn ) R n (r) = J (λ n r) = J l r, n = 1, 2,..., T n (t) = A n cos cλ n t + B n sin cλ n t General solution and coefficients u(r, t) = (A n cos cλ n t + B n sin cλ n t)j (λ n r), A n = cλ n B n = n=1 2 l 2 J 2 1 (α n ) 2 l 2 J 2 1 (α n ) l l f(r)j (λ n r)r dr, g(r)j (λ n r)r dr.
37 The Vibrating Drumhead (Asymmetric I.C.) The vibrating of a thin circular membrane with uniform density with asymmetric initial conditions is given by ( PDE: 2 u 2 t = u 2 c2 r + 1 u 2 r r + 1 ) 2 u, r 2 φ 2 < r < l, φ < 2π, t >. B.C.: u(l, φ, t) =, φ < 2π, t ; u I.C.: u(r, φ, ) = f(r, φ), (r, φ, ) = g(r, φ), r < r < l, φ < 2π. The initial conditions are said to be asymmetric as f and g are depend on r and φ.
38 The Vibrating Drumhead (Asymmetric I.C.) Separation of variables: u(r, φ, t) = R(r)Φ(φ)T (t) = T = R + R + Φ = c 2 T R rr r 2 Φ λ2 ; and r 2 R +rr +λr 2 R = Φ = R Φ µ2. The ODEs and corresponding B.C. Φ + µ 2 Φ =, Φ() = Φ(2π), Φ () = Φ (2π) r 2 R + rr + (λ 2 r 2 µ 2 )R =, R(l) = T + c 2 λ 2 T = Solutions to ODEs λ mn = αmn l and α mn is n th root of J m. Φ(φ) = Φ m (φ) = A m cos mφ + B m sin mφ, m =, 1, 2,... R(r) = R mn (r) = J m (λ mn r), m =, 1, 2,..., n = 1, 2,... T (t) = T mn (t) = A mn cos cλ mn t + B mn sin cλ mn t.
39 The Vibrating Drumhead (Asymmetric I.C.) General solution u(r, φ, t) and coefficients u = J m (λ mn r)(a mn cos mφ + b mn sin mφ) cos cλ mn t + a n = a mn = b mn = m= n=1 m= n=1 J m (λ mn r)(a mn cos mφ + b mn sin mφ) sin cλ mn t 1 πl 2 J 2 1 (α n ) l 2π 2 πl 2 J 2 m+1(α mn ) 2 πl 2 J 2 m+1(α mn ) l 2π l 2π f(r, φ)j (λ n r) r dr dφ f(r, φ) cos mφj m (λ mn r) r dr dφ f(r, φ) sin mφj m (λ mn r) r dr dφ
40 The Vibrating Drumhead (Asymmetric I.C.) a n = a mn = b mn = 1 πl 2 J 2 1 (α n ) l 2π 2 πl 2 J 2 m+1(α mn ) 2 πl 2 J 2 m+1(α mn ) l 2π l 2π g(r, φ)j (λ n r) r dr dφ g(r, φ) cos mφj m (λ mn r) r dr dφ g(r, φ) sin mφj m (λ mn r) r dr dφ
41 Heat Flow in the Infinite Cylinder (Radial Sym.) The radial flow of heat in a infinite cylinder or on a disk with lateral surface is kept at zero temperature is given by PDE: u ( 2 t = u c2 r + 1 ) u, < r < l, t >. 2 r r B.C.: u(l, t) =, t ; I.C.: u(r, ) = f(r), < r < l. The initial conditions are said to be radial symmetric as f is depend only on r.
42 Heat Flow in the Infinite Cylinder (Radial Sym.) Separation of variables: u(r, t) = R(r)T (t) = T c 2 T = 1 R (R + 1 r R ) = λ 2. The ODEs and corresponding B.C. Solutions to ODEs rr + R + λ 2 rr =, R(l) = R n (r) = A n J (λ n r) T n (t) = C n e c2 λ 2 n t T + c 2 λ 2 T =. λ n = α n /l, n = 1, 2,..., α n are roots of J.
43 Heat Flow in the Infinite Cylinder (Radial Sym.) General solution and coefficients u(r, t) = a n = a n J (λ n r)e c2 λ 2 n t, n=1 2 l 2 J 2 1 (α n ) l f(r)j (λ n r)r dr.
44 Heat Flow in a Finite Cylinder The heat flow in a finite cylinder of radius l and height 2h with lateral surface and bases are kept at zero temperature is given by PDE: u ( 2 t = u c2 r + 1 u 2 r r + 1 ) 2 u r 2 φ + 2 u, 2 z 2 < r < l, φ < 2π, h < z < h, t >. B.C.: u(l, φ, z, t) = u(r, φ, h, t) = u(r, φ, h, t) =, I.C.: u(r, φ, z, ) = f(r, φ, z).
45 Heat Flow in a Finite Cylinder Separation of variables: u(r, φ, z, t) = R(r)Φ(φ)Z(z)T (t) The general solution u(r, φ, z, t) = m= n=1 k=1 { e c2 (λ 2 mn +µ2 k )t J m (λ mn r) sin µ k (z + h) (A mnk cos mφ + B mnk sin mφ) }
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