Definitions, Theorems, Examples, Topics for Asmar 2nd edition
Chapter 4: 4.5 only

 4.5 Laplace's Equation in a Cylinder
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  EXAMPLE 1. STEADY STATE TEMPERATURE in a CYLINDER: Radial Symmetry Case
    u_rr + (1/r) u_r + u_zz = 0,  
    0 < r < a, 0 < z < h,
    u(a,z) = 0 [zero on lateral surface)
    u(r,0) = 0 [zero on cylinder bottom] 
    u(r,h) = f(r) [temperature given on cylinder top]

  PRODUCT SOLUTION 
    u=R(r)Z(z)  r=radial variable, z=vertical variable

    THEOREM
      Z'' + mu Z = 0, Z(0)=0
      r^2 R'' + r R' + (-mu)r^2 R = 0, R(a)=0
      Separation constant mu to be determined

    THEOREM    
      The problem r^2 R'' + r R' + (-mu)r^2 R = 0, R(a)=0 has no
      solution for mu > 0. 

      PROOF. This result depends on section 4.7, problems 29-30. See the
      definitions and theorems in EXAMPLE 2, below, where R is written 
      as R=c1 I0 + c2 K0. Because K0 is unbounded then c2=0. Series I0 has
      only positive terms, so I0(a)>0. Then R(a)=0 implies c1=0. Hence R=0.

    THEOREM    
      The problem r^2 R'' + r R' + (-mu)r^2 R = 0, R(a)=0 has no
      solution for mu = 0. In this case, the DE is an Euler-Cauchy 
      equation with solution R=c1 + c2 ln|r|.

    THEOREM    
      R(r) = J_0(alpha_n r/a) for n=1,2,3, ... [section 4.8, Thm 3]
      Symbol alpha_n = nth positive zero of Bessel function J_0
      Symbol mu = -(alpha_n / a)^2.

    THEOREM
      Z(z) = linear combination of the Euler solution atoms
             cosh(alpha_n z/a), sinh(alpha_n z/a), for n=1,2,3, ...
                        
    THEOREM
      The solution of the Dirichlet problem is the sum n=1 to infinity of
      the terms 

        J0(alpha_n r/a)(a_n cosh(alpha_n z/a) + b_n sinh(alpha_n z/a)) 

      The coefficients are Bessel series coefficients of f(r)=u(r,h).


  EXAMPLE 2. STEADY STATE TEMPERATURE in a CYLINDER: Radial Symmetry Case
    u_rr + (1/r) u_r + u_zz = 0,  
    0 < r < a, 0 < z < h,
    u(a,z) = f(z) [non-zero on lateral surface)
    u(r,0) = 0 [zero on cylinder bottom] 
    u(r,h) = 0 [zero on cylinder top]

 PRODUCT SOLUTION 
    u=R(r)Z(z)  r=radial variable, z=vertical variable

    THEOREM
      Z'' + mu Z = 0, Z(0)=0, Z(h)=0
      r^2 R'' + r R' + (-mu)r^2 R = 0, R bounded
      Separation constant mu to be determined

    THEOREM
      The Z-BVP has a solution <==> mu = (n Pi/h)^2, n=1,2,3, ...
      Z(z) = linear combination of the Euler solution atoms
             cos(n Pi z/h), sin(n Pi z/h), for n=1,2,3, ...

    DEF. [background section 4.7]
      BesselI and BesselK are the modified Bessel functions of the first
      and second kinds, respectively. They satisfy the modified Bessel
      equation: x^2 y'' + x y' - (x^2 + p^2) y = 0. The series expansion
      of I0(x) is the same as J0(x), except for the factor (-1)^k, which
      is replaced by 1.

      # maple solve
      de:=r^2*diff(R(r),r,r)+r*diff(R(r),r)- q^2*r^2*R(r)=0;
      dsolve(de,R(r));
      # Answer:  R(r) = _C1 BesselI(0, q r) + _C2 BesselK(0, q r)

    THEOREM [Background section 4.7, problems 29-30]
      The equation r^2 R'' + r R' + (-mu)r^2 R = 0 with mu = q^2 > 0
      has general solution R(r) = c1 I0(q r) + c2 K0(q r) where I0, K0
      are modified Bessel functions of the first and second kind.

    THEOREM 
      R(r) = I0(n Pi r/h) for n=1,2,3, ...
                        
    THEOREM
      The solution of the Dirichlet problem is the sum n=1 to infinity of
      the terms 

                   I0(n Pi r/h)(b_n sin(n Pi z/h)) 

      The coefficients are Bessel series coefficients of f(z)=u(a,z).