Find a solution to the Laplace equation in cylindrical coordinates with specific boundary conditions on a half-infinite cylinder | Step-by-Step Solution

🌟 Solving the Laplace Equation

1. What We're Solving:

We need to find a function u(r,θ,z) that satisfies Laplace's equation ∇²u = 0 in cylindrical coordinates, specifically outside a half-infinite cylinder (r > a, z > 0) with given boundary conditions. This is a classic problem in mathematical physics!

2. The Approach:

We'll use the separation of variables method because:

• Cylindrical coordinates are perfect for cylindrical geometry
• The linearity of Laplace's equation allows us to build solutions from simpler parts
• We can match our general solution to the specific boundary conditions

Laplace's equation describes steady-state situations (like temperature distributions or electric potentials), and separation of variables lets us break the 3D problem into three easier 1D problems.

3. Step-by-Step Solution:

Step 1: Set up the equation in cylindrical coordinates ∇²u = (1/r)(∂/∂r)(r∂u/∂r) + (1/r²)(∂²u/∂θ²) + ∂²u/∂z² = 0

Step 2: Assume separated solution Let u(r,θ,z) = R(r)Θ(θ)Z(z)

Step 3: Substitute and separate After substituting and dividing by RΘZ:

• (1/r)(d/dr)(r dR/dr)/R + (1/r²)(d²Θ/dθ²)/Θ + (d²Z/dz²)/Z = 0

Multiply by r² and rearrange:

• r(d/dr)(r dR/dr)/R + (d²Θ/dθ²)/Θ + r²(d²Z/dz²)/Z = 0

Step 4: Identify separation constants Since each term depends on a different variable, each must equal a constant:

• d²Z/dz² = k²Z (choose k² for exponential solutions in z)
• d²Θ/dθ² = -n²Θ (choose -n² for periodic solutions in θ)
• r(d/dr)(r dR/dr) = (k²r² + n²)R

Step 5: Solve each ODE

For Z(z): Z(z) = A₁e^(kz) + A₂e^(-kz)

For Θ(θ): Θ(θ) = B₁cos(nθ) + B₂sin(nθ)

For R(r): This gives us Bessel's equation!

• If k ≠ 0: R(r) = C₁I_n(kr) + C₂K_n(kr)
• If k = 0: R(r) = C₃r^n + C₄r^(-n)

Step 6: Apply boundary conditions You'll need to:

• Use boundedness conditions (u must not blow up as r→∞ or z→∞)
• Apply the given boundary values
• Use orthogonality to find coefficients

4. The General Framework:

Your final solution will typically look like:

u(r,θ,z) = Σ[A_n K_n(k_n r)(B_n cos(nθ) + C_n sin(nθ))e^(-k_n z)]

The specific coefficients A_n, B_n, C_n and values k_n depend on your exact boundary conditions!

5. Memory Tip: 🎯

Remember "SBC" - Separate variables, solve the Bessel equation for r, then apply Conditions!

The key insight is that problems with cylindrical symmetry naturally lead to Bessel functions - they're the "sine and cosine" of cylindrical coordinates!