---
title: "Laplace equation with u(x,∞) = sin(nπx/l)"
subject: "Applied Mathematics"
chapter: 3
key: "semester_4/applied_mathematics:laplace-eq-u-x-inf-sin"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/laplace-eq-u-x-inf-sin
site: BCT Engineering
---

# Laplace equation with u(x,∞) = sin(nπx/l)

Applied Mathematics · Chapter 3: Partial Differential Equations

- Asked in: 2073 Magh, 10 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/laplace-eq-u-x-inf-sin
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-3

## Question

Solve completely the Laplace equation ∂²u/∂x² + ∂²u/∂y² = 0 under the conditions: u(0, y) = u(l, y) = u(x, 0) = 0, u(x, ∞) = sin(nπx/l).

## Short answer

The bounded solution as y→∞ with u(x,0)=sin(nπx/l) is u(x,y) = sin(nπx/l) e^{−nπy/l}. If data is on a finite top y=b with u(x,0)=0, then u(x,y)= sin(nπx/l) sinh(nπy/l)/sinh(nπb/l).

## Answer

**Reading of the problem.** The conditions are u(0, y) = u(l, y) = 0 and u(x, 0) = 0 with u(x, ∞) = sin(nπx/l). Taken literally this cannot hold: the solution that vanishes at y = 0 is proportional to sinh(nπy/l), which tends to ∞ rather than to a finite profile. The usual consistent reading in the strip 0 ≤ x ≤ l, y ≥ 0 is: u is bounded as y → ∞, with u = 0 on x = 0 and x = l, and the non-zero data sin(nπx/l) on the edge y = 0. I solve that problem, and also give the finite-top case y = b that the literal wording leads to.

**Step 1: separate variables.** u = X(x)Y(y): X″/X = −Y″/Y = −p².

```text
X″ + p² X = 0  →  X = A cos px + B sin px
Y″ − p² Y = 0  →  Y = C e^{py} + D e^{−py}
```

**Step 2: x-conditions.** X(0) = 0 gives A = 0; X(l) = 0 gives p = mπ/l. So Xₘ = sin(mπx/l).

**Step 3: y-condition at infinity.** u must stay bounded as y → ∞, so C = 0:

u(x, y) = Σ_{m=1}^{∞} Bₘ sin(mπx/l) e^{−mπy/l}

**Step 4: data on the edge y = 0.** u(x, 0) = sin(nπx/l) = Σ Bₘ sin(mπx/l). By orthogonality, Bₙ = 1 and all other Bₘ = 0.

**Answer (bounded strip):**

**u(x, y) = sin(nπx/l) e^{−nπy/l}**

**If the data sin(nπx/l) is on a finite top edge y = b** with u(x, 0) = 0, the solution is

u(x, y) = sin(nπx/l) sinh(nπy/l) / sinh(nπb/l)

which has the same form as the earlier rectangle problem.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why can the problem not be taken literally?
   A. Because Laplace's equation has no solution in a strip
   B. Because sin(n pi x/l) does not vanish at x = 0
   C. A solution vanishing at y=0 is proportional to sinh(n pi y/l), which tends to infinity, not a finite profile
   D. Because X must be exponential
   Answer: C. sinh grows without bound, so the data at y infinity cannot be finite. (Key point: Problem reading)

2. With X'' + p^2 X = 0 and zero values at x=0 and x=l, what is X?
   A. cos(m pi x/l)
   B. sinh(m pi x/l)
   C. e^{m pi x/l}
   D. sin(m pi x/l), with p = m pi/l
   Answer: D. X(0)=0 gives A=0 and X(l)=0 gives sin pl = 0. (Key point: Separation)

3. What does boundedness as y tends to infinity imply?
   A. C = 0, leaving e^{-p y}
   B. D = 0, leaving e^{p y}
   C. p = 0
   D. C = D
   Answer: A. e^{py} would blow up, so only the decaying exponential is kept. (Key point: Bounded condition)

4. What is the bounded-strip answer?
   A. u = sin(n pi x/l) e^{n pi y/l}
   B. u = sin(n pi x/l) e^{-n pi y/l}
   C. u = sin(n pi x/l) sinh(n pi y/l)
   D. u = sin(n pi y/l) e^{-n pi x/l}
   Answer: B. Data sin(n pi x/l) on y=0 picks out B_n = 1 only. (Key point: Final answer)

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