---
title: "Poisson: ∇²u = -10(x²+y²+10)"
subject: "Numerical Method"
chapter: 7
key: "semester_4/numerical_method:poisson-10-x2-y2-10"
source: https://www.bctengineer.com/q/semester_4/numerical_method/poisson-10-x2-y2-10
site: BCT Engineering
---

# Poisson: ∇²u = -10(x²+y²+10)

Numerical Method · Chapter 7: Numerical solution of Partial differential Equation

- Asked in: 2068 Bhadra, 10 marks
- Video: [NUMERICAL SOL OF POISSON EQUATION Eg.problem2 | Solution for Poisson Eqn by finite difference method](https://www.youtube.com/watch?v=jTptSWQYlSg) — THINK MASTERMIND
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/poisson-10-x2-y2-10
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7

## Question

Solve the equation ∇²u = -10(x² + y² + 10) over a square mesh with sides x = 0, y = 0, x = 3, y = 3 with u = 0 on the boundary and mesh length = 1.

## Short answer

Applying the five-point formula for Poisson's equation with source term f(x,y) = -10(x²+y²+10) and Gauss-Seidel iteration, the solution converges to u1 = 75.0000, u2 = 82.5000, u3 = 67.5000, u4 = 75.0000.

## Answer

Poisson's equation ∇²u = f with f(x,y) = −10(x² + y² + 10) is solved with the five-point formula (h = 1):

u(i−1,j) + u(i+1,j) + u(i,j−1) + u(i,j+1) − 4u(i,j) = h²f(i,j)

so u(i,j) = ¼[ sum of four neighbours − f(i,j) ].

#### Set up
The square 0 ≤ x, y ≤ 3 with h = 1 has four interior points: u₁ = u(1,2), u₂ = u(2,2), u₃ = u(1,1), u₄ = u(2,1). All boundary values are 0.

```text
f(1,2) = −10(1+4+10) = −150      f(2,2) = −10(4+4+10) = −180
f(1,1) = −10(1+1+10) = −120      f(2,1) = −10(4+1+10) = −150

u₁ = ¼(0 + 0 + u₂ + u₃ + 150) = ¼(150 + u₂ + u₃)
u₂ = ¼(180 + u₁ + u₄)
u₃ = ¼(120 + u₁ + u₄)
u₄ = ¼(150 + u₂ + u₃)
```
(u₁ and u₄ have the same equation, so u₁ = u₄.)

#### Iteration (Gauss–Seidel)
Start with u = 30 for all points.

| iteration | u₁ | u₂ | u₃ | u₄ |
|---|---|---|---|---|
| 0 | 30.0000 | 30.0000 | 30.0000 | 30.0000 |
| 1 | 52.5000 | 65.6250 | 50.6250 | 66.5625 |
| 2 | 66.5625 | 78.2812 | 63.2812 | 72.8906 |
| 3 | 72.8906 | 81.4453 | 66.4453 | 74.4727 |
| 4 | 74.4727 | 82.2363 | 67.2363 | 74.8682 |
| 5 | 74.8682 | 82.4341 | 67.4341 | 74.9670 |
| 6 | 74.9670 | 82.4835 | 67.4835 | 74.9918 |
| 7 | 74.9918 | 82.4959 | 67.4959 | 74.9979 |
| 8 | 74.9979 | 82.4990 | 67.4990 | 74.9995 |
| 9 | 74.9995 | 82.4997 | 67.4997 | 74.9999 |
| 10 | 74.9999 | 82.4999 | 67.4999 | 75.0000 |
| 11 | 75.0000 | 82.5000 | 67.5000 | 75.0000 |
| 12 | 75.0000 | 82.5000 | 67.5000 | 75.0000 |

Converged to 4 decimals after 12 iterations.

#### Direct solution (check)
Put u₁ = u₄ = a, u₂ = b, u₃ = c: a = ¼(150 + b + c), b = ¼(180 + 2a), c = ¼(120 + 2a). Then 4a = 150 + (300 + 4a)/4 → 3a = 225 → a = 75.0000, b = 82.5000, c = 67.5000.

**Answer: u₁ = 75.0000, u₂ = 82.5000, u₃ = 67.5000, u₄ = 75.0000**

## Check yourself

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

1. For Poisson with h = 1, what is u(i,j)?
   A. (sum of four neighbours + f(i,j))/4
   B. (sum of four neighbours - f(i,j))/4
   C. (sum of neighbours)/4
   D. (sum of neighbours - 4f)/4
   Answer: B. From sum - 4u = h^2 f. (Key point: Formula)

2. At (1,2), f = -150. What is the contribution to u1's equation?
   A. -150
   B. +37.5 as u
   C. +150 inside the bracket
   D. 0
   Answer: C. -f = +150, so u1 = (150 + u2 + u3)/4. (Key point: Sign of f)

3. How do boundary values affect the equations?
   A. They add 100 each
   B. They add f
   C. They make u1 = 0
   D. They are zero, so only interior neighbours appear
   Answer: D. All boundary values are 0. (Key point: Zero boundary)

4. What are u1 and u3?
   A. 75 and 67.5
   B. 82.5 and 75
   C. 75 and 82.5
   D. 67.5 and 75
   Answer: A. a = 75, b = 82.5, c = 67.5. (Key point: Result)

## Related in Numerical Method

- [Elliptic: 0≤x,y≤3 boundary](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-uxx-0-3-boundary) — Numerical Method, chapter 7
- [Elliptic: 2x2 grid 10/20/40/50](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aelliptic-2x2-10-20-40-50) — Numerical Method, chapter 7, asked 5×
- [Elliptic: 8 cm plate, 50/30/10](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Apoisson-8cm-plate) — Numerical Method, chapter 7
- [Laplace: 2x2 grid 10/15/20](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-2x2-10-15-20-25) — Numerical Method, chapter 7, asked 6×
- [Poisson: uxx + uyy = -81xy](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Apoisson-81xy) — Numerical Method, chapter 7
- [Laplace: 3x3 grid 500/1000/500](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-3x3-0-500-1000) — Numerical Method, chapter 7, asked 6×

## Related in other subjects

- [Derive or solve wave equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aderive-or-solve-wave-equation) — Mathematics III, chapter 4
- [Chi-square goodness of fit: Poisson errors](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Achi-square-goodness-fit-poisson-pages) — Probability and Statistics, chapter 7, asked 2×
- [Derive 1D wave equation and solve](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Awave-eq-derive-and-solve) — Applied Mathematics, chapter 3, asked 5×
- [Derive 1D heat equation and solve](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aheat-eq-derive-and-solve) — Applied Mathematics, chapter 3, asked 4×