--- title: "Elliptic: 0≤x,y≤3 boundary" subject: "Numerical Method" chapter: 7 key: "semester_4/numerical_method:laplace-uxx-0-3-boundary" source: https://www.bctengineer.com/q/semester_4/numerical_method/laplace-uxx-0-3-boundary site: BCT Engineering --- # Elliptic: 0≤x,y≤3 boundary Numerical Method · Chapter 7: Numerical solution of Partial differential Equation - Asked in: 2073 Bhadra, 10 marks - Video: [Solution of Elliptic Equation - Part 4 | 9 Mesh Squares | Laplace Equation](https://www.youtube.com/watch?v=VODlaV2AiHc) — Mathematics Tutor - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/numerical_method/laplace-uxx-0-3-boundary - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7 ## Question Solve the elliptic equation uxx + uyy = 0 on the square mesh bounded by 0 ≤ x ≤ 3, 0 ≤ y ≤ 3. The boundary values are u(x, 0) = 10, u(x, 3) = 90, 0 ≤ x ≤ 3 and u(0, y) = 70, u(3, y) = 0, 0 < y < 3. ## Short answer Using standard five-point finite difference formula for Laplace's equation, solved by Gauss-Seidal iteration. Converged interior values: u(1,2)=61.2500, u(2,2)=43.7500, u(1,1)=41.2500, u(2,1)=23.7500. ## Answer With h = 1 the interior points are (1,1), (2,1), (1,2), (2,2). Name them u₁ = u(1,2), u₂ = u(2,2), u₃ = u(1,1), u₄ = u(2,1) (y increases upward). For Laplace's equation ∇²u = 0 on a square mesh, each interior value equals the mean of its four neighbours (standard five-point formula): u(i,j) = ¼[u(i−1,j) + u(i+1,j) + u(i,j−1) + u(i,j+1)] Boundary values: top (y = 3) = 90, bottom (y = 0) = 10, left (x = 0) = 70, right (x = 3) = 0. #### Equations ```text u₁ = ¼(160 + u₂ + u₃) u₂ = ¼(90 + u₁ + u₄) u₃ = ¼(80 + u₁ + u₄) u₄ = ¼(10 + u₂ + u₃) ``` #### Iteration (Gauss–Seidel / Liebmann) Initial value 42 for every unknown (close to the mean of the boundary values); each new value is used at once in the next equation. | iteration | u₁ | u₂ | u₃ | u₄ | |---|---|---|---|---| | 0 | 42.0000 | 42.0000 | 42.0000 | 42.0000 | | 1 | 61.0000 | 48.2500 | 45.7500 | 26.0000 | | 2 | 63.5000 | 44.8750 | 42.3750 | 24.3125 | | 3 | 61.8125 | 44.0312 | 41.5312 | 23.8906 | | 4 | 61.3906 | 43.8203 | 41.3203 | 23.7852 | | 5 | 61.2852 | 43.7676 | 41.2676 | 23.7588 | | 6 | 61.2588 | 43.7544 | 41.2544 | 23.7522 | | 7 | 61.2522 | 43.7511 | 41.2511 | 23.7505 | | 8 | 61.2505 | 43.7503 | 41.2503 | 23.7501 | | 9 | 61.2501 | 43.7501 | 41.2501 | 23.7500 | | 10 | 61.2500 | 43.7500 | 41.2500 | 23.7500 | | 11 | 61.2500 | 43.7500 | 41.2500 | 23.7500 | Converged to 4 decimals after 11 iterations. #### Direct solution of the equations (check) ```text u₁ = 61.2500 u₂ = 43.7500 u₃ = 41.2500 u₄ = 23.7500 ``` **Answer: u₁ = 61.2500, u₂ = 43.7500, u₃ = 41.2500, u₄ = 23.7500** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With h = 1 on 0<=x,y<=3, how many interior points? A. 9 B. 2 C. 4 D. 16 Answer: C. Interior x,y = 1,2. (Key point: Interior points) 2. u1 = u(1,2): what is its equation? A. u1 = (90 + u2 + u3)/4 B. u1 = (70 + u2 + u3)/4 C. u1 = (80 + u2 + u3)/4 D. u1 = (160 + u2 + u3)/4 Answer: D. Top 90 + left 70 = 160. (Key point: Equation u1) 3. u4 = u(2,1): which boundary values enter? A. Right 0 and bottom 10, so u4 = (10 + u2 + u3)/4 B. Right 0 and top 90 C. Left 70 and bottom 10 D. Right 0 only Answer: A. (2,1) touches x = 3 (value 0) and y = 0 (value 10). (Key point: Equation u4) 4. What is u1? A. 43.75 B. 61.25 C. 41.25 D. 23.75 Answer: B. The solved values are 61.25, 43.75, 41.25, 23.75. (Key point: Result) ## Related in Numerical Method - [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× - [Poisson: uxx + uyy = -81xy](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Apoisson-81xy) — Numerical Method, chapter 7 - [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: 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× - [Laplace: 3x3 grid 400/800/600/1200](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-3x3-0-400-800) — Numerical Method, chapter 7, asked 6× - [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× ## Related in other subjects - [Cable size MDB to SDB1 (70 kW) and SDB2 (90 kW)](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_power_distribution_and_illumination%3Acable-mdb-sdb-70kw-90kw) — Industrial Power Distribution and Illumination, chapter 5, asked 5× - [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 - [Hill road earthwork, 0 m to 90 m, 10/8 m widths](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-e-0-90) — Estimating and Costing, chapter 6, 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×