--- title: "Poisson: uxx + uyy = -81xy" subject: "Numerical Method" chapter: 7 key: "semester_4/numerical_method:poisson-81xy" source: https://www.bctengineer.com/q/semester_4/numerical_method/poisson-81xy site: BCT Engineering --- # Poisson: uxx + uyy = -81xy Numerical Method · Chapter 7: Numerical solution of Partial differential Equation - Asked in: 2071 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-81xy - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7 ## Question Solve the Poisson equation uxx + uyy = -81xy, 0 < x < 1, 0 < y < 1 given that u(0, y) = 0, u(x, 0) = 0, u(1, y) = 100, u(x, 1) = 100 and h = 1/3. ## Short answer Poisson equation with f=-81xy, h=1/3 gives h²f = -9xy. Applying five-point Poiston formula and Gauss-Seidal iteration, converged interior values: u(1/3,2/3)=51.0833, u(2/3,2/3)=76.5417, u(1/3,1/3)=25.7917, u(2/3,1/3)=51.0833. ## Answer For Poisson's equation ∇²u = f, the five-point formula is u(i−1,j) + u(i+1,j) + u(i,j−1) + u(i,j+1) − 4u(i,j) = h²f(x,y) Here f = −81xy and h = 1/3, so h²f = −9xy and u(i,j) = ¼[ sum of four neighbours + 9xy ] #### Set up Interior points (h = 1/3, y increases upward): u₁ = u(1/3, 2/3), u₂ = u(2/3, 2/3), u₃ = u(1/3, 1/3), u₄ = u(2/3, 1/3). Boundary: u = 0 on x = 0 and y = 0; u = 100 on x = 1 and y = 1. ```text 9xy at u₁ = 9(1/3)(2/3) = 2, u₂ = 9(2/3)(2/3) = 4, u₃ = 9(1/3)(1/3) = 1, u₄ = 9(2/3)(1/3) = 2 u₁ = ¼(100 + 0 + u₂ + u₃ + 2) = ¼(102 + u₂ + u₃) u₂ = ¼(100 + 100 + u₁ + u₄ + 4) = ¼(204 + u₁ + u₄) u₃ = ¼(0 + 0 + u₁ + u₄ + 1) = ¼(1 + u₁ + u₄) u₄ = ¼(100 + 0 + u₂ + u₃ + 2) = ¼(102 + u₂ + u₃) ``` (u₁ and u₄ have the same equation, so u₁ = u₄.) #### Iteration (Gauss–Seidel) Start with u = 51 for all four points. | iteration | u₁ | u₂ | u₃ | u₄ | |---|---|---|---|---| | 0 | 51.0000 | 51.0000 | 51.0000 | 51.0000 | | 1 | 51.0000 | 76.5000 | 25.7500 | 51.0625 | | 2 | 51.0625 | 76.5312 | 25.7812 | 51.0781 | | 3 | 51.0781 | 76.5391 | 25.7891 | 51.0820 | | 4 | 51.0820 | 76.5410 | 25.7910 | 51.0830 | | 5 | 51.0830 | 76.5415 | 25.7915 | 51.0833 | | 6 | 51.0833 | 76.5416 | 25.7916 | 51.0833 | | 7 | 51.0833 | 76.5417 | 25.7917 | 51.0833 | | 8 | 51.0833 | 76.5417 | 25.7917 | 51.0833 | Converged to 4 decimals after 8 iterations. #### Direct solution (check) With u₁ = u₄ = a, u₂ = b, u₃ = c: a = ¼(102 + b + c), b = ¼(204 + 2a), c = ¼(1 + 2a). Substituting gives a = 51.0833, b = 76.5417, c = 25.7917. **Answer: u₁ = 51.0833, u₂ = 76.5417, u₃ = 25.7917, u₄ = 51.0833** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the five-point formula modified for Poisson's equation? A. Sum of neighbours - 4u = 0 B. Sum of neighbours + 4u = h^2 f C. Sum of neighbours - 4u = h^2 f(x,y) D. Sum of neighbours - 4u = f/h^2 Answer: C. The right side h^2 f replaces zero. (Key point: Poisson formula) 2. With f = -81xy and h = 1/3, what is h^2 f? A. -81xy B. -27xy C. -xy/9 D. -9xy Answer: D. (1/9)(-81xy) = -9xy. (Key point: h^2 f) 3. For u3 = u(1/3,1/3) the equation is A. u3 = (1 + u1 + u4)/4 B. u3 = (100 + u1 + u4)/4 C. u3 = (-1 + u1 + u4)/4 D. u3 = (2 + u1 + u4)/4 Answer: A. Neighbouring boundaries are 0; 9xy = 1 is added since u = (sum + 9xy)/4. (Key point: Equation u3) 4. What is u2? A. 51.0833 B. 76.5417 C. 25.7917 D. 100 Answer: B. a = 51.0833, b = 76.5417, c = 25.7917. 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