{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-boundary-value-problem"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-boundary-value-problem-01","title":{"zh-Hant":"邊界值問題：矩形上 Poisson–Dirichlet 問題的內部與四邊分開核對","en":"Boundary-value problem: checking the interior and all four edges separately for a Poisson–Dirichlet problem on a rectangle"},"summary":{"zh-Hant":"在矩形 Ω=(0,2)×(0,1) 上宣告 BVP：內部 Δu=u_xx+u_yy=4；Dirichlet 資料為左邊 u(0,y)=y²、右邊 u(2,y)=y²+2y+4（0≤y≤1）、下邊 u(x,0)=x²、上邊 u(x,1)=x²+x+1（0≤x≤2）。對二次多項式候選 u=x²+xy+y² 與 v=2x²+xy−2x+y，分別記錄內部 Poisson 義務（符號 Laplacian 恆等式）與四條邊的 Dirichlet 義務（精確限制到整條邊後逐係數比較），判定誰滿足此宣告的 BVP，並說明 v 為何在內部方程成立、四個角點也都吻合的情況下仍不滿足。","en":"On the rectangle Ω=(0,2)×(0,1) declare the BVP: in the interior Δu=u_xx+u_yy=4; Dirichlet data u(0,y)=y² on the left, u(2,y)=y²+2y+4 on the right (0≤y≤1), u(x,0)=x² on the bottom and u(x,1)=x²+x+1 on the top (0≤x≤2). For the quadratic polynomial candidates u=x²+xy+y² and v=2x²+xy−2x+y, record separately the interior Poisson obligation (a symbolic Laplacian identity) and the Dirichlet obligation on all four edges (exact restriction to each whole edge, then coefficient comparison); decide which candidate satisfies this declared BVP, and explain why v fails although its interior equation holds and it matches the data at all four corners."},"mko_ids":["mko-boundary-value-problem"]}],"title":"概念推導導讀"}