第 11701 題：邊界值問題

原題與已知條件

內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={"coefficients":[-12,-2,-3,1,-1,-2],"bounds":[-2,2,-1,3],"edge":"left","boundary_coefficients":[-3,-1,-2]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。

• G001 — 內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={"coefficients":[-12,-2,-3,1,-1,-2],"bounds":[-2,2,-1,3],"edge":"left","boundary_coefficients":[-3,-1,-2]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-boundary-value-problem

逐步推導

original coefficients=[-12, -2, -3, 1, -1, -2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-12)+(-2)*x+(-3)*y+(1)*x^2+(-1)*x*y+(-2)*y^2

selected boundary fixed coordinate squared := mul(-2, -2)

selected boundary fixed coordinate squared = 4

selected boundary fixed linear term := mul(-2, -2)

selected boundary fixed linear term = 4

selected boundary fixed square term := mul(1, 4)

selected boundary fixed square term = 4

selected boundary constant coefficient := sum(-12, 4, 4)

selected boundary constant coefficient = -4

selected boundary cross contribution := mul(-1, -2)

selected boundary cross contribution = 2

selected boundary linear coefficient := sum(-3, 2)

selected boundary linear coefficient = -1

selected boundary:vary y;fix x=-2;original coefficients=[-12, -2, -3, 1, -1, -2]

ascending y slice coefficients=[-4, -1, -2]

selected boundary coefficient residual0 := sub(-4, -3)

selected boundary coefficient residual0 = -1

selected boundary coefficient residual1 := sub(-1, -1)

selected boundary coefficient residual1 = 0

selected boundary coefficient residual2 := sub(-2, -2)

selected boundary coefficient residual2 = 0

selected boundary:edge=left;bounds=[-2, 2, -1, 3];remaining variable=y;restriction=[-4, -1, -2];supplied=[-3, -1, -2]

all coefficient residuals=[-1, 0, 0];polynomial identity=否

逐步反向核對

• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 selected boundary coefficient residual2，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 selected boundary coefficient residual1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 selected boundary coefficient residual0，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 selected boundary linear coefficient，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 selected boundary cross contribution，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 selected boundary constant coefficient，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 selected boundary fixed square term，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 selected boundary fixed linear term，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 selected boundary fixed coordinate squared，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([-2, 2, -1, 3], left, y, [-4, -1, -2], [-3, -1, -2], 否, declared_edge_only)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。