第 11709 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":[-4,1,-2,1,1,1],"bounds":[-10,10,-9,11],"edge":"left","boundary_coefficients":[86,-12,1]}。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":[-4,1,-2,1,1,1],"bounds":[-10,10,-9,11],"edge":"left","boundary_coefficients":[86,-12,1]}。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=[-4, 1, -2, 1, 1, 1];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-4)+(1)*x+(-2)*y+(1)*x^2+(1)*x*y+(1)*y^2

selected boundary fixed coordinate squared := mul(-10, -10)

selected boundary fixed coordinate squared = 100

selected boundary fixed linear term := mul(1, -10)

selected boundary fixed linear term = -10

selected boundary fixed square term := mul(1, 100)

selected boundary fixed square term = 100

selected boundary constant coefficient := sum(-4, -10, 100)

selected boundary constant coefficient = 86

selected boundary cross contribution := mul(1, -10)

selected boundary cross contribution = -10

selected boundary linear coefficient := sum(-2, -10)

selected boundary linear coefficient = -12

selected boundary:vary y;fix x=-10;original coefficients=[-4, 1, -2, 1, 1, 1]

ascending y slice coefficients=[86, -12, 1]

selected boundary coefficient residual0 := sub(86, 86)

selected boundary coefficient residual0 = 0

selected boundary coefficient residual1 := sub(-12, -12)

selected boundary coefficient residual1 = 0

selected boundary coefficient residual2 := sub(1, 1)

selected boundary coefficient residual2 = 0

selected boundary:edge=left;bounds=[-10, 10, -9, 11];remaining variable=y;restriction=[86, -12, 1];supplied=[86, -12, 1]

all coefficient residuals=[0, 0, 0];polynomial identity=是

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

([-10, 10, -9, 11], left, y, [86, -12, 1], [86, -12, 1], 是, 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。