第 11760 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-joint-candidate-audit；資料={"coefficients":[-3,2,-1,2,-1,2],"bounds":[-11,11,-10,12],"source":8,"boundary_data":[["left",[217,10,2]],["right",[261,-12,2]],["bottom",[207,12,2]],["top",[273,-10,2]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。

• G001 — 內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={"coefficients":[-3,2,-1,2,-1,2],"bounds":[-11,11,-10,12],"source":8,"boundary_data":[["left",[217,10,2]],["right",[261,-12,2]],["bottom",[207,12,2]],["top",[273,-10,2]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,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=[-3, 2, -1, 2, -1, 2];basis=[1,x,y,x^2,xy,y^2]

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

u_xx constant := mul(2, 2)

u_xx constant = 4

u_yy constant := mul(2, 2)

u_yy constant = 4

u coefficients=[-3, 2, -1, 2, -1, 2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y

u_xx=4;u_xy=-1;u_yy=4

interior Laplacian := sum(4, 4)

interior Laplacian = 8

interior constant signed residual := sub(8, 8)

interior constant signed residual = 0

u coefficients=[-3, 2, -1, 2, -1, 2];constant Laplacian=8;source=8

constant residual=0;interior polynomial identity=是

ordered bounds=[-11, 11, -10, 12];supplied edge order=[left, right, bottom, top]

all four distinct edges present;preserve original record order

boundary record0 fixed coordinate squared := mul(-11, -11)

boundary record0 fixed coordinate squared = 121

boundary record0 fixed linear term := mul(2, -11)

boundary record0 fixed linear term = -22

boundary record0 fixed square term := mul(2, 121)

boundary record0 fixed square term = 242

boundary record0 constant coefficient := sum(-3, -22, 242)

boundary record0 constant coefficient = 217

boundary record0 cross contribution := mul(-1, -11)

boundary record0 cross contribution = 11

boundary record0 linear coefficient := sum(-1, 11)

boundary record0 linear coefficient = 10

boundary record0:vary y;fix x=-11;original coefficients=[-3, 2, -1, 2, -1, 2]

ascending y slice coefficients=[217, 10, 2]

boundary record0 coefficient residual0 := sub(217, 217)

boundary record0 coefficient residual0 = 0

boundary record0 coefficient residual1 := sub(10, 10)

boundary record0 coefficient residual1 = 0

boundary record0 coefficient residual2 := sub(2, 2)

boundary record0 coefficient residual2 = 0

boundary record0:edge=left;bounds=[-11, 11, -10, 12];remaining variable=y;restriction=[217, 10, 2];supplied=[217, 10, 2]

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

boundary record1 fixed coordinate squared := mul(11, 11)

boundary record1 fixed coordinate squared = 121

boundary record1 fixed linear term := mul(2, 11)

boundary record1 fixed linear term = 22

boundary record1 fixed square term := mul(2, 121)

boundary record1 fixed square term = 242

boundary record1 constant coefficient := sum(-3, 22, 242)

boundary record1 constant coefficient = 261

boundary record1 cross contribution := mul(-1, 11)

boundary record1 cross contribution = -11

boundary record1 linear coefficient := sum(-1, -11)

boundary record1 linear coefficient = -12

boundary record1:vary y;fix x=11;original coefficients=[-3, 2, -1, 2, -1, 2]

ascending y slice coefficients=[261, -12, 2]

boundary record1 coefficient residual0 := sub(261, 261)

boundary record1 coefficient residual0 = 0

boundary record1 coefficient residual1 := sub(-12, -12)

boundary record1 coefficient residual1 = 0

boundary record1 coefficient residual2 := sub(2, 2)

boundary record1 coefficient residual2 = 0

boundary record1:edge=right;bounds=[-11, 11, -10, 12];remaining variable=y;restriction=[261, -12, 2];supplied=[261, -12, 2]

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

boundary record2 fixed coordinate squared := mul(-10, -10)

boundary record2 fixed coordinate squared = 100

boundary record2 fixed linear term := mul(-1, -10)

boundary record2 fixed linear term = 10

boundary record2 fixed square term := mul(2, 100)

boundary record2 fixed square term = 200

boundary record2 constant coefficient := sum(-3, 10, 200)

boundary record2 constant coefficient = 207

boundary record2 cross contribution := mul(-1, -10)

boundary record2 cross contribution = 10

boundary record2 linear coefficient := sum(2, 10)

boundary record2 linear coefficient = 12

boundary record2:vary x;fix y=-10;original coefficients=[-3, 2, -1, 2, -1, 2]

ascending x slice coefficients=[207, 12, 2]

boundary record2 coefficient residual0 := sub(207, 207)

boundary record2 coefficient residual0 = 0

boundary record2 coefficient residual1 := sub(12, 12)

boundary record2 coefficient residual1 = 0

boundary record2 coefficient residual2 := sub(2, 2)

boundary record2 coefficient residual2 = 0

boundary record2:edge=bottom;bounds=[-11, 11, -10, 12];remaining variable=x;restriction=[207, 12, 2];supplied=[207, 12, 2]

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

boundary record3 fixed coordinate squared := mul(12, 12)

boundary record3 fixed coordinate squared = 144

boundary record3 fixed linear term := mul(-1, 12)

boundary record3 fixed linear term = -12

boundary record3 fixed square term := mul(2, 144)

boundary record3 fixed square term = 288

boundary record3 constant coefficient := sum(-3, -12, 288)

boundary record3 constant coefficient = 273

boundary record3 cross contribution := mul(-1, 12)

boundary record3 cross contribution = -12

boundary record3 linear coefficient := sum(2, -12)

boundary record3 linear coefficient = -10

boundary record3:vary x;fix y=12;original coefficients=[-3, 2, -1, 2, -1, 2]

ascending x slice coefficients=[273, -10, 2]

boundary record3 coefficient residual0 := sub(273, 273)

boundary record3 coefficient residual0 = 0

boundary record3 coefficient residual1 := sub(-10, -10)

boundary record3 coefficient residual1 = 0

boundary record3 coefficient residual2 := sub(2, 2)

boundary record3 coefficient residual2 = 0

boundary record3:edge=top;bounds=[-11, 11, -10, 12];remaining variable=x;restriction=[273, -10, 2];supplied=[273, -10, 2]

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

interior identity=是;boundary row flags=[是, 是, 是, 是]

all boundary identities=是;joint certificate=是;state=both_pass

逐步反向核對

• S053 · 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。
• S052 · 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。
• S051 · premise_and_result_audit — 以精確有理數重算 boundary record3 coefficient residual2，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 boundary record3 coefficient residual1，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 boundary record3 coefficient residual0，不以小數近似替代。
• S048 · 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。
• S047 · premise_and_result_audit — 以精確有理數重算 boundary record3 linear coefficient，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 boundary record3 cross contribution，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 boundary record3 constant coefficient，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 boundary record3 fixed square term，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 boundary record3 fixed linear term，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 boundary record3 fixed coordinate squared，不以小數近似替代。
• S041 · 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。
• S040 · premise_and_result_audit — 以精確有理數重算 boundary record2 coefficient residual2，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 boundary record2 coefficient residual1，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 boundary record2 coefficient residual0，不以小數近似替代。
• S037 · 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。
• S036 · premise_and_result_audit — 以精確有理數重算 boundary record2 linear coefficient，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 boundary record2 cross contribution，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 boundary record2 constant coefficient，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 boundary record2 fixed square term，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 boundary record2 fixed linear term，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 boundary record2 fixed coordinate squared，不以小數近似替代。
• S030 · 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。
• S029 · premise_and_result_audit — 以精確有理數重算 boundary record1 coefficient residual2，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 boundary record1 coefficient residual1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 boundary record1 coefficient residual0，不以小數近似替代。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 boundary record1 linear coefficient，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 boundary record1 cross contribution，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 boundary record1 constant coefficient，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 boundary record1 fixed square term，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 boundary record1 fixed linear term，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 boundary record1 fixed coordinate squared，不以小數近似替代。
• S019 · 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。
• S018 · premise_and_result_audit — 以精確有理數重算 boundary record0 coefficient residual2，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 boundary record0 coefficient residual1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 boundary record0 coefficient residual0，不以小數近似替代。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 boundary record0 linear coefficient，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 boundary record0 cross contribution，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 boundary record0 constant coefficient，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 boundary record0 fixed square term，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 boundary record0 fixed linear term，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 boundary record0 fixed coordinate squared，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 interior constant signed residual，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 interior Laplacian，不以小數近似替代。
• S004 · 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。
• S003 · premise_and_result_audit — 以精確有理數重算 u_yy constant，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 u_xx constant，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

([-3, 2, -1, 2, -1, 2], 8, 8, 是, [[left, y, [217, 10, 2], [217, 10, 2], 是], [right, y, [261, -12, 2], [261, -12, 2], 是], [bottom, x, [207, 12, 2], [207, 12, 2], 是], [top, x, [273, -10, 2], [273, -10, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_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。