第 11791 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-obligation-separation；資料={"coefficients":[3,-2,-2,4,-1,-2],"bounds":[-17,17,-16,18],"source":4,"boundary_data":[["left",[1194,15,-2]],["right",[1125,-19,-2]],["bottom",[-477,14,4]],["top",[-681,-20,4]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。

• G001 — 內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={"coefficients":[3,-2,-2,4,-1,-2],"bounds":[-17,17,-16,18],"source":4,"boundary_data":[["left",[1194,15,-2]],["right",[1125,-19,-2]],["bottom",[-477,14,4]],["top",[-681,-20,4]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 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, -2, 4, -1, -2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(3)+(-2)*x+(-2)*y+(4)*x^2+(-1)*x*y+(-2)*y^2

u_xx constant := mul(2, 4)

u_xx constant = 8

u_yy constant := mul(2, -2)

u_yy constant = -4

u coefficients=[3, -2, -2, 4, -1, -2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y

u_xx=8;u_xy=-1;u_yy=-4

interior Laplacian := sum(8, -4)

interior Laplacian = 4

interior constant signed residual := sub(4, 4)

interior constant signed residual = 0

u coefficients=[3, -2, -2, 4, -1, -2];constant Laplacian=4;source=4

constant residual=0;interior polynomial identity=是

ordered bounds=[-17, 17, -16, 18];supplied edge order=[left, right, bottom, top]

all four distinct edges present;preserve original record order

boundary record0 fixed coordinate squared := mul(-17, -17)

boundary record0 fixed coordinate squared = 289

boundary record0 fixed linear term := mul(-2, -17)

boundary record0 fixed linear term = 34

boundary record0 fixed square term := mul(4, 289)

boundary record0 fixed square term = 1156

boundary record0 constant coefficient := sum(3, 34, 1156)

boundary record0 constant coefficient = 1193

boundary record0 cross contribution := mul(-1, -17)

boundary record0 cross contribution = 17

boundary record0 linear coefficient := sum(-2, 17)

boundary record0 linear coefficient = 15

boundary record0:vary y;fix x=-17;original coefficients=[3, -2, -2, 4, -1, -2]

ascending y slice coefficients=[1193, 15, -2]

boundary record0 coefficient residual0 := sub(1193, 1194)

boundary record0 coefficient residual0 = -1

boundary record0 coefficient residual1 := sub(15, 15)

boundary record0 coefficient residual1 = 0

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

boundary record0 coefficient residual2 = 0

boundary record0:edge=left;bounds=[-17, 17, -16, 18];remaining variable=y;restriction=[1193, 15, -2];supplied=[1194, 15, -2]

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

boundary record1 fixed coordinate squared := mul(17, 17)

boundary record1 fixed coordinate squared = 289

boundary record1 fixed linear term := mul(-2, 17)

boundary record1 fixed linear term = -34

boundary record1 fixed square term := mul(4, 289)

boundary record1 fixed square term = 1156

boundary record1 constant coefficient := sum(3, -34, 1156)

boundary record1 constant coefficient = 1125

boundary record1 cross contribution := mul(-1, 17)

boundary record1 cross contribution = -17

boundary record1 linear coefficient := sum(-2, -17)

boundary record1 linear coefficient = -19

boundary record1:vary y;fix x=17;original coefficients=[3, -2, -2, 4, -1, -2]

ascending y slice coefficients=[1125, -19, -2]

boundary record1 coefficient residual0 := sub(1125, 1125)

boundary record1 coefficient residual0 = 0

boundary record1 coefficient residual1 := sub(-19, -19)

boundary record1 coefficient residual1 = 0

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

boundary record1 coefficient residual2 = 0

boundary record1:edge=right;bounds=[-17, 17, -16, 18];remaining variable=y;restriction=[1125, -19, -2];supplied=[1125, -19, -2]

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

boundary record2 fixed coordinate squared := mul(-16, -16)

boundary record2 fixed coordinate squared = 256

boundary record2 fixed linear term := mul(-2, -16)

boundary record2 fixed linear term = 32

boundary record2 fixed square term := mul(-2, 256)

boundary record2 fixed square term = -512

boundary record2 constant coefficient := sum(3, 32, -512)

boundary record2 constant coefficient = -477

boundary record2 cross contribution := mul(-1, -16)

boundary record2 cross contribution = 16

boundary record2 linear coefficient := sum(-2, 16)

boundary record2 linear coefficient = 14

boundary record2:vary x;fix y=-16;original coefficients=[3, -2, -2, 4, -1, -2]

ascending x slice coefficients=[-477, 14, 4]

boundary record2 coefficient residual0 := sub(-477, -477)

boundary record2 coefficient residual0 = 0

boundary record2 coefficient residual1 := sub(14, 14)

boundary record2 coefficient residual1 = 0

boundary record2 coefficient residual2 := sub(4, 4)

boundary record2 coefficient residual2 = 0

boundary record2:edge=bottom;bounds=[-17, 17, -16, 18];remaining variable=x;restriction=[-477, 14, 4];supplied=[-477, 14, 4]

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

boundary record3 fixed coordinate squared := mul(18, 18)

boundary record3 fixed coordinate squared = 324

boundary record3 fixed linear term := mul(-2, 18)

boundary record3 fixed linear term = -36

boundary record3 fixed square term := mul(-2, 324)

boundary record3 fixed square term = -648

boundary record3 constant coefficient := sum(3, -36, -648)

boundary record3 constant coefficient = -681

boundary record3 cross contribution := mul(-1, 18)

boundary record3 cross contribution = -18

boundary record3 linear coefficient := sum(-2, -18)

boundary record3 linear coefficient = -20

boundary record3:vary x;fix y=18;original coefficients=[3, -2, -2, 4, -1, -2]

ascending x slice coefficients=[-681, -20, 4]

boundary record3 coefficient residual0 := sub(-681, -681)

boundary record3 coefficient residual0 = 0

boundary record3 coefficient residual1 := sub(-20, -20)

boundary record3 coefficient residual1 = 0

boundary record3 coefficient residual2 := sub(4, 4)

boundary record3 coefficient residual2 = 0

boundary record3:edge=top;bounds=[-17, 17, -16, 18];remaining variable=x;restriction=[-681, -20, 4];supplied=[-681, -20, 4]

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

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

all boundary identities=否;joint certificate=否;state=interior_pass_boundary_fail

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

(4, 4, 是, [[left, y, [1193, 15, -2], [1194, 15, -2], 否], [right, y, [1125, -19, -2], [1125, -19, -2], 是], [bottom, x, [-477, 14, 4], [-477, 14, 4], 是], [top, x, [-681, -20, 4], [-681, -20, 4], 是]], 否, interior_pass_boundary_fail)

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。