第 11781 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":[-7,-2,2,2,1,-2],"bounds":[-7,7,-6,8],"source":0,"boundary_data":[["left",[106,-5,-2]],["right",[77,9,-2]],["bottom",[-91,-8,2]],["top",[-119,6,2]]]}。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":[-7,-2,2,2,1,-2],"bounds":[-7,7,-6,8],"source":0,"boundary_data":[["left",[106,-5,-2]],["right",[77,9,-2]],["bottom",[-91,-8,2]],["top",[-119,6,2]]]}。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=[-7, -2, 2, 2, 1, -2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-7)+(-2)*x+(2)*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=[-7, -2, 2, 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 = 0

interior constant signed residual := sub(0, 0)

interior constant signed residual = 0

u coefficients=[-7, -2, 2, 2, 1, -2];constant Laplacian=0;source=0

constant residual=0;interior polynomial identity=是

ordered bounds=[-7, 7, -6, 8];supplied edge order=[left, right, bottom, top]

all four distinct edges present;preserve original record order

boundary record0 fixed coordinate squared := mul(-7, -7)

boundary record0 fixed coordinate squared = 49

boundary record0 fixed linear term := mul(-2, -7)

boundary record0 fixed linear term = 14

boundary record0 fixed square term := mul(2, 49)

boundary record0 fixed square term = 98

boundary record0 constant coefficient := sum(-7, 14, 98)

boundary record0 constant coefficient = 105

boundary record0 cross contribution := mul(1, -7)

boundary record0 cross contribution = -7

boundary record0 linear coefficient := sum(2, -7)

boundary record0 linear coefficient = -5

boundary record0:vary y;fix x=-7;original coefficients=[-7, -2, 2, 2, 1, -2]

ascending y slice coefficients=[105, -5, -2]

boundary record0 coefficient residual0 := sub(105, 106)

boundary record0 coefficient residual0 = -1

boundary record0 coefficient residual1 := sub(-5, -5)

boundary record0 coefficient residual1 = 0

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

boundary record0 coefficient residual2 = 0

boundary record0:edge=left;bounds=[-7, 7, -6, 8];remaining variable=y;restriction=[105, -5, -2];supplied=[106, -5, -2]

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

boundary record1 fixed coordinate squared := mul(7, 7)

boundary record1 fixed coordinate squared = 49

boundary record1 fixed linear term := mul(-2, 7)

boundary record1 fixed linear term = -14

boundary record1 fixed square term := mul(2, 49)

boundary record1 fixed square term = 98

boundary record1 constant coefficient := sum(-7, -14, 98)

boundary record1 constant coefficient = 77

boundary record1 cross contribution := mul(1, 7)

boundary record1 cross contribution = 7

boundary record1 linear coefficient := sum(2, 7)

boundary record1 linear coefficient = 9

boundary record1:vary y;fix x=7;original coefficients=[-7, -2, 2, 2, 1, -2]

ascending y slice coefficients=[77, 9, -2]

boundary record1 coefficient residual0 := sub(77, 77)

boundary record1 coefficient residual0 = 0

boundary record1 coefficient residual1 := sub(9, 9)

boundary record1 coefficient residual1 = 0

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

boundary record1 coefficient residual2 = 0

boundary record1:edge=right;bounds=[-7, 7, -6, 8];remaining variable=y;restriction=[77, 9, -2];supplied=[77, 9, -2]

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

boundary record2 fixed coordinate squared := mul(-6, -6)

boundary record2 fixed coordinate squared = 36

boundary record2 fixed linear term := mul(2, -6)

boundary record2 fixed linear term = -12

boundary record2 fixed square term := mul(-2, 36)

boundary record2 fixed square term = -72

boundary record2 constant coefficient := sum(-7, -12, -72)

boundary record2 constant coefficient = -91

boundary record2 cross contribution := mul(1, -6)

boundary record2 cross contribution = -6

boundary record2 linear coefficient := sum(-2, -6)

boundary record2 linear coefficient = -8

boundary record2:vary x;fix y=-6;original coefficients=[-7, -2, 2, 2, 1, -2]

ascending x slice coefficients=[-91, -8, 2]

boundary record2 coefficient residual0 := sub(-91, -91)

boundary record2 coefficient residual0 = 0

boundary record2 coefficient residual1 := sub(-8, -8)

boundary record2 coefficient residual1 = 0

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

boundary record2 coefficient residual2 = 0

boundary record2:edge=bottom;bounds=[-7, 7, -6, 8];remaining variable=x;restriction=[-91, -8, 2];supplied=[-91, -8, 2]

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

boundary record3 fixed coordinate squared := mul(8, 8)

boundary record3 fixed coordinate squared = 64

boundary record3 fixed linear term := mul(2, 8)

boundary record3 fixed linear term = 16

boundary record3 fixed square term := mul(-2, 64)

boundary record3 fixed square term = -128

boundary record3 constant coefficient := sum(-7, 16, -128)

boundary record3 constant coefficient = -119

boundary record3 cross contribution := mul(1, 8)

boundary record3 cross contribution = 8

boundary record3 linear coefficient := sum(-2, 8)

boundary record3 linear coefficient = 6

boundary record3:vary x;fix y=8;original coefficients=[-7, -2, 2, 2, 1, -2]

ascending x slice coefficients=[-119, 6, 2]

boundary record3 coefficient residual0 := sub(-119, -119)

boundary record3 coefficient residual0 = 0

boundary record3 coefficient residual1 := sub(6, 6)

boundary record3 coefficient residual1 = 0

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

boundary record3 coefficient residual2 = 0

boundary record3:edge=top;bounds=[-7, 7, -6, 8];remaining variable=x;restriction=[-119, 6, 2];supplied=[-119, 6, 2]

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

(0, 0, 是, [[left, y, [105, -5, -2], [106, -5, -2], 否], [right, y, [77, 9, -2], [77, 9, -2], 是], [bottom, x, [-91, -8, 2], [-91, -8, 2], 是], [top, x, [-119, 6, 2], [-119, 6, 2], 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。