第 11765 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":[2,2,-3,3,1,2],"bounds":[-16,16,-15,17],"source":11,"boundary_data":[["left",[738,-19,2]],["right",[802,13,2]],["bottom",[497,-13,3]],["top",[529,19,3]]]}。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":[2,2,-3,3,1,2],"bounds":[-16,16,-15,17],"source":11,"boundary_data":[["left",[738,-19,2]],["right",[802,13,2]],["bottom",[497,-13,3]],["top",[529,19,3]]]}。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=[2, 2, -3, 3, 1, 2];basis=[1,x,y,x^2,xy,y^2]

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

u_xx constant := mul(2, 3)

u_xx constant = 6

u_yy constant := mul(2, 2)

u_yy constant = 4

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

u_xx=6;u_xy=1;u_yy=4

interior Laplacian := sum(6, 4)

interior Laplacian = 10

interior constant signed residual := sub(10, 11)

interior constant signed residual = -1

u coefficients=[2, 2, -3, 3, 1, 2];constant Laplacian=10;source=11

constant residual=-1;interior polynomial identity=否

ordered bounds=[-16, 16, -15, 17];supplied edge order=[left, right, bottom, top]

all four distinct edges present;preserve original record order

boundary record0 fixed coordinate squared := mul(-16, -16)

boundary record0 fixed coordinate squared = 256

boundary record0 fixed linear term := mul(2, -16)

boundary record0 fixed linear term = -32

boundary record0 fixed square term := mul(3, 256)

boundary record0 fixed square term = 768

boundary record0 constant coefficient := sum(2, -32, 768)

boundary record0 constant coefficient = 738

boundary record0 cross contribution := mul(1, -16)

boundary record0 cross contribution = -16

boundary record0 linear coefficient := sum(-3, -16)

boundary record0 linear coefficient = -19

boundary record0:vary y;fix x=-16;original coefficients=[2, 2, -3, 3, 1, 2]

ascending y slice coefficients=[738, -19, 2]

boundary record0 coefficient residual0 := sub(738, 738)

boundary record0 coefficient residual0 = 0

boundary record0 coefficient residual1 := sub(-19, -19)

boundary record0 coefficient residual1 = 0

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

boundary record0 coefficient residual2 = 0

boundary record0:edge=left;bounds=[-16, 16, -15, 17];remaining variable=y;restriction=[738, -19, 2];supplied=[738, -19, 2]

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

boundary record1 fixed coordinate squared := mul(16, 16)

boundary record1 fixed coordinate squared = 256

boundary record1 fixed linear term := mul(2, 16)

boundary record1 fixed linear term = 32

boundary record1 fixed square term := mul(3, 256)

boundary record1 fixed square term = 768

boundary record1 constant coefficient := sum(2, 32, 768)

boundary record1 constant coefficient = 802

boundary record1 cross contribution := mul(1, 16)

boundary record1 cross contribution = 16

boundary record1 linear coefficient := sum(-3, 16)

boundary record1 linear coefficient = 13

boundary record1:vary y;fix x=16;original coefficients=[2, 2, -3, 3, 1, 2]

ascending y slice coefficients=[802, 13, 2]

boundary record1 coefficient residual0 := sub(802, 802)

boundary record1 coefficient residual0 = 0

boundary record1 coefficient residual1 := sub(13, 13)

boundary record1 coefficient residual1 = 0

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

boundary record1 coefficient residual2 = 0

boundary record1:edge=right;bounds=[-16, 16, -15, 17];remaining variable=y;restriction=[802, 13, 2];supplied=[802, 13, 2]

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

boundary record2 fixed coordinate squared := mul(-15, -15)

boundary record2 fixed coordinate squared = 225

boundary record2 fixed linear term := mul(-3, -15)

boundary record2 fixed linear term = 45

boundary record2 fixed square term := mul(2, 225)

boundary record2 fixed square term = 450

boundary record2 constant coefficient := sum(2, 45, 450)

boundary record2 constant coefficient = 497

boundary record2 cross contribution := mul(1, -15)

boundary record2 cross contribution = -15

boundary record2 linear coefficient := sum(2, -15)

boundary record2 linear coefficient = -13

boundary record2:vary x;fix y=-15;original coefficients=[2, 2, -3, 3, 1, 2]

ascending x slice coefficients=[497, -13, 3]

boundary record2 coefficient residual0 := sub(497, 497)

boundary record2 coefficient residual0 = 0

boundary record2 coefficient residual1 := sub(-13, -13)

boundary record2 coefficient residual1 = 0

boundary record2 coefficient residual2 := sub(3, 3)

boundary record2 coefficient residual2 = 0

boundary record2:edge=bottom;bounds=[-16, 16, -15, 17];remaining variable=x;restriction=[497, -13, 3];supplied=[497, -13, 3]

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

boundary record3 fixed coordinate squared := mul(17, 17)

boundary record3 fixed coordinate squared = 289

boundary record3 fixed linear term := mul(-3, 17)

boundary record3 fixed linear term = -51

boundary record3 fixed square term := mul(2, 289)

boundary record3 fixed square term = 578

boundary record3 constant coefficient := sum(2, -51, 578)

boundary record3 constant coefficient = 529

boundary record3 cross contribution := mul(1, 17)

boundary record3 cross contribution = 17

boundary record3 linear coefficient := sum(2, 17)

boundary record3 linear coefficient = 19

boundary record3:vary x;fix y=17;original coefficients=[2, 2, -3, 3, 1, 2]

ascending x slice coefficients=[529, 19, 3]

boundary record3 coefficient residual0 := sub(529, 529)

boundary record3 coefficient residual0 = 0

boundary record3 coefficient residual1 := sub(19, 19)

boundary record3 coefficient residual1 = 0

boundary record3 coefficient residual2 := sub(3, 3)

boundary record3 coefficient residual2 = 0

boundary record3:edge=top;bounds=[-16, 16, -15, 17];remaining variable=x;restriction=[529, 19, 3];supplied=[529, 19, 3]

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

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

all boundary identities=是;joint certificate=否;state=boundary_pass_interior_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

([2, 2, -3, 3, 1, 2], 10, 11, 否, [[left, y, [738, -19, 2], [738, -19, 2], 是], [right, y, [802, 13, 2], [802, 13, 2], 是], [bottom, x, [497, -13, 3], [497, -13, 3], 是], [top, x, [529, 19, 3], [529, 19, 3], 是]], 是, 否, 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。