第 11771 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":[8,-2,3,1,1,-2],"bounds":[-22,22,-21,23],"source":-2,"boundary_data":[["left",[536,-19,-2]],["right",[448,25,-2]],["bottom",[-937,-23,1]],["top",[-981,21,1]]]}。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":[8,-2,3,1,1,-2],"bounds":[-22,22,-21,23],"source":-2,"boundary_data":[["left",[536,-19,-2]],["right",[448,25,-2]],["bottom",[-937,-23,1]],["top",[-981,21,1]]]}。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=[8, -2, 3, 1, 1, -2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(8)+(-2)*x+(3)*y+(1)*x^2+(1)*x*y+(-2)*y^2

u_xx constant := mul(2, 1)

u_xx constant = 2

u_yy constant := mul(2, -2)

u_yy constant = -4

u coefficients=[8, -2, 3, 1, 1, -2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y

u_xx=2;u_xy=1;u_yy=-4

interior Laplacian := sum(2, -4)

interior Laplacian = -2

interior constant signed residual := sub(-2, -2)

interior constant signed residual = 0

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

constant residual=0;interior polynomial identity=是

ordered bounds=[-22, 22, -21, 23];supplied edge order=[left, right, bottom, top]

all four distinct edges present;preserve original record order

boundary record0 fixed coordinate squared := mul(-22, -22)

boundary record0 fixed coordinate squared = 484

boundary record0 fixed linear term := mul(-2, -22)

boundary record0 fixed linear term = 44

boundary record0 fixed square term := mul(1, 484)

boundary record0 fixed square term = 484

boundary record0 constant coefficient := sum(8, 44, 484)

boundary record0 constant coefficient = 536

boundary record0 cross contribution := mul(1, -22)

boundary record0 cross contribution = -22

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

boundary record0 linear coefficient = -19

boundary record0:vary y;fix x=-22;original coefficients=[8, -2, 3, 1, 1, -2]

ascending y slice coefficients=[536, -19, -2]

boundary record0 coefficient residual0 := sub(536, 536)

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=[-22, 22, -21, 23];remaining variable=y;restriction=[536, -19, -2];supplied=[536, -19, -2]

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

boundary record1 fixed coordinate squared := mul(22, 22)

boundary record1 fixed coordinate squared = 484

boundary record1 fixed linear term := mul(-2, 22)

boundary record1 fixed linear term = -44

boundary record1 fixed square term := mul(1, 484)

boundary record1 fixed square term = 484

boundary record1 constant coefficient := sum(8, -44, 484)

boundary record1 constant coefficient = 448

boundary record1 cross contribution := mul(1, 22)

boundary record1 cross contribution = 22

boundary record1 linear coefficient := sum(3, 22)

boundary record1 linear coefficient = 25

boundary record1:vary y;fix x=22;original coefficients=[8, -2, 3, 1, 1, -2]

ascending y slice coefficients=[448, 25, -2]

boundary record1 coefficient residual0 := sub(448, 448)

boundary record1 coefficient residual0 = 0

boundary record1 coefficient residual1 := sub(25, 25)

boundary record1 coefficient residual1 = 0

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

boundary record1 coefficient residual2 = 0

boundary record1:edge=right;bounds=[-22, 22, -21, 23];remaining variable=y;restriction=[448, 25, -2];supplied=[448, 25, -2]

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

boundary record2 fixed coordinate squared := mul(-21, -21)

boundary record2 fixed coordinate squared = 441

boundary record2 fixed linear term := mul(3, -21)

boundary record2 fixed linear term = -63

boundary record2 fixed square term := mul(-2, 441)

boundary record2 fixed square term = -882

boundary record2 constant coefficient := sum(8, -63, -882)

boundary record2 constant coefficient = -937

boundary record2 cross contribution := mul(1, -21)

boundary record2 cross contribution = -21

boundary record2 linear coefficient := sum(-2, -21)

boundary record2 linear coefficient = -23

boundary record2:vary x;fix y=-21;original coefficients=[8, -2, 3, 1, 1, -2]

ascending x slice coefficients=[-937, -23, 1]

boundary record2 coefficient residual0 := sub(-937, -937)

boundary record2 coefficient residual0 = 0

boundary record2 coefficient residual1 := sub(-23, -23)

boundary record2 coefficient residual1 = 0

boundary record2 coefficient residual2 := sub(1, 1)

boundary record2 coefficient residual2 = 0

boundary record2:edge=bottom;bounds=[-22, 22, -21, 23];remaining variable=x;restriction=[-937, -23, 1];supplied=[-937, -23, 1]

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

boundary record3 fixed coordinate squared := mul(23, 23)

boundary record3 fixed coordinate squared = 529

boundary record3 fixed linear term := mul(3, 23)

boundary record3 fixed linear term = 69

boundary record3 fixed square term := mul(-2, 529)

boundary record3 fixed square term = -1058

boundary record3 constant coefficient := sum(8, 69, -1058)

boundary record3 constant coefficient = -981

boundary record3 cross contribution := mul(1, 23)

boundary record3 cross contribution = 23

boundary record3 linear coefficient := sum(-2, 23)

boundary record3 linear coefficient = 21

boundary record3:vary x;fix y=23;original coefficients=[8, -2, 3, 1, 1, -2]

ascending x slice coefficients=[-981, 21, 1]

boundary record3 coefficient residual0 := sub(-981, -981)

boundary record3 coefficient residual0 = 0

boundary record3 coefficient residual1 := sub(21, 21)

boundary record3 coefficient residual1 = 0

boundary record3 coefficient residual2 := sub(1, 1)

boundary record3 coefficient residual2 = 0

boundary record3:edge=top;bounds=[-22, 22, -21, 23];remaining variable=x;restriction=[-981, 21, 1];supplied=[-981, 21, 1]

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

([8, -2, 3, 1, 1, -2], -2, -2, 是, [[left, y, [536, -19, -2], [536, -19, -2], 是], [right, y, [448, 25, -2], [448, 25, -2], 是], [bottom, x, [-937, -23, 1], [-937, -23, 1], 是], [top, x, [-981, 21, 1], [-981, 21, 1], 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。