Question 11633: A Symbolic Heat Residual Identity, Not a Point Sample

Original question and givens

Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-5, 0, 16, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].

• G001 — Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-5, 0, 16, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-partial-differential-equation

Step-by-step derivation

original coefficients=[-5, 0, 16, 2, 0, 0];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-5)+(0)*x+(16)*y+(2)*x^2+(0)*x*y+(0)*y^2

u_xx constant := mul(2, 2)

u_xx constant = 4

u_yy constant := mul(2, 0)

u_yy constant = 0

u coefficients=[-5, 0, 16, 2, 0, 0];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y

u_xx=4;u_xy=0;u_yy=0

heat diffusivity times u_xx := mul(4, 4)

heat diffusivity times u_xx = 16

heat constant residual := sub(16, 16)

heat constant residual = 0

u_y=(16)+(0)*x+(0)*y;diffusivity*u_xx=16

residual coefficients in [1,x,y]=[0, 0, 0];identically zero=Yes

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute heat constant residual with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute heat diffusivity times u_xx with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute u_yy constant with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute u_xx constant with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([-5, 0, 16, 2, 0, 0], 4, [0, 0, 0], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.