Problem 1
A nonconstant harmonic function reaches its extrema on the boundary, not strictly inside.
Faculty use
Laplace Equation and Numerical PDE Bridge
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Grid-point differential equations become a sparse linear system for unknown field values.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind connecting steady-state fields to harmonic functions and discretization. Use the formulas u_xx+u_yy=0; five-point stencil u_ij≈average of four neighbors.
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to heat conduction, vibrating strings, structural modes, diffusion, electrostatics, acoustics, and field simulation, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.