Problem 1
Equation, parameters, initial conditions, interval, method, tolerances, software/version, outputs, error evidence, and code or pseudocode.
Faculty use
Reproducible Numerical Experiments
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
An unstable or inaccurate computation can look plausible; refinement, residuals, invariants, and independent comparisons are required.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind documenting solver, tolerance, initial data, units, diagnostics, and independent checks. Use the formulas report method + tolerance + step statistics; compare invariants/residuals/reference solutions.
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 simulation, control, orbital motion, chemical kinetics, heat transfer, and computational engineering, 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.