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The class of LINEAR SECOND-ORDER PDEs (parabolic, elliptic, hyperbolic) appearing in heat transfer & fluid flow classifies as:
A{'text': 'Unrelated to physics', 'label': 'A'}
B{'text': 'Heat conduction in time = PARABOLIC; steady conduction = ELLIPTIC (Laplace); wave equation = HYPERBOLIC — driven by the discriminant of the highest-order terms', 'label': 'B'}
C{'text': 'All are linear', 'label': 'C'}
D{'text': 'All are nonlinear', 'label': 'D'}
Answer & Solution
Correct answer: B. {'text': 'Heat conduction in time = PARABOLIC; steady conduction = ELLIPTIC (Laplace); wave equation = HYPERBOLIC — driven by the discriminant of the highest-order terms', 'label': 'B'}
Classification by discriminant: $B^2 - 4AC$. Parabolic (=0, e.g. heat eqn) needs IC + boundary in space. Elliptic (<0, Laplace) needs boundary only. Hyperbolic (>0, wave) needs IC + boundary. Numerical methods differ for each.
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