A temperature profile and a vibrating string can have the same initial shape, yet they need different data to predict their futures. Write down what a snapshot omits: a string also has velocity. Temperature in the simplest diffusion model needs only its current temperature and boundary information. This distinction comes from the order of the time derivative, not from the appearance of the initial graph.
This original course uses the breadth of MIT 18.152, Fall 2011 and Stanford Math 220, Autumn 2018. These historical syllabi guide topic selection; reading these pages does not establish equivalent university assessment.
Predict
Consider uₓₓ + 6uₓᵧ + 5uᵧᵧ = 0. Before classifying it, identify the coefficient convention: A uₓₓ + 2B uₓᵧ + C uᵧᵧ. Is B equal to 6 or 3? Explain why overlooking the factor of two can change the classification.
Experiment
Here A = 1, B = 3 and C = 5, so B² − AC = 9 − 5 = 4. Change only the mixed coefficient from 6 to 2 and recompute the discriminant.
Principal part
B² − AC
Type at that point
uₓₓ + uᵧᵧ
−1
Elliptic
uₓₓ + 2uₓᵧ + uᵧᵧ
0
Parabolic
uₓₓ − uᵧᵧ
1
Hyperbolic
The table concerns a real second-order equation in two independent variables with a nonzero principal part. Classification in more variables uses the eigenvalue signature of the principal matrix. Coefficients may vary, so an equation can change type across its domain.
Understand
Linearity is a separate question. A linear equation contains u and its derivatives only to first power, with coefficients independent of u. The equation uₜ − uₓₓ = u² is semilinear: its highest derivatives are linear, but superposition fails because of the lower-order square. A quasilinear equation allows highest-derivative coefficients to depend on the unknown or lower derivatives under the stated convention.
A problem includes the equation, domain, data and function class. Well-posedness means existence, uniqueness and continuous dependence in specified norms over a specified time interval. None follows merely from attaching a familiar type label. Boundary data imposed on the wrong part of a transport boundary can overdetermine a problem; incompatible endpoint data can prevent a continuous solution.
Backward heat shows why continuous dependence matters. A forward sine mode decays by e^(−n²t). Reversing time multiplies it by e^(n²t). At a fixed positive time T, initial perturbations e^(−n²T) sin(nx) tend to zero in L² but their backward outputs have fixed size. Thus the inverse evolution is not continuous in that norm, even though each individual mode can be evolved algebraically.
Apply
Write complete problem specifications for a string fixed at both ends and a rod held at zero temperature at both ends. Include the spatial interval, positive constants, initial functions and corner compatibility. For a classical string solution, displacement must match the endpoint values; higher regularity can impose further compatibility on velocity and derivatives. State the desired solution class before declaring that the data are sufficient.
Continue
Next, characteristics make data dependence visible along curves. Keep three questions separate throughout the course: does a formula solve the equation, does it satisfy the data, and is it stable under small changes in those data? A numerical plot alone answers none of these universally.
Apply · explain · revise
Practice and reasoning
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For uₓₓ+4uₓᵧ+3uᵧᵧ=0, use A uₓₓ+2B uₓᵧ+C uᵧᵧ. What is B²−AC?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For a wave equation second order in time, how many independent initial functions normally specify displacement and velocity?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Explain why uₜ−uₓₓ=u² is semilinear rather than linear.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
State the three parts of well-posedness and why a norm must accompany continuous dependence.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Use sin(nx) heat modes to explain instability of backward heat evolution over a fixed positive time T.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For homogeneous Dirichlet heat data on [0,L], explain the endpoint compatibility needed for a solution continuous at t=0 on the closed cylinder.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
