Math · 13
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Solve \(u_t=u_{xx}\) for \(0
Timer not started. Twenty minutes of real work before a solution.
Separate: \(X''+\lambda X=0\), \(X(0)=X(\pi)=0\), so \(\lambda=n^2\), \(X=\sin nx\). Time part \(T'= -n^2 T\). The initial condition is already a single mode, so \(u(x,t)=e^{-t}\sin x\).
Attack 2
For the wave equation \(u_{tt}=c^2 u_{xx}\) on the line, state d’Alembert’s formula and the domain of dependence of a point \((x,t)\).
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\(u(x,t)=\frac{1}{2}(f(x+ct)+f(x-ct))+\frac{1}{2c}\int_{x-ct}^{x+ct} g\), for initial position \(f\) and velocity \(g\). The value at \((x,t)\) depends only on the initial data in \([x-ct,x+ct]\).
Attack 3
Why does a maximum principle for the heat equation say a hot spot in the interior cannot appear later if it was not on the boundary or in the initial data?
Timer not started. Twenty minutes of real work before a solution.
For the heat equation on a bounded interval, the maximum of a solution on a closed time-space rectangle is attained on the parabolic boundary: the initial segment and the lateral sides. An interior maximum at a positive time would force, by the equation and the second-derivative test, a contradiction unless the solution is constant. So a new interior hot spot cannot be born.
Partial differential equations ask for functions of several variables whose partial derivatives satisfy a relation. The introductory trinity is the Laplace equation, the heat equation, and the wave equation. They are the models of equilibrium, diffusion, and propagation. Separation of variables turns them into Sturm–Liouville problems you already met in ordinary differential equations. Energy estimates and maximum principles are the tools that survive after separation of variables stops working, which is almost immediately on a domain that is not a rectangle.
MIT 18.152 is a real introduction with assignments, and it is the course to work. Strauss’s Partial Differential Equations is the standard paid book; no edition was verified. Evans is the graduate ceiling and is paid, same situation. You are not ready for Evans until maximum principles and energy methods are boring. Fourier series from the ODE page are a prerequisite you must actually be able to use, not a topic you once saw.
A formal series that does not satisfy the boundary conditions is not a solution. Write the conditions first.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Classification. A second-order linear equation \(au_{xx}+2bu_{xy}+cu_{yy}+\cdots=0\) is elliptic, parabolic, or hyperbolic according to the sign of \(b^2-ac\), the same discriminant as a conic. Laplace, heat, and wave are the models.
A linear change of variables diagonalizes the principal symbol. The sign of the discriminant is invariant and decides whether you get a sum of squares, a heat-type degeneracy, or a difference of squares.
Separation. On a rectangle or an interval, product solutions \(X(x)T(t)\) reduce the PDE to ODEs. The spatial operator with the boundary conditions produces an orthogonal basis of eigenfunctions. The initial condition becomes a coefficient problem.
Plug in, divide, and equate a function of \(x\) to a function of \(t\). Orthogonality of sines on \([0,\pi]\) is a calculus computation. Completeness of the basis is a theorem you may quote from Fourier analysis after you have used the coefficients correctly.
Wave equation. d’Alembert’s formula solves the one-dimensional wave equation on the line. Finite propagation speed is the domain of dependence. Reflections come from the method of images when the boundary condition is Dirichlet or Neumann on a half-line.
Factor \(\partial_{tt}-c^2\partial_{xx}=(\partial_t-c\partial_x)(\partial_t+c\partial_x)\) and change to characteristic coordinates \(\xi=x+ct\), \(\eta=x-ct\). The general solution is \(F(\xi)+G(\eta)\). Match initial conditions to get the formula.
Maximum principle. A harmonic function on a bounded domain attains its maximum and minimum on the boundary. A consequence is uniqueness for the Dirichlet problem.
If the Laplacian is positive, a maximum cannot occur at an interior point because the Hessian would have negative trace. For harmonic functions, apply that to \(u+\varepsilon|x|^2\) and let \(\varepsilon\to 0\). Uniqueness: the difference of two solutions is zero on the boundary and harmonic, hence zero.
Energy. For the wave equation with suitable boundary conditions, the sum of kinetic and potential energy is constant. That identity gives uniqueness of the initial-boundary-value problem.
Multiply the equation by \(u_t\) and integrate by parts. The boundary terms vanish because of the boundary condition. What remains is the derivative of the energy.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
MIT 18.152, the course description
Read once, do not study from this. See that the course is Laplace, heat, and wave, not a catalogue of tricks. Then use the assignments as the work.
The thing you work through.
MIT 18.152 assignments and notes
The main text. Strauss is the usual paid book; no edition was verified.
The problem book.
18.152 problem sets, every one
Separation, maximum principles, and energy estimates are all in the sets. Do them in order.
The hardest questions that exist at this level.
Evans is the graduate ceiling; the public ceiling is a full 18.152 exam plus d’Alembert from memory
Evans is paid and unnumbered here. Do not start it until the maximum principle proof is yours.
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: solve the heat equation on an interval for a general sine series initial condition, derive d’Alembert’s formula, and prove uniqueness for the Dirichlet problem via the maximum principle. Written. Fail if the series ignores the boundary conditions.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Classify the three second-order equations, solve the heat equation for a sine initial condition, and state the maximum principle. No notes.