Partial Differential Equations
I took these notes in fall 2016 for CME 303, taught by Professor Lenya Ryzhik.
Method of characteristics
Suppose we have a first-order linear PDE such as with the boundary condition . We should think of this as which we interpret as saying that the directional derivative of at is given by . Let be an integral curve (a characteristic) starting at of the vector field , i.e. satisfying . Then along this integral curve, the PDE becomes the ODE in the variable . Once we solve it, we have a solution to the PDE given by .
Separation of variables
Suppose we have the PDE on . Then if we have the fundamental solution that satisfies , then we have a solution to the inhomogeneous PDE with the last equality assumes is translation-invariant, allowing us to define . We can think of it as the impulse response at . Note that also satisfies the homogeneous equation everywhere except . Suppose we have the boundary-value problem on with the boundary condition on . Then we can construct a Green’s function , which satisfies for and for . We can write it as an interior term minus a boundary term correction: so that for and for . The homogeneous PDE for is solvable by some other method. Thus we can think of it as the impulse response at minus some homogeneous solution to make it zero at the boundary. We can then often write derived from a Green’s identity of the form and substituting and on the left. Relatedly, we can use Duhamel’s principle to convert an inhomogeneous initial value problem into an integral of solutions to the homogeneous initial value problem. Suppose we have the initial value problem with zero initial condition. Then we can decompose the inhomogeneous part as a convolution solve the family of PDEs, indexed by , also solvable as the initial value problem and arrive at the solution
The Fourier transform transform of a function is the function It relates multiplication to derivatives: It relates convolution to multiplication: It relates smoothness to decay: if , then as . The Schwartz space is the space of smooth, rapidly-decreasing functions on , i.e. those for which for all and . It is closed under addition and pointwise multiplication. The Fourier transform becomes a linear isomorphism on with inverse The Gaussian is its own Fourier transform. The Plancherel theorem says that the Fourier transform is unitary on :
If a function minimizes the functional over some set , with then it satisfies the PDE given by the Euler-Lagrange equation Noether’s theorem gives conserved quantities in certain cases, such as the energy
Types of PDEs
Conservation law: Advection equation: Burgers’ equation: These come from the integral conservation law
They can frequently be solved using the method of characteristics. They present two difficulties: if characteristics cross (a shock) and if characteristics fail to cover the domain (rarefaction).
In the first case, if there is a discontinuity at time , then the speed at which the discontinuity propagates is given by the Rankine-Hugoniot jump condition:
In the second case, we seek an entropy solution, which means that we fill the missing part with characteristics that must not intersect backwards in time.
Elliptic PDEs: Laplace’s equation
Laplace’s equation is , where . Poisson’s equation is . A function is harmonic if it satisfies Laplace’s equation .
The unique solution to Poisson’s equation with a prescribed boundary uniquely minimizes
The fundamental solution for , , and dimensions is respectively where is the volume of an -ball. For a Dirichlet boundary condition ( on ), the unique solution is where we interpret the normal derivative as
Properties of harmonic functions
Parabolic PDEs: the heat equation
The heat equation is .
The energy decreases as a function of time.
The fundamental solution, a.k.a. the heat kernel, is defined for as We may obtain this from the Fourier transform. For the initial value problem with , the solution is since .
Instantaneous regularity: If is bounded and continuous, then is smooth for . \proof Write , and notice that is smooth for . Maximum principle for a bounded domain: Define and its parabolic boundary . Then only achieves its maximum and minimum over on , unless is constant. \proof Set , so that it satisfies . The maximum of occurs on , because if the maximum is in , then or ( if the maximum is at ) and . Then Maximum principle: only achieves its maximum and minimum over at , assuming that , unless is constant. Infinite speed of propagation: If is not identically and is bounded and continuous, then for all and . Uniqueness: Given an initial condition (and a boundary condition if the domain is bounded), there is at most one solution to the inhomogeneous heat equation (assuming the growth estimate for case). \proof Apply the maximum principle to , implying that . \proof The energy of is identically , implying that . Estimates: Self-similarity: As , \proof From the formula make the change of variable , so we have
Hyperbolic PDEs: the wave equation
The wave equation is .
Solutions to the wave equation minimize the functional The energy is constant as a function of time.
For the one-dimensional wave equation with Cauchy initial conditions and , we have the d’Alembert formula (for example, use Fourier transform).
Finite speed of propagation: The value only depends on the values of the initial conditions within a ball around of radius . \proof Suppose and have the same initial conditions inside the ball, and consider the energy of within the ball. The energy is non-negative, and it is initially . Show that it decreases (not obvious); conclude that in the ball. Huygens principle: In odd dimensions, the value only depends on the values of the initial conditions on the sphere around of radius . Uniqueness: There is at most one solution to the Cauchy initial value problem (with and specified). \proof satisfies the wave equation with zero initial condition, so its energy is zero.