Partial Differential Equations

Wave equation initial value problem

The wave equation u_{tt}=c^2u_{xx} describes disturbances that travel with speed c>0. An initial value problem on the whole real line needs two data: the displacement u(x,0)=f(x) and the velocity u_t(x,0)=g(x). Its solution is given by d'Alembert's formula $u(x,t)=\frac12\left[f(x+ct)+f(x-ct)\right]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds.$ Two consequences are essential. First, the value at (x_0,t_0) depends only on the data on the interval [x_0-ct_0,\,x_0+ct_0], which is called the domain of dependence. Second, propagation speed is finite: data supported in a bounded set can affect a point only after the time needed to travel from the support to that point.

Several pitfalls recur. The factor c must appear in the limits x\pm ct and in the prefactor \frac{1}{2c}; for u_{tt}=9u_{xx} one has c=3, not c^2=3. The antiderivative of g must be evaluated at both endpoints, and when the velocity is zero the formula reduces to the average of f at x\pm ct. When g=0 and f is even, u(0,t)=f(ct), so a box-shaped displacement keeps its central height only while ct stays inside the box.

Example. For u_{tt}=u_{xx} with u(x,0)=0 and u_t(x,0)=2s, the formula gives u=\frac12\int_{x-t}^{x+t}2s\,ds=2xt, which satisfies both the equation and the initial data. If the velocity is g(s)=se^{-s^2} with zero displacement, the antiderivative -\frac12e^{-s^2} gives u=\frac14\left(e^{-(x-t)^2}-e^{-(x+t)^2}\right); evaluating such closed forms at a given point is the core computation.

On a bounded interval 0<x<\pi with fixed ends u(0,t)=u(\pi,t)=0, one expands the data in sine series instead. With c=1, each mode \sin nx oscillates in time as \cos nt, so the solution with zero initial velocity is u=\sum_n b_n\cos(nt)\sin(nx), where b_n are the sine coefficients of the initial displacement.

First-order linear PDE by characteristics

A first-order linear PDE a(x,y)u_x+b(x,y)u_y=c(x,y)u+d(x,y) is solved by the method of characteristics. The characteristic curves satisfy \frac{dx}{ds}=a and \frac{dy}{ds}=b, and along each of them \frac{du}{ds}=cu+d. A quantity \xi(x,y) that is constant on characteristics satisfies a\xi_x+b\xi_y=0, and the general solution is an arbitrary function of such first integrals.

For a Cauchy problem, the data are prescribed on a curve \Gamma. If \Gamma is transverse to the characteristics, the data determine a unique solution near \Gamma, obtained by tracing each characteristic back to its starting point on \Gamma.

Pitfalls. For a family u=\Phi(x,y,f(\xi)) with arbitrary f, one checks that it solves the PDE by forming a combination of u_x and u_y in which the term containing f' cancels, with the weights taken from the coefficients of the PDE. If the initial curve is itself a characteristic, the vector (a,b) is tangent to \Gamma. Then the data must satisfy a compatibility condition along \Gamma: if it holds there are infinitely many solutions, and if it fails there is none.

Example. For xu_x+u_y=0 the characteristics satisfy \frac{dx}{dy}=x, so xe^{-y} is constant along them, and the general solution is u=F(xe^{-y}). Direct substitution confirms that xu_x+u_y=0 holds for every differentiable F.

For a source term, u_t+cu_x=\lambda u with u(x,0)=\phi(x) has the solution u=e^{\lambda t}\phi(x-ct). A nonlinear relative v_t+cv_x=v^2 becomes a Riccati equation \frac{dv}{dt}=v^2 along characteristics, with solution v=\frac{v_0}{1-tv_0}. This is undefined when tv_0=1, so a solution with smooth data can fail to exist at some points at a given time.

Classification of second-order PDEs

For Au_{xx}+2Bu_{xy}+Cu_{yy}+(\text{lower-order terms})=0 with coefficients that may depend on (x,y), the type is decided pointwise by the discriminant B^2-AC. The equation is hyperbolic where B^2-AC>0, parabolic where B^2-AC=0, and elliptic where B^2-AC<0. Hyperbolic equations have two real families of characteristics and wave-like behaviour, parabolic equations have one family and heat-like behaviour, and elliptic equations have none and Laplace-like behaviour. Be careful with the factor 2 convention: if the mixed coefficient is written as \beta, the discriminant is \beta^2-4AC.

Because the coefficients depend on position, the type can change from region to region. The curve where the discriminant vanishes is a parabolic curve. For example, u_{xx}+2xu_{xy}+u_{yy} has B^2-AC=x^2-1, so it is hyperbolic for |x|>1, parabolic on |x|=1, and elliptic for |x|<1.

The type is invariant under a change of independent variables. Under a linear change with Jacobian determinant J, the discriminant of the principal part is multiplied by J^2, so its sign is preserved. The canonical form is obtained by the chain rule: if \eta=x^2/2 then u_x=xu_\eta and u_{xx}=u_\eta+x^2u_{\eta\eta}, and after dividing by the common factor one replaces x^2 by 2\eta to read off the first-order coefficients.

For variable coefficients the discriminant often factors, and its sign then depends on a factor that can be inspected directly. For constant coefficients au_{xx}+bu_{yy} with a,b>0 the equation is elliptic. Exponential solutions e^{px+qy} then exist only when ap^2+bq^2 equals the constant on the right-hand side, which is a dispersion relation.

Dirichlet problem for Laplace equation

For \nabla^2u=0 in a bounded domain \Omega with u=g on \partial\Omega, the Dirichlet problem has at most one solution. The difference of two solutions is harmonic with zero boundary values, so by the maximum principle it vanishes identically. Consequently the solution lies between the minimum and maximum of the boundary data, and for continuous data on a reasonable domain a solution exists.

In practice one often finds the solution by guessing a harmonic function that already matches the boundary data. Linear functions and polynomials such as x^2-y^2 or xy are harmonic, and their restriction to a curve can serve as data. Uniqueness then guarantees that the guessed function is the answer.