Partial differential equation - Classification and Model Equations

Classification of PDEs Introduction Partial differential equations come in many varieties, and understanding how to classify them is essential for predicting their behavior and choosing appropriate solution methods. The classification of PDEs addresses fundamental questions: Will solutions be smooth or can they have discontinuities? Do we expect solutions to smooth out over time, or will disturbances propagate? Does specifying initial conditions determine a unique solution? This guide covers the main classification schemes: separating linear from nonlinear PDEs, and then dividing second-order equations into three fundamental types (elliptic, parabolic, and hyperbolic). Each type behaves fundamentally differently, and this classification guides us toward the right mathematical tools. Linear versus Nonlinear PDEs The most basic distinction is whether a PDE is linear in the dependent variable and its derivatives. Linear PDEs A linear PDE has the form where every term involving $u$ or its derivatives appears to the first power only. In two variables, the general second-order linear PDE looks like: $$a(x,y)u{xx}+b(x,y)u{xy}+c(x,y)u{yy}+d(x,y)ux+e(x,y)uy+f(x,y)u+g(x,y)=0$$ The key property is that all coefficients can depend on the independent variables ($x$ and $y$), but $u$ and its derivatives appear linearly. Example: $u{xx} + u{yy} = e^{xy}$ is linear. Constant Coefficient Linear PDEs When the coefficients $a, b, c, d, e, f$ are all constants, we have a linear PDE with constant coefficients. These are typically easier to solve because we can use techniques like Fourier analysis and separation of variables. Example: $u{xx} + 2u{xy} + u{yy} = 0$ has constant coefficients. Homogeneous versus Inhomogeneous A homogeneous linear PDE has no forcing term—meaning the right-hand side is zero. An inhomogeneous (or non-homogeneous) linear PDE has a nonzero forcing term. Examples: Homogeneous: $u{xx} + u{yy} = 0$ (Laplace's equation) Inhomogeneous: $u{xx} + u{yy} = f(x,y)$ (Poisson's equation) Beyond Fully Linear: Semi-Linear and Quasilinear PDEs Real physical problems often involve nonlinearities, but they may enter in restricted ways: Semi-linear PDEs have the highest-order derivatives appearing linearly, but lower-order terms (or source terms) can be nonlinear. Example: $u{xx} + u{yy} = e^u$ is semi-linear because the $u{xx}$ and $u{yy}$ terms are linear, but the right side is nonlinear in $u$. Quasilinear PDEs have the highest-order derivatives linear, but their coefficients can depend on $u$ or lower-order derivatives. Example: $ux \cdot u{xx} + u{yy} = 0$ is quasilinear; the coefficient of $u{xx}$ depends on $ux$. Fully Nonlinear PDEs Fully nonlinear PDEs have nonlinearities in the highest-order derivatives themselves, making them substantially more difficult to analyze. Example: The Monge–Ampère equation, $u{xx}u{yy} - u{xy}^2 = f(x,y)$, is fully nonlinear. Second-Order Classification: Elliptic, Parabolic, Hyperbolic For second-order linear PDEs in two independent variables, there is a classical and powerful classification based on the discriminant of the highest-order terms. The Discriminant Test Consider a general second-order linear PDE in two variables: $$A(x,y)u{xx}+2B(x,y)u{xy}+C(x,y)u{yy}+\text{(lower order terms)}=0$$ Define the discriminant: $$\Delta = B^2 - AC$$ This single quantity determines everything about the character of the equation: Elliptic: $\Delta < 0$ (or equivalently, $B^2 < AC$) Parabolic: $\Delta = 0$ (or equivalently, $B^2 = AC$) Hyperbolic: $\Delta > 0$ (or equivalently, $B^2 > AC$) Why This Matters: Physical Behavior Each type has fundamentally different solution behavior: Elliptic PDEs describe equilibrium or steady-state phenomena. Solutions are smooth inside the domain and their values throughout the region depend on all boundary conditions. There is no preferred direction of propagation. Think of these as describing systems that have "settled down" to a stable state. Parabolic PDEs describe diffusive processes where information spreads smoothly over time. Solutions tend to smooth out irregularities as evolution progresses. There is an asymmetry: time points in one direction (forward or backward), and the equation behaves differently in that direction. Hyperbolic PDEs describe wave-like propagation. Discontinuities or sharp features can propagate along characteristics (paths in spacetime). Information travels at finite speed. Solutions can maintain sharp discontinuities. Prototype Examples Each class has a canonical prototype: Laplace's equation $\Delta u = u{xx} + u{yy} = 0$: $A=1, B=0, C=1$, so $\Delta = 0 - 1 = -1 < 0$ → Elliptic Heat equation $ut - \kappa u{xx} = 0$: Write as $-\kappa u{xx} + ut = 0$. In the $(x,t)$ plane: $A=-\kappa, B=0, C=0$, so $\Delta = 0 - 0 = 0$ → Parabolic Wave equation $u{tt} - c^2 u{xx} = 0$: $A = -c^2, B=0, C=1$, so $\Delta = 0 - (-c^2) = c^2 > 0$ → Hyperbolic Important Caveat: PDEs with More Variables The classification can become more complex in three or more spatial dimensions. For systems and higher-order equations, we must consider more subtle structures (hyperbolicity, symmetry, etc.), but the discriminant test applies directly to second-order scalar equations in two variables. Common PDEs and Their Classification Elliptic PDEs (Steady State) Laplace's equation: $\Delta u = 0$, where $\Delta = u{xx} + u{yy} + \cdots$ This describes harmonic functions: the value at any point is the average of values around it. Arises in electrostatics, gravitational potential, and steady-state heat distribution. Poisson's equation: $\Delta u = f(x,y)$ Same as Laplace's but with a source term. Still elliptic and steady-state. Parabolic PDEs (Diffusion) Heat equation: $ut - \kappa \Delta u = 0$ Describes heat diffusion with diffusivity $\kappa$. Initial conditions at $t=0$ propagate forward in time, and discontinuities smooth out exponentially fast. Parabolic structure means time and space are fundamentally asymmetric. Hyperbolic PDEs (Waves) Wave equation: $u{tt} - c^2 \Delta u = 0$ Describes wave propagation at speed $c$. Discontinuities propagate along characteristic cones in spacetime. The solution at a point depends only on initial data within a finite "domain of dependence." Complex PDEs Schrödinger equation: $i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\Delta\psi + V(x)\psi$ This quantum mechanical equation has an imaginary unit multiplying the time derivative, giving it parabolic-like structure but with complex-valued solutions. The evolution is unitary and reversible (unlike real parabolic equations). Nonlinear Systems Navier–Stokes equations: Describe fluid flow with momentum conservation for viscous fluids. These are a nonlinear system (often quasilinear) that couples spatial and temporal evolution. Classification becomes subtle for such systems. Maxwell's equations: A system of first-order linear PDEs governing electromagnetic fields. This is a hyperbolic system; electromagnetic waves propagate at the speed of light. Systems of First-Order PDEs Many of the PDEs above can be rewritten as systems of first-order equations. For example, the wave equation $u{tt} = c^2 u{xx}$ becomes a system by introducing $v = ut$: $$ut = v, \quad vt = c^2 u{xx}$$ For a general system of first-order PDEs: $$\sum{\nu=1}^{n}A{\nu}(x,u)\,\frac{\partial u}{\partial x{\nu}} + B(x,u)=0$$ where $A\nu$ and $B$ are matrices, we can classify the system using the concept of characteristic surfaces. Characteristic Surfaces A characteristic surface is an implicit surface $\phi(x)=0$ (with $\nabla\phi \neq 0$) on which the system degenerates in a specific way. Intuitively, information propagates along characteristics. On a non-characteristic surface, boundary data uniquely determine the normal derivative of the solution. On a characteristic surface, this regularity breaks down, and discontinuities can propagate. Elliptic and Hyperbolic Systems Elliptic system: No real characteristic surfaces exist. All boundary data everywhere determine the solution everywhere. Hyperbolic system: Real characteristic surfaces exist, and they typically have a timelike structure—disturbances propagate in discrete directions at finite speeds. Summary PDE classification is your first diagnostic tool: Identify the type (linear/nonlinear) and compute the discriminant $\Delta = B^2 - AC$ for second-order equations. Recognize the physical meaning: elliptic = steady state (all boundaries matter), parabolic = diffusion (one-way time evolution), hyperbolic = waves (finite propagation speed). Choose your method: Elliptic problems often use Green's functions or Fourier series; parabolic problems use heat kernel techniques; hyperbolic problems use characteristics and d'Alembert's formula. Understanding classification guides both analytical techniques and intuition about solution behavior.