Partial differential equation Study Guide

📖 Core Concepts Partial Differential Equation (PDE) – an equation involving an unknown multivariable function \(u(x{1},\dots ,x{n})\) and its partial derivatives. Order – highest total derivative order \(|\alpha|\) that appears (e.g., second‑order PDE contains \(u{xx},u{xy},u{yy}\)). Linear vs. Nonlinear – Linear if every term is linear in \(u\) and its derivatives; nonlinear otherwise. Sub‑categories: Semi‑linear – highest‑order terms linear, lower‑order may be nonlinear. Quasilinear – highest‑order terms linear but coefficients may depend on \(u\) or lower derivatives. Fully nonlinear – nonlinearity also appears in highest‑order terms. Second‑order classification (via discriminant \(D = B^{2}-AC\) for \(A u{xx}+2B u{xy}+C u{yy}\)): Elliptic: \(D<0\) (e.g., Laplace’s equation). Parabolic: \(D=0\) (e.g., heat equation). Hyperbolic: \(D>0\) (e.g., wave equation). Well‑posed problem – satisfies existence, uniqueness, and continuous dependence on data. Weak solution – satisfies the PDE after integration against smooth test functions; essential when classical derivatives do not exist. Superposition principle – any linear combination of solutions of a linear PDE is again a solution. --- 📌 Must Remember Formal PDE definition \[ F\bigl(x,u,\partial u,\partial^{2}u,\dots ,\partial^{\alpha}u\bigr)=0,\qquad |\alpha|\le k . \] Notation: \(u{i}= \partial u/\partial x{i}\), \(u{ij}= \partial^{2}u/\partial x{i}\partial x{j}\), \(\Delta u = \sum{i=1}^{n} u{ii}\). Discriminant rule for second‑order linear PDEs: \(D<0\) → elliptic, \(D=0\) → parabolic, \(D>0\) → hyperbolic. Typical prototypes Elliptic: \(\Delta u = 0\). Parabolic: \(u{t}-\kappa\Delta u = 0\). Hyperbolic: \(u{tt}-c^{2}\Delta u = 0\). Boundary/initial condition types Dirichlet: prescribe \(u\) on \(\partial\Omega\). Neumann: prescribe \(\partial u/\partial n\) on \(\partial\Omega\). Robin: prescribe \(\alpha u + \beta \partial u/\partial n\). Cauchy: prescribe both \(u\) and its normal derivative on a surface (hyperbolic). Well‑posedness checklist – verify existence, uniqueness, and stability before attempting a solution. --- 🔄 Key Processes Separation of Variables Assume \(u(x,t)=X(x)T(t)\). Substitute into linear PDE → divide by \(XT\) to isolate a constant \(-\lambda\). Obtain two ODEs: one for \(X\) (spatial eigenvalue problem) and one for \(T\) (temporal). Solve eigenvalue problem → orthogonal eigenfunctions. Expand arbitrary data as series in eigenfunctions → construct full solution. Method of Characteristics (first‑order PDE) Write PDE in form \(a(x,u)u{x}+b(x,u)u{y}=c(x,u)\). Set up characteristic ODE system: \(\dot{x}=a,\; \dot{y}=b,\; \dot{u}=c\). Integrate to obtain characteristic curves \((x(s),y(s))\). Along each curve, solve the ODE for \(u\). Impose initial data on a non‑characteristic curve to determine integration constants. Finite Difference Method (FDM) Choose a grid \((x{i},t^{n})\). Replace derivatives by difference quotients, e.g. \[ u{t}\approx \frac{u{i}^{\,n+1}-u{i}^{\,n}}{\Delta t},\qquad u{xx}\approx \frac{u{i+1}^{\,n}-2u{i}^{\,n}+u{i-1}^{\,n}}{\Delta x^{2}} . \] Assemble algebraic equations for all interior nodes. Apply boundary/initial conditions → solve linear system (explicit, implicit, or Crank–Nicolson). Finite Element Method (FEM) (variational form) Write weak formulation: find \(u\in V\) such that \[ a(u,v)=\ell(v)\quad\forall v\in V, \] where \(a(\cdot,\cdot)\) is a bilinear form and \(\ell\) a linear functional. Choose a finite‑dimensional subspace \(Vh\subset V\) (piecewise polynomials on mesh). Express \(uh=\sum{j}Uj\phij\); insert into weak form → matrix system \(KU=F\). Solve for coefficients \(Uj\). Constructing Green’s Function Find fundamental solution \(G(x,\xi)\) solving \(L G = \delta(x-\xi)\). Solution of \(L u = f\) with homogeneous boundary conditions: \[ u(x)=\int{\Omega} G(x,\xi) f(\xi)\,d\xi . \] --- 🔍 Key Comparisons Linear vs. Nonlinear Linear: superposition holds; coefficients may be functions of \(x\) but not of \(u\). Nonlinear: superposition fails; solution behavior can change dramatically with data. Semi‑linear vs. Quasilinear vs. Fully Nonlinear Semi‑linear: highest‑order terms linear, lower terms may be nonlinear. Quasilinear: coefficients of highest‑order terms depend on \(u\) or its lower derivatives. Fully nonlinear: nonlinearity appears in the highest‑order derivatives themselves. Elliptic vs. Parabolic vs. Hyperbolic (second order) Elliptic ( \(D<0\) ): no real characteristics, solutions smooth, boundary‑value problems. Parabolic ( \(D=0\) ): one family of real characteristics, diffusion‑type, initial‑boundary value. Hyperbolic ( \(D>0\) ): two families of real characteristics, wave‑type, Cauchy problems. Dirichlet vs. Neumann vs. Robin Dirichlet: prescribe value → often ensures uniqueness for elliptic problems. Neumann: prescribe normal derivative → uniqueness up to an additive constant (need compatibility). Robin: linear combo → models convection or impedance; requires both value and flux information. FDM vs. FEM vs. FVM FDM: easy on structured grids, best for simple geometries. FEM: flexible mesh, naturally handles weak form, good for variable coefficients. FVM: conserves fluxes locally, ideal for conservation laws and fluid dynamics. --- ⚠️ Common Misunderstandings “All PDEs have explicit formulas.” Most PDEs require numerical or approximate analytic methods. Confusing discriminant sign with coefficient signs. Classification depends on \(B^{2}-AC\), not on the sign of \(A\) or \(C\) alone. Assuming weak = classical solution. Weak solutions only satisfy an integrated form; they may lack pointwise derivatives. Using superposition for nonlinear equations. It works only for linear PDEs. Neumann problem always unique. It is unique only up to an additive constant and requires \(\int{\Omega} f = 0\) (compatibility). --- 🧠 Mental Models / Intuition “Steady‑state vs. evolving” – Elliptic ≈ equilibrium (no time), Parabolic ≈ smoothing over time, Hyperbolic ≈ propagation of sharp fronts. Characteristics as “information highways.” Data travel along characteristic curves/surfaces; outside them the solution is unaffected. Weak formulation = moving the derivative onto a smooth test function (integration by parts). Think of it as “testing” the PDE with harmless probes. Green’s function = impulse response – just like a linear time‑invariant system, the response to any source is the convolution with the impulse (point) response. --- 🚩 Exceptions & Edge Cases Degenerate elliptic/parabolic: when \(A\) or \(C\) vanish on part of the domain, classification can change locally. Ultrahyperbolic equations: \(D>0\) but with mixed‑sign signature; standard energy methods fail. Non‑characteristic surface requirement for uniqueness in first‑order systems; data on a characteristic surface may not determine a unique solution. Cauchy–Kowalevski theorem only guarantees local analytic solutions when all data are analytic – not applicable to general smooth data. Neumann compatibility: need \(\int{\partial\Omega} \partial u/\partial n = \int{\Omega} f\). --- 📍 When to Use Which Separation of Variables – linear, homogeneous PDE on a product (rectangular) domain with separable boundary conditions. Method of Characteristics – first‑order PDE (or first‑order system) with given data on a non‑characteristic curve/surface. Fourier Transform – infinite or periodic domains, constant coefficients, especially for heat or wave equations. Laplace Transform – initial‑value problems on semi‑infinite time intervals (e.g., heat equation with time‑dependent BC). Finite Difference – simple geometry, uniform grid, moderate accuracy needs. Finite Element – complex geometry, variable coefficients, need of higher‑order accuracy or adaptive refinement. Finite Volume – conservation laws, fluid dynamics, when local flux balance is critical. Green’s Function – linear PDE with simple geometry and known fundamental solution; ideal for point‑source problems. --- 👀 Patterns to Recognize \(D = B^{2}-AC\) appears → immediately classify elliptic/parabolic/hyperbolic. Presence of \(\Delta u\) without time derivative → elliptic (steady state). \(u{t}\) + \(\Delta u\) → parabolic (diffusion). \(u{tt}\) – \(\Delta u\) → hyperbolic (wave). Boundary condition only on part of \(\partial\Omega\) → suspect Robin or mixed problem. Coefficients depending on \(u\) → quasilinear (watch for blow‑up). Linear PDE + homogeneous boundary conditions → superposition and eigenfunction expansion are viable. --- 🗂️ Exam Traps Choosing the wrong type based on sign of a single coefficient – always compute the discriminant \(D\). Assuming Dirichlet data guarantees uniqueness for any PDE – only true for well‑posed elliptic problems; hyperbolic Cauchy data need both value and normal derivative. Applying separation of variables when the domain is not a product – results in incorrect eigenfunctions. Neglecting the compatibility condition for Neumann problems – leads to “no solution” or non‑unique constant offset. Mixing up \(\Delta\) and \(\nabla^{2}\) notation – they are the same operator; inconsistency can cause sign errors in physics conventions. Treating a semi‑linear PDE as fully linear – lower‑order nonlinear terms break superposition; check the highest‑order linearity only. Using FDM on highly irregular geometries – grid misalignment produces large truncation errors; FEM is safer. ---