Partial differential equations basics

In short

A partial differential equation (PDE) is an equation relating an unknown function of two or more independent variables to its partial derivatives — for example the heat equation ∂u/∂t = c²∂²u/∂x². Unlike ordinary differential equations, a PDE's general solution contains arbitrary functions rather than arbitrary constants, so boundary and initial conditions decide the answer. The B.Sc / CBCS syllabus covers formation, first-order equations (Lagrange, Charpit, Clairaut), second-order linear equations with constant coefficients, classification into elliptic, parabolic and hyperbolic types, and solution of the wave, heat and Laplace equations by separation of variables. It carries heavy weight in GATE MA, CSIR-UGC NET and JAM, and underlies every modelling or simulation role — CFD and FEA at L&T, Tata Consultancy Research, ISRO and Ansys users, option pricing in BFSI analytics, and image/signal processing in Indian product firms.

Explained

Definitions, order, degree and classification by linearity

• A PDE involves partial derivatives of a dependent variable z (or u) with respect to two or more independent variables x, y, t.
• Standard Monge notation for z = z(x, y): p = ∂z/∂x, q = ∂z/∂y, r = ∂²z/∂x², s = ∂²z/∂x∂y, t = ∂²z/∂y².
• Order = order of the highest derivative; degree = power of the highest-order derivative after clearing radicals and fractions.
• Linear: dependent variable and all its derivatives appear to the first power with no products among them, coefficients depending only on the independent variables — e.g. xr + yt + z = 0.
• Semi-linear: linear in the highest derivatives, coefficients free of z and its lower derivatives; quasi-linear: linear in highest derivatives but coefficients may contain z, p, q (e.g. zp + q = 1); nonlinear: none of these, e.g. p² + q² = 1.
• Homogeneous if every term contains the dependent variable (RHS zero); otherwise non-homogeneous, and the general solution = complementary function + particular integral.
• Solutions are classed as general (contains arbitrary functions), particular (functions fixed by data) and singular (not obtainable from the general solution by any choice of constants — arises in Charpit/Clairaut problems).

Formation of a PDE

Two routes, both standard two-mark starters:

1. By eliminating arbitrary constants. Given z = f(x, y, a, b), differentiate once with respect to x and y to get p and q, then eliminate a and b among the three equations → a first-order PDE.
2. By eliminating an arbitrary function. Given φ(u, v) = 0 where u, v are functions of x, y, z, differentiating and eliminating ∂φ/∂u, ∂φ/∂v gives the Lagrange form Pp + Qq = R with P, Q, R the Jacobians ∂(u,v)/∂(y,z), ∂(u,v)/∂(z,x), ∂(u,v)/∂(x,y).
3. Rule of thumb: eliminating n arbitrary constants gives order n at most; eliminating one arbitrary function of two arguments gives a first-order PDE; eliminating two arbitrary functions typically gives a second-order PDE.

First-order PDEs: linear and nonlinear methods

• Lagrange's linear equation Pp + Qq = R. Solve the auxiliary (characteristic) system dx/P = dy/Q = dz/R. If two independent integrals are u = c₁, v = c₂, the general solution is φ(u, v) = 0 or u = f(v).
• Useful tricks for the auxiliary system: pick two ratios that integrate directly; use multipliers (l, m, n) so that lP + mQ + nR = 0, giving l dx + m dy + n dz = 0; add numerators and denominators.
• Standard nonlinear forms (special types):
• Type I f(p, q) = 0: take z = ax + by + c with f(a, b) = 0.
• Type II (Clairaut form) z = px + qy + f(p, q): solution z = ax + by + f(a, b); the singular solution comes from eliminating a, b between the partial derivatives with respect to a and b.
• Type III f(z, p, q) = 0: put u = x + ay, p = dz/du, q = a dz/du, reducing to an ODE.
• Type IV separable f₁(x, p) = f₂(y, q): set each side = a and integrate dz = p dx + q dy.
• Charpit's method (general first-order f(x, y, z, p, q) = 0): solve

dp/(f_x + p f_z) = dq/(f_y + q f_z) = dz/(−p f_p − q f_q) = dx/(−f_p) = dy/(−f_q). Any one integral giving a relation between p, q and the variables is combined with the original equation to get p and q, then z follows from dz = p dx + q dy.

• Jacobi's method is the analogue for three or more independent variables (often optional in B.Sc, examined in M.Sc/GATE).

Second-order linear PDEs with constant coefficients

• Write with operators D = ∂/∂x, D′ = ∂/∂y: f(D, D′) z = F(x, y).
• Homogeneous case (all terms of the same order): put D/D′ = m (equivalently D = m, D′ = 1), solve the auxiliary equation; for roots m₁, m₂, … the CF is φ₁(y + m₁x) + φ₂(y + m₂x) + … ; a repeated root m gives φ₁(y + mx) + x φ₂(y + mx).
• Non-homogeneous operator factorising as ∏(D − mᵢD′ − cᵢ): each factor contributes e^cᵢx φᵢ(y + mᵢx).
• Particular integral by the inverse operator 1/f(D, D′):
• For F = e^ax+by: substitute D → a, D′ → b (if f(a, b) ≠ 0).
• For F = sin/cos(ax + by): substitute D² → −a², DD′ → −ab, D′² → −b².
• For F = x^m yⁿ: expand 1/f(D, D′) as a series in D′/D and operate term by term.
• If the direct substitution gives zero, the case fails and multiplication by x (or the general rule for repeated factors) applies, exactly as in ODEs.
• Monge's method handles variable-coefficient second-order equations Rr + Ss + Tt = V through the Monge equations R dy² − S dx dy + T dx² = 0 and R dp dy + T dq dx = V dx dy.

Classification, canonical forms and characteristics

• For A u_xx + B u_xy + C u_yy + (lower-order terms) = 0, the discriminant is B² − 4AC:
• B² − 4AC > 0 → hyperbolic (two real characteristic families; wave equation; signals travel at finite speed).
• B² − 4AC = 0 → parabolic (one family; heat/diffusion equation; smoothing, infinite signal speed).
• B² − 4AC < 0 → elliptic (no real characteristics; Laplace/Poisson equation; steady state, boundary-value problem).
• Type can change with position: y u_xx + u_yy = 0 (Tricomi) is elliptic for y > 0, parabolic on y = 0, hyperbolic for y < 0.
• Characteristic ODE: A dy² − B dx dy + C dx² = 0; its integrals ξ = const, η = const are the new coordinates that give the canonical forms u_ξη = … (hyperbolic), u_ξξ = … (parabolic), u_ξξ + u_ηη = … (elliptic).
• Well-posedness (Hadamard): a problem needs existence, uniqueness and continuous dependence on data. Dirichlet data suit elliptic equations; initial + boundary data suit parabolic and hyperbolic. The backward heat equation and the Cauchy problem for Laplace's equation are the standard ill-posed examples.
• Boundary-condition names: Dirichlet (u given), Neumann (∂u/∂n given), Robin/mixed (αu + β∂u/∂n given). Pure Neumann problems for Laplace need the compatibility condition ∮ ∂u/∂n ds = 0 and fix u only up to a constant.

The three classical equations and separation of variables

• One-dimensional wave equation u_tt = c² u_xx (vibrating string, c² = T/ρ with T tension in N and ρ linear density in kg/m).
• One-dimensional heat (diffusion) equation u_t = α u_xx, α = k/(ρc_p) in m²/s (thermal diffusivity).
• Two-dimensional Laplace equation u_xx + u_yy = 0 (steady temperature, electrostatic potential, ideal fluid flow); with a source term it becomes Poisson's equation ∇²u = f.
• Method of separation of variables (Fourier method):
• Assume u(x, t) = X(x)T(t) and substitute.
• Separate so that each side depends on one variable only; equate to a constant −λ.
• Solve the two ODEs; apply the homogeneous boundary conditions to the X-equation — this is an eigenvalue (Sturm–Liouville) problem giving λₙ and eigenfunctions Xₙ.
• Superpose: u = Σ cₙ Xₙ(x) Tₙ(t), and find cₙ from the initial condition using Fourier sine/cosine coefficients.
• Only three sign cases for λ need testing; for u(0, t) = u(L, t) = 0 the positive case alone survives, with λₙ = (nπ/L)².
• D'Alembert's solution of the infinite-string Cauchy problem shows the hyperbolic character directly: the solution at (x, t) depends only on data in the interval [x − ct, x + ct] (domain of dependence).
• Other techniques met at this level: Laplace transform in t for semi-infinite heat problems, Fourier transform for infinite domains, the method of images, and numerical schemes (explicit/implicit finite differences, Crank–Nicolson) in the computational-methods paper.
• Limitations: separation of variables needs a linear equation, a separable geometry (rectangle, disc, cylinder, sphere) and homogeneous boundary conditions — non-homogeneous conditions must first be shifted by a steady-state part; nonlinear PDEs (Burgers, Navier–Stokes) have no superposition and are handled numerically.

Key points

• PDE - equation in partial derivatives of a function of two or more independent variables.
• Order and degree - highest derivative present; its power after clearing radicals.
• General solution - contains arbitrary *functions*, not constants, so PDE solution sets are far larger than ODE ones.
• Monge notation - p, q for first derivatives and r, s, t for the three second derivatives.
• Formation - eliminate arbitrary constants (first order) or arbitrary functions (first or second order).
• Lagrange's equation - Pp + Qq = R solved via dx/P = dy/Q = dz/R, answer φ(u, v) = 0.
• Charpit's method - the general first-order technique; the special types I–IV are shortcuts from it.
• Clairaut form - z = px + qy + f(p, q) gives z = ax + by + f(a, b) plus a singular solution.
• Homogeneous linear PDE - CF from roots m of the auxiliary equation as arbitrary functions φ(y + mx).
• Repeated root m - adds x φ(y + mx), exactly parallel to the ODE rule.
• Discriminant B² − 4AC - positive hyperbolic, zero parabolic, negative elliptic.
• Characteristics - real curves along which information propagates; exist only for hyperbolic (two) and parabolic (one) types.
• Wave, heat, Laplace - the canonical hyperbolic, parabolic and elliptic equations respectively.
• Separation of variables - converts a linear PDE into an eigenvalue ODE problem plus a Fourier series.
• D'Alembert - u = ½[f(x + ct) + f(x − ct)] + (1/2c)∫ g; finite domain of dependence [x − ct, x + ct].
• Boundary types - Dirichlet (value), Neumann (flux), Robin (mixed).
• Well-posed problem - existence + uniqueness + continuous dependence on the data; backward heat equation is ill-posed.
• Maximum principle - a harmonic function attains its extremes on the boundary, which gives uniqueness for the Dirichlet problem.

Formulas and examples

Example 1 — Formation by eliminating constants. Form the PDE from z = ax + by + ab.

• p = ∂z/∂x = a, q = ∂z/∂y = b.
• Substituting: z = px + qy + pq — a Clairaut-form first-order PDE. (Check: the given surface is recovered by a = p, b = q.)

Example 2 — Lagrange's equation. Solve (y − z)p + (z − x)q = x − y.

• Auxiliary: dx/(y − z) = dy/(z − x) = dz/(x − y).
• Multipliers (1, 1, 1): denominators add to 0 → dx + dy + dz = 0 → x + y + z = c₁.
• Multipliers (x, y, z): x(y − z) + y(z − x) + z(x − y) = 0 → x dx + y dy + z dz = 0 → x² + y² + z² = c₂.
• General solution: φ(x + y + z, x² + y² + z²) = 0.

Example 3 — Second-order linear PDE, CF and PI. Solve (D² − 3DD′ + 2D′²) z = e^2x+3y.

• Auxiliary equation m² − 3m + 2 = 0 → m = 1, 2 → CF = φ₁(y + x) + φ₂(y + 2x).
• PI: put D → 2, D′ → 3 in 1/(D² − 3DD′ + 2D′²): 4 − 3(2)(3) + 2(9) = 4 − 18 + 18 = 4 ≠ 0.
• So PI = e^2x+3y/4, and z = φ₁(x + y) + φ₂(2x + y) + e^2x+3y/4.

Example 4 — Classification. Classify u_xx + 4u_xy + 4u_yy + u_x = 0 and x²u_xx − y²u_yy = 0 (x, y ≠ 0).

• First: A = 1, B = 4, C = 4 → Δ = 16 − 16 = 0 → parabolic everywhere. Characteristic ODE dy² − 4 dx dy + 4 dx² = 0 → (dy − 2dx)² = 0 → the single family y − 2x = const.
• Second: A = x², B = 0, C = −y² → Δ = 0 − 4(x²)(−y²) = 4x²y² > 0 → hyperbolic for all x ≠ 0, y ≠ 0 (degenerate on the axes).

Example 5 — Vibrating string, numbers. A steel wire of length L = 0.60 m under tension T = 120 N has linear density ρ = 3.0 × 10⁻³ kg/m; it is plucked so that u(x, 0) = sin(πx/L) m (idealised unit-amplitude first mode) with zero initial velocity, ends fixed.

• c = √(T/ρ) = √(120 / 0.003) = √40,000 = 200 m/s.
• Only the n = 1 term survives (b₁ = 1, all other bₙ = 0), so u(x, t) = sin(πx/0.60) cos(200πt/0.60).
• Fundamental frequency f₁ = c/(2L) = 200/(2 × 0.60) = 166.67 Hz; period T₁ = 1/f₁ ≈ 6.0 ms.
• Harmonics: f₂ = 333.3 Hz, f₃ = 500 Hz.

Example 6 — Heat equation, decay of modes. A copper bar L = 1.0 m, α = 1.11 × 10⁻⁴ m²/s, both ends held at 0 °C, initial u(x, 0) = 100 sin(πx/L) °C.

• u(x, t) = 100 sin(πx) e^−απ²t with απ² = 1.11 × 10⁻⁴ × 9.8696 ≈ 1.0955 × 10⁻³ s⁻¹.
• Mid-point temperature after t = 600 s: exponent = −1.0955 × 10⁻³ × 600 = −0.6573 → e^−0.6573 ≈ 0.5183, so u ≈ 51.8 °C.
• Time to fall to 50 °C: t = ln 2 / 1.0955 × 10⁻³ ≈ 633 s.
• The n = 3 mode would decay 9 times faster (exponent ∝ n²) — high-frequency detail is smoothed out quickly, the signature of a parabolic equation.

# Explicit finite-difference check of the heat equation (stability r <= 0.5)
import numpy as np
L, T_end, alpha = 1.0, 600.0, 1.11e-4
nx = 21; dx = L/(nx-1)
dt = 0.4*dx*dx/alpha                 # r = 0.4 < 0.5, stable
nt = int(T_end/dt)
u = 100*np.sin(np.pi*np.linspace(0, L, nx))
u[0] = u[-1] = 0.0
for _ in range(nt):
    u[1:-1] += 0.4*(u[2:] - 2*u[1:-1] + u[:-2])
print(u[nx//2])                      # ~51.8 degC, matches the series solution

Where it is used

• CFD and FEA engineering: Navier–Stokes, heat conduction and structural equations solved in Ansys, OpenFOAM and Abaqus — used daily at L&T, Tata Motors, Mahindra R&D, ISRO/VSSC, HAL, BARC and NPCIL for thermal and stress analysis.
• Weather, ocean and climate modelling: IMD, NCMRWF, INCOIS and IITM Pune run primitive-equation models (hyperbolic–parabolic systems) on HPC clusters; roles for M.Sc/Ph.D. mathematicians.
• Quantitative finance and BFSI analytics: the Black–Scholes equation is a parabolic PDE; derivative-pricing and risk desks at NSE-linked firms, banks and GCCs in Mumbai, Bengaluru and Pune test it in quant interviews.
• Petroleum, chemical and power plants: reservoir simulation (ONGC, Reliance), reactor diffusion models, groundwater flow studies for CGWB and irrigation design.
• Imaging, graphics and ML: anisotropic diffusion and level-set methods for de-noising and segmentation in medical-imaging products; diffusion-based generative models rest on the same stochastic–parabolic ideas.
• Teaching and exams: PDE basics are a scoring unit in GATE MA, CSIR-UGC NET (Unit 3), IIT JAM MA and state PSC/assistant-professor papers, and the standard first-year engineering paper (Engineering Mathematics II/III) under AICTE's model curriculum.

Interview questions

• Q: How does a PDE's general solution differ from an ODE's? A: An nth-order ODE's general solution has n arbitrary constants, while a PDE's contains arbitrary *functions*, so far more data (conditions on curves or surfaces) is needed to pin down a unique solution.
• Q: What does the discriminant B² − 4AC tell you? A: It classifies a second-order linear PDE — positive means hyperbolic, zero parabolic, negative elliptic — and thereby fixes what data the problem needs and which numerical method is stable.
• Q: Why does the heat equation need only an initial condition in time while the wave equation needs two? A: The heat equation is first order in t, the wave equation second order, so the latter needs both the initial displacement and the initial velocity.
• Q: What are characteristics physically? A: Curves along which information propagates and across which discontinuities can travel without smoothing; for u_tt = c²u_xx they are x ± ct = const, giving a finite signal speed c.
• Q: State Lagrange's method in one line. A: For Pp + Qq = R, solve dx/P = dy/Q = dz/R; two independent integrals u = c₁ and v = c₂ give the general solution φ(u, v) = 0.
• Q: When would you use Charpit's method instead of a standard type? A: When the first-order nonlinear equation does not fall into the four special forms — Charpit is the general procedure and the special types are its shortcuts.
• Q: What is a singular solution and where does it come from? A: A solution that is the envelope of the family of general solutions and cannot be obtained by choosing constants; in Clairaut's form it follows from eliminating a, b between ∂/∂a = 0 and ∂/∂b = 0.
• Q: What are the three sign cases checked in separation of variables, and why does only one survive for a fixed string? A: λ negative, zero and positive; with u(0, t) = u(L, t) = 0 the non-positive cases force the trivial solution, leaving λₙ = (nπ/L)².
• Q: Why is the backward heat equation ill-posed? A: Forward diffusion damps high-frequency modes by e^{−αn²π²t/L²}; reversing time amplifies them without bound, so tiny data errors blow up and continuous dependence fails.
• Q: What is the maximum principle and one use of it? A: A harmonic (or sub-/super-) solution attains its extreme values on the boundary; it proves uniqueness and stability for the Dirichlet problem and bounds numerical solutions.
• Q: What condition must a pure Neumann problem for Laplace's equation satisfy? A: The net normal flux over the boundary must vanish (∮ ∂u/∂n ds = 0), and even then u is determined only up to an additive constant.
• Q: Why is quasi-linear different from linear, with an example? A: A quasi-linear PDE is linear in its highest derivatives but its coefficients may depend on the solution, e.g. u u_x + u_t = 0 — superposition fails and shocks can form.
• Q: What stability limit applies to the explicit scheme for u_t = αu_xx? A: r = αΔt/Δx² ≤ ½; beyond that the explicit (FTCS) scheme oscillates and diverges, which is why implicit or Crank–Nicolson schemes are used for large time steps.

Common mistakes

• Writing arbitrary *constants* instead of arbitrary *functions* in the general solution, or giving only a particular solution when the general one was asked for.
• Mis-signing the discriminant by not writing the equation in the standard A u_xx + B u_xy + C u_yy form — note B is the full coefficient of u_xy, so "2u_xy" means B = 2, not 1.
• In homogeneous constant-coefficient PDEs, writing φ(y − mx) instead of φ(y + mx) for a root m of the auxiliary equation.
• Substituting D → a, D′ → b for the particular integral without checking f(a, b) ≠ 0; the failure case needs the repeated-factor rule.
• Dropping sign cases in separation of variables, or forgetting to apply the boundary conditions *before* the initial condition — the boundary conditions are what quantise λ.
• Using the wrong Fourier coefficient: sine series for Dirichlet ends, cosine series for insulated (Neumann) ends; and using half-range formulas with the wrong interval length.
• Mixing units in the heat equation — thermal diffusivity α = k/(ρc_p) is in m²/s, not k in W/m·K; and losing track of °C versus K in temperature differences (differences are equal in both).
• Applying separation of variables to non-homogeneous boundary conditions without first subtracting the steady-state solution, or to a nonlinear equation where superposition does not hold.

Related topics

Ordinary differential equations · Fourier series and transforms · Laplace transforms · Numerical solution of PDEs (finite differences) · Vector calculus and Green's/Stokes' theorems · Sturm–Liouville and boundary-value problems