Partial Differential Equations Notes

Last reviewed 16 Sept 2026 · 6 min read

Definitions

A partial differential equation (PDE) involves partial derivatives of a function of two or more independent variables.
Notation for $z = z (x, y)$ : $p = \frac{\partial z}{\partial x}$ , $q = \frac{\partial z}{\partial y}$ , $r = \frac{\partial ^{2} z}{\partial x ^{2}}$ , $s = \frac{\partial ^{2} z}{\partial x \partial y}$ , $t = \frac{\partial ^{2} z}{\partial y ^{2}}$ .
Order, degree and linearity defined as for ODEs.

Formation of PDEs

Eliminating arbitrary constants — differentiate partially and eliminate (if the number of constants equals the number of independent variables, a first-order PDE results).
Eliminating arbitrary functions — e.g. $z = f (x + y)$ , $z = f (x^{2} + y^{2})$ .

Lagrange's linear equation

P p + Qq = R (P, Q, R functions of x, y, z)

Auxiliary (subsidiary) equations:

\frac{d x}{P} = \frac{d y}{Q} = \frac{d z}{R}

Find two independent solutions $u (x, y, z) = c_{1}$ and $v (x, y, z) = c_{2}$ ; the general solution is $ϕ (u, v) = 0$ .

(Methods: grouping, and multipliers $l, m, n$ with $l P + m Q + n R = 0$ .)

Classification of second-order linear PDEs

For $A u_{xx} + B u_{x y} + C u_{y y} + (lower-order terms) = 0$ :

Discriminant $B^{2} - 4 A C$	Type	Standard example
$< 0$	Elliptic	Laplace equation $u_{xx} + u_{y y} = 0$ (steady state)
$= 0$	Parabolic	Heat (diffusion) equation $u_{t} = c^{2} u_{xx}$
$> 0$	Hyperbolic	Wave equation $u_{tt} = c^{2} u_{xx}$

Fourier series

A periodic function $f (x)$ of period $2 L$ can be represented as

f (x) = \frac{a _{0}}{2} + n = 1 \sum \infty (a_{n} cos \frac{nπ x}{L} + b_{n} sin \frac{nπ x}{L})

FormulaEuler coefficients (interval −L to L)

a_{0} = \frac{1}{L} \int_{- L}^{L} f (x) d x, a_{n} = \frac{1}{L} \int_{- L}^{L} f (x) cos \frac{nπ x}{L} d x, b_{n} = \frac{1}{L} \int_{- L}^{L} f (x) sin \frac{nπ x}{L} d x

Even function ( $f (- x) = f (x)$ ): $b_{n} = 0$ — cosine series only
Odd function ( $f (- x) = - f (x)$ ): $a_{0} = a_{n} = 0$ — sine series only
Half-range series on $(0, L)$ : sine series $b_{n} = \frac{2}{L} \int_{0}^{L} f (x) sin \frac{nπ x}{L} d x$ ; cosine series $a_{n} = \frac{2}{L} \int_{0}^{L} f (x) cos \frac{nπ x}{L} d x$

Dirichlet conditions — a function that is bounded, has finitely many discontinuities and extrema in a period has a convergent Fourier series; at a jump it converges to the average of left and right limits.

Method of separation of variables

Assume $u (x, t) = X (x) T (t)$ , substitute, separate into ODEs set equal to a separation constant (chosen negative, $- λ^{2}$ , for oscillatory/decaying physically meaningful solutions), solve with boundary conditions, and superpose using Fourier series for initial conditions.

One-dimensional heat (diffusion) equation

\frac{\partial u}{\partial t} = c^{2} \frac{\partial ^{2} u}{\partial x ^{2}}, c^{2} = \frac{k}{ρ s} (diffusivity)

FormulaRod of length L with ends kept at zero temperature

Boundary conditions $u (0, t) = u (L, t) = 0$ ; initial temperature $u (x, 0) = f (x)$ :

u (x, t) = n = 1 \sum \infty b_{n} sin \frac{nπ x}{L} e^{- c^{2} n^{2} π^{2} t / L^{2}}, b_{n} = \frac{2}{L} \int_{0}^{L} f (x) sin \frac{nπ x}{L} d x

Higher modes decay faster; the steady state is $u = 0$ (or linear for fixed unequal end temperatures).

Terzaghi's one-dimensional consolidation equation $\frac{\partial u}{\partial t} = c_{v} \frac{\partial ^{2} u}{\partial z ^{2}}$ (excess pore pressure) has exactly this form (see Consolidation in Geotechnical Engineering).

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Partial Differential Equations