Engineering Mathematics

Partial Differential Equations

Laplace, heat and wave equations and their classification — the field equations governing steady state, diffusion and vibration.

PART 1

Topic Breakdown & Traps

A PDE involves partial derivatives of a function of several variables. A second-order linear PDE

A u_{xx} + B u_{x y} + C u_{y y} + \dots = 0

is classified by the discriminant

B^{2} - 4 A C

: elliptic (

< 0

, e.g. Laplace — steady state), parabolic (

= 0

, e.g. heat — diffusion) or hyperbolic (

> 0

, e.g. wave — propagation). Boundary/initial conditions plus separation of variables give the solution.

Classification:

B^{2} - 4 A C < 0

elliptic,

= 0

parabolic,

> 0

hyperbolic

Laplace (elliptic):

u_{xx} + u_{y y} = 0

Heat (parabolic):

u_{t} = c^{2} u_{xx}

Wave (hyperbolic):

u_{tt} = c^{2} u_{xx}

⚠
Read coefficients of the second-order terms only for $B^{2} - 4 A C$ ; first-order and source terms don't affect the type.
⚠
** $B$ is the coefficient of the mixed term $u_{x y}$ **, and the discriminant uses $B^{2}$ , not $(B /2)^{2}$ in the standard GATE convention — be consistent.
⚠
Heat is parabolic, wave is hyperbolic — both look similar but the time order differs ( $u_{t}$ vs $u_{tt}$ ).

Advanced Engineering Mathematics — Erwin Kreyszig · Partial Differential Equations

PART 2

Q1BASIC1 Mark · MCQ

The Laplace equation

u_{xx} + u_{y y} = 0

is classified as:

Q2MEDIUM2 Marks · MCQ

The one-dimensional heat equation

u_{t} = c^{2} u_{xx}

is:

Q3HARD2 Marks · NAT

For the PDE

u_{xx} + 4 u_{x y} + 4 u_{y y} = 0

, the value of the discriminant

B^{2} - 4 A C

is _____.