GATE MA: Partial Differential Equations

Partial Differential Equations in GATE MA covers first-order characteristics, classification of second-order equations, and the classical wave, heat, and Laplace equations, usually via separation of variables and Fourier series.

Core rules

• First-order quasilinear: a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u); solve the characteristic system dx/a=dy/b=du/c.
• Second-order Au_xx+2Bu_xy+Cu_yy+...: discriminant B²−AC gives hyperbolic (>0), parabolic (=0), elliptic (<0).
• Wave equation u_tt=c²u_xx: D'Alembert solution u(x,t)=F(x−ct)+G(x+ct).
• Heat equation u_t=α²u_xx with Dirichlet boundaries: separation u=X(x)T(t) gives decaying exponentials and sine eigenfunctions.
• Laplace equation u_xx+u_yy=0 has no interior extrema (maximum principle).
• The method of characteristics solves u_x+u_y=0 as u=f(y−x).

Worked example

Solve u_t=u_xx on 0<x<π, t>0, with u(0,t)=u(π,t)=0 and u(x,0)=sin x. Separation u=X(x)T(t) gives X″+λX=0, T′+λT=0. Boundary conditions force X=sin(nx), λ=n². The initial data selects n=1, so u(x,t)=e^{−t}sin x.

Traps

• Do not mix initial and boundary conditions; the heat equation needs one initial condition and two boundary conditions.
• Fourier coefficients depend on the interval length and normalization; a missing factor 2/π changes the answer.
• Characteristics for first-order PDEs can cross, causing shocks; do not assume a global smooth solution.
• Classification is pointwise if the coefficients vary.