An Elementary Course on Partial Differential Equations

We now introduce the idea of partial differentialequations formally as well as technically. A partialdifferential equation (often abbreviated in thesequel as PDE) is defined as an equation involvingone or more partial derivatives of an unknownfunction of several variables. Thus, for an unknownfunction u of severalindependent variables x, y, z,t,…, the general form of PDE will be

In Eq. (2.1), we assume that at least one of thepartial derivatives of the unknown function u must be involved. We alsoassume that the mixed partial derivatives of u are independent of theorder of differentiation, for example, uxyx = uxxy = uyxx.

The following examples of well-known PDEs naturallyevolve in various physical considerations.

• (i) Linear Transport Equation:ut + cux = 0

• (ii) Inviscid Burgers’ Equation(Shock Waves): ut + uux = 0

• (iii) Eikonal Equation: u2x + u2y = 1

• (iv) Laplace Equation or PotentialEquation: uxx + uyy + uzz = 0

• (v) Heat Equation: ut =𝛼2(uxx + uyy + uzz)

• (vi) Wave Equation: utt = c2(uxx + uyy + uzz)

• (vii) Poisson Equation: uxx + uyy + uzz = f(x, y,z)

• (viii) Helmholtz Equation: uxx + uyy + uzz + ƛu = 0

• (ix) Telegraph Equation: utt + aut + bu = c2uxx

• (x) Burgers’ Equation: ut + uux = 𝜇uxx

• (xi) Minimal Surface Equation: (1+ u2y) uxx - 2uxuyuxy+ (1 + u2x) uyy = 0

• (xii) Born–Infeld Equation: (1 -u2t) uxx + 2uxutuxt- (1 + u2x) utt = 0

• (xiii) Korteweg-de Vries (KdV)Equation: ut + uux + uxxx = 0

• (xiv) Biharmonic Equation: uxxxx + 2uxxyy + uyyyy = 0.

All these PDEs are classical and each of them isprofoundly significant in theoretical physics. Here,generally, u remains afunction of time t andone/two/three rectangular coordinate(s) of apoint.

The theory of PDEs is quite different as compared tothat of ordinary differential equations (ODEs) andis relatively difficult and cumbersome in everyrespect. PDEs form a subject of vigorousmathematical research for over three centuries andstill continue to be so. It remains to be a veryactive and young area for critical investigationsand research for mathematicians, scientists, andengineers.