An Elementary Course on Partial Differential Equations

Many problems in mathematical, physical, andengineering sciences deal with the formulation andthe solution of first-order partial differentialequations (PDEs), for example, Brownian motion,glacier motion, mechanical transport of solvents influids, propagation of wavefronts in optics,stochastic process, waves in shallow water, thermalefficiency of heat exchangers, heat propagationbetween two superconducting cables, radioactivedisintegration, flood waves, acoustics, gasdynamics, traffic flow, noise in communicationsystems, population growth, telephone traffic, andso on. Nevertheless, first-order PDEs appear lessfrequently than second-order equations. From amathematical point of view, first-order PDEs havethe advantage of providing a conceptual frameworkthat can be utilised for second- and higher orderequations.

In this chapter, we discuss various methods for findingthe different types of solutions, namely, general,complete, singular, and particular solutions oflinear/semilinear/quasilinear/non-linear PDEs oforder one. Recall that the most general PDE of firstorder can be written in symbolic form as

4.1 Linear PDEs: Reduction to CanonicalForm

In Chapter 2, we have seen that the general linear PDEof order one in two independent variables x and y is of the form

Here, the coefficients Pand Q are continuouslydifferentiable functions and do not vanishsimultaneously. Without loss of generality, we shallassume that P ≠ 0.

In Chapter 3, we have discussed the method for findingthe general solution of standard forms of Eq. (4.1)in which one of P andQ remainsidentically zero, so that Eq. (4.1) contains asingle derivative. Now, we discuss a technique tosolve the general form of Eq. (4.1). In thistechnique, we consider certain transformation ofindependent variables of the form

under which Eq. (4.1) is transformed into standard formand hence it can be solved using the techniquediscussed in Section 3.2. The obtained standard formis called the canonical form of Eq.(4.1). Such new coordinates (𝜉;𝜂), under which theoriginal equation reduces to its canonical form, arecalled the canonical coordinates. Thismethod of solving linear PDEs is sometimes referredto as coordinate method orcanonical method.

Under consideration, we assume that the Jacobian of thetransformation (4.2) does not vanish at (x, y), i:e:,

so that by inverse function theorem, x and y can be determined uniquely from thesystem (4.2).