An Elementary Course on Partial Differential Equations

In this chapter, we discuss various methods for solvingthe partial differential equations (PDEs) of ordertwo. Second-order PDEs play a fascinating role inapplied sciences describing a variety of physicalproblems such as transverse vibrations of a stringas well as a membrane, longitudinal vibrations in abar, sound waves, electromagnetic waves, electricsignals in cables, gravitational potential,electrostatic potential, magnetostatic potential,irrotational motion of a fluid, steady currents,steady flow of heat, surface waves on a fluid,conduction of heat in a bar, diffusion in isotropicsubstances, slowing down of neutrons in a matter,transmission line, nuclear reactors, etc.

We shall begin this chapter with two standard forms oflinear PDEs, whose solutions are relatively morenatural. In the next section, first we classify thesemilinear PDEs and then for each class, we reducethe equation to its canonical form in order toobtain the general solution. Thereafter, awell-known general method, namely, Monge’s method,has been discussed for solving quasilinear andnon-linear equations. We also describe the Fouriermethod to solve homogeneous linear PDEs associatedwith boundary conditions (BCs). Finally, we presentthe derivations and solutions of some well-knownfundamental PDEs of mathematical physics.

5.1 Linear PDEs: Standard Forms

As discussed in Chapter 2, the most general linear PDEof order two in two independent variables x and y is of the form

Here, the coefficients R,S, and T donot vanish simultaneously.

In this section, by particularising the coefficientssuitably in Eq. (5.1), we obtain some standardforms, which on integrating reduce easily to thefirst-order PDEs. Then, solving these equations forp or q, we obtain the generalsolution as desired. In the following lines, wediscuss two types of such equations.

Type I (Equations Reducible to First-Order LinearEquations): In this type, we adopt thefollowing forms:

Such forms can be rewritten, respectively, as

These are linear PDEs of order one, in which eitherp or q is the dependent variableand hence can be solved by the method discussed inSection 3.2.

Now, we adopt some examples of these forms.

Example 5.1.Solve xr + p = 9x2y2.