ABSTRACT
As illustrated in the preceding discussion, a wide variety of partial differential equations exists. Each problem has its own special governing equation or equations and its own peculiarities which must be considered individually. However, useful insights into the general features of POEs can be obtained by studying three special cases. The first special case is the general quasi linear (i.e., linear in the highest-order derivative) second-order nonhomogeneous POE in two independent variables, which is
IAfrx + Bhy + C/yl' + Df~ + EJ;. + Ff = G I (III. 18) where the coefficients A to C may depend on x, y, ix, and /y, the coefficients D to F may depend on x, y, and f, and the nonhomogeneous term G may depend on x and y. The second special case is the general quasi linear first-order nonhomogeneous POE in two independent variables, which is
where a, b, and c may depend on x, t, and f. The third special case is the system of two general quasi linear first-order nonhomogeneous POEs in two independent variables, which can be written as
(III.20a) (IIL20b)
where the coefficients a to d and A to D and the nonhomogeneous terms e and E may depend on x, t,j, and g. The general features of these three special cases are similar to the general features of all the POEs discussed in this book. Consequently, these three special cases are studied thoroughly in the following sections.
