First-Order or Linear Equations

First we derive the method of characteristic lines for solving first-order quasi-linear partial differential equations, including boundary value problems; then we discuss the more sophisticated characteristic strip method. Exact first-order partial differential equations are also treated. We use the grading technique from representation theory to solve flag partial differential equations and find the complete set of polynomial solutions. Using the method of characteristic lines, we prove a Campbell–Hausdorff-type factorization of exponential differential operators and then solve the initial value problem of flag evolution partial differential equations. The factorization is also used to solve initial value problems of generalized wave equations of flag type. We prove that a two-parameter generalization of the Weyl function of type A is a solution of the Calogero–Sutherland model for quantum n-body systems in one dimension. If n=2, we find a connection between the Calogero–Sutherland model and the Gauss hypergeometric function. When n>2, we have a new class of multivariable hypergeometric functions. Finally, we use matrix differential operators and Fourier expansions to solve the Maxwell equations, the free Dirac equations, and a generalized acoustic system.