Higher‐Order PDEs With PCGAs: A Multi‐Characteristic Approach

This paper addresses a multi‐characteristic problem for a higher‐order hyperbolic partial differential equation with a piecewise‐constant generalized argument. By introducing appropriate auxiliary unknowns, the original formulation is transformed into a parameter‐dependent problems for a system of first‐order differential equations with a piecewise‐constant generalized argument and accompanying integral relations. The analysis and resolution of this auxiliary set of problems are carried out by means of the parametrization Dzhumabaev method. A novel procedure for constructing solutions on subdomains is proposed. Sufficient conditions guaranteeing the existence and uniqueness of solutions to the resulting parameter‐dependent systems with piecewise‐constant generalized argument are established, and algorithms for their computation are developed; their convergence is rigorously justified. On this basis, conditions ensuring the unique solvability of the initial multi‐characteristic problem for higher‐order partial differential equations with piecewise‐constant generalized argument are derived.