The existence and uniqueness of solutions to the Cauchy problem for a first-order quasi-linear partial differential equation is studied in this paper using the Henstock-Kurzweil integral. The classical theory requires certain differentiability and continuity conditions on the coefficients of the derivatives in the equation. It is shown here that in the Henstock-Kurzweil integral setting these conditions can be relaxed and that the resulting solution is differentiable though the derivatives need not be continuous. This sharpens the classical result and provides a bridge between classical and weak solutions in the linear case.
"First-order partial differential equations and Henstock-Kurzweil integrals." Differential Integral Equations 10 (5) 947 - 960, 1997. https://doi.org/10.57262/die/1367438627