Differential and Integral Equations

First-order partial differential equations and Henstock-Kurzweil integrals

Tuan Seng Chew, B. Van-Brunt, and G. C. Wake

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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.

Article information

Differential Integral Equations, Volume 10, Number 5 (1997), 947-960.

First available in Project Euclid: 1 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35F25: Initial value problems for nonlinear first-order equations
Secondary: 26A39: Denjoy and Perron integrals, other special integrals


Chew, Tuan Seng; Van-Brunt, B.; Wake, G. C. First-order partial differential equations and Henstock-Kurzweil integrals. Differential Integral Equations 10 (1997), no. 5, 947--960. https://projecteuclid.org/euclid.die/1367438627

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