On linear partial equations of first-order (Journal of Mathematics Mechanics and Computer Science, 2018)
We study a linear differential equation with first-order partial derivatives, where the coefficientsof the equation are given on an unbounded set and have continuous first-order partial derivatives.Each partial differential equation is closely related to a system of ordinary differential equations,a system of so-called characteristic equations of a given first-order partial differential equation.Each first-order partial differential equation under certain conditions has a fundamental system ofintegrals or an integral basis. We note that for a general linear partial differential equation of thefirst order there can be no nontrivial integral. For a linear first-order partial differential equation,where the coefficients of the equation are given on an unbounded set and have continuous firstorderpartial derivatives, with the first coefficient equal to one, an integral basis exists. For alinear first-order partial differential equation, where the coefficients of the equation are given onan unbounded set and have continuous first-order partial derivatives, with the first coefficientequal to one, an integral basis exists. For a linear first-order partial differential equation, we definethe asymptotic stability of a linear first-order partial differential equation. A sufficient conditionfor the asymptotic stability of a linear partial differential equation of the first order is given. Atpresent, the theory of partial differential equations finds its application in various fields of naturalscience.
Keywords: equation; first order partial derivatives