Solution of the cauchy problem for a hyperbolic equation with constant coefficients in the case of two independent variables

Abstract

On the plane, we consider a linear partial differential equation of arbitrary order of hyperbolic type. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. For the equation, we obtain an analytic form of the general solution, from which we single out the unique classical solution of the Cauchy problem.

Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

“…x к нулю, получим, что области (1) , Q (3) , Q (4) Q и (5) Q уменьшаются и в пределе становятся пустыми множествами, но их значения будут влиять на значения решения на характеристике 0, x at -= поскольку замыкание множеств (3) Q и (5) Q станет характеристикой 0, x at -= а замыкание (1) Q и (4) Q станет точкой (0, 0). В то же время области (2) Q и (6) Q останутся, и решение будет иметь вид (10) где функции (2) , u (3) , u (5) u и (6) u определены формулами (8) и (9) при * 0.…”

“…Существование классических решений многих задач зависит не только от правильного выбора вида граничных условий для дифференциальных уравнений с частными производными, но и от выполнения условий согласования заданных функций в угловых точках области [4; 5]. Как показано в [6][7][8][9][10], от вида условий согласования зависят гладкость решений и постановка задач.…”

Cauchy problem for some fourth-order nonstrictly hyperbolic equations

“…We have coefficients of equation (5) satisfy a (i) = a (j) with ∀i, j = 1, 4 and (b 2) ). According to equation (7), the general solution of equation (5) in this case has the form:…”

“…The operator appearing in the equation involves a composition of first-order differential operators. The Cauchy problem for such an equation was previously considered in [1,2] in the case of a strictly hyperbolic equation (a Petrovskii hyperbolic equation [3,4]). The general solution for both strictly and nonstrictly hyperbolic equations of arbitrary order was constructed there as well.…”

“…The general solution for both strictly and nonstrictly hyperbolic equations of arbitrary order was constructed there as well. The case of a nonstrictly hyperbolic equation with the coincidence of all characteristics was considered in [5], and the solutions of the Cauchy problem in all cases of a nonstrictly hyperbolic third-order equation of such a form were obtained in [6].…”