Differential equations with highest partial derivatives (Q2784293)

The reviewed monograph gives the development of the known Riemann's method for the solving of Cauchy and Goursat problems for hyperbolic equations of the second-order to equations with highest partial derivatives \(D\equiv D_n= \partial^n/\partial x_1\partial x_2\cdots\partial x_n\) of the form NEWLINE\[NEWLINE(D+ M)u= f(x),\tag{\(*\)}NEWLINE\]NEWLINE where \(M\) is the differential operator of the lower than \(n\) order. The suggested modification consists in the usage of Riemann's identity in another form. Moreover the Riemann's function is determined as the solution of some integral equation, opposite to usual situation of its determination as the solution of adjoint equation satisfying the boundary conditions, the number of which is increasing with the growth of the order \(n\) of the equation. This development is given in the Ch. 1 for the cases of \(n= 2,3,4\) with various variants of integral equations for the determination of Riemann's function (RF). Here the main attention is payed to the construction of the solutions of Cauchy and Goursat problems in terms of RF.NEWLINENEWLINE Ch. 2 is devoted to various characteristical problems (i.e. of Goursat type) with normal derivatives in boundary conditions, at first for the cases of \(n= 2\) and \(3\), and then for the case of any finite dimensions.NEWLINENEWLINE In Ch. 3 the equations of the form \((*)\) are studied, where the operator \(D\) is changed by the derivative of it on one or two independent variables. Also are considered the polylinear equations with iterations of the operator \(D\) or some others, close to \(D\) on their properties.NEWLINENEWLINE In Ch. 4 the general characteristical problems for PDE systems of the first-order are investigated by the Riemann's method modification. For every of the considered problems the existence and uniqueness of solution theorems are preliminary proved.