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We consider linear evolutionary systems of partial differential equations with constant coefficients of general form. We suppose that the matrix of operators at the higher time derivative of the sought vector-function is degenerate. These systems are called partial differential-algebraic equations (DAEs). The index is the most important characteristic defining the structure complexity of these equations. We discuss the ways of approach to the definition of index for partial DAEs and some related questions.

We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.

We study solvability conditions for a system of Volterra equations with some identically degenerate or rectangular matrix at the main term. Connection is discussed of the solvability conditions and applicability of numerical methods for solving these systems. In particular, the conditions of the convergence of the least squares method with the error functional defined in Sobolev spaces are presented.

We obtain an integral formula for a solution to a general algebraic equation. In this formula the integrand is an elementary function and integration is carried out over an interval. The advantage of this formula over the well-known Mellin formula is that the integral has a broader convergence domain. This circumstance makes it possible to describe the monodromy of a solution for trinomial equations.

Consider a system of equations ( a I A ) z = f , ( 1 ) where a is some number, I is the unity matrix, A is a symmetric matrix, z is the sought vector, and F is a known vector. A solution has the form z = ( a I A ) ' F . ( 2 ) A p p r o x i m a t i o n o f t h e i n v e r s e m a t r i x by an o p e r a t o r p o l y n o m i a l Pk(A) o f d e g r e e k y i e l d s an a p p r o x i m a t e s o l u t i o n z~ = P ~ ( A ) F . ( 3 ) The error ~k satisfies the inequality It~kll = Itz zh!l ~ II (al -A) i _ P~(A)11 ItF[I. ( 4 ) The no rm o f t h e s y m m e t r i c m a t r i x ( a I I ) -~ Pk (A) i s c o m p u t a b l e i n t h e m e t r i c o f Rn by means of the spectrum S(A) of the matrix A: E[ (aI -A)-* Ph (:1)][ = m,qx l (a )0 -1 Pk (k) I" ( 5 ) ~s(A) The best estimate of ]lekJ] is yielded by that operator polynomial Pk(A) which minimizes the quantity max I(a--k) -1-1p~(k)[ on the spectrum of the operator A. Usually the spectrum S(A) is unknown, and it is impossible to construct that polynomial. The error (4) can be minimized also under the conditions of incomplete information on the spectrum S(A). Let m, M be lower and upper estimates of the bounds of the spectrum S(A), and let a ~ [m, M]. From S ( A ) ~ [ m , M ] i t f o l l o w s t h a t m a x I (a ) . ) l P ~ ( X ) I < ~ max I ( a z ) -1 P h ( ~ ) [ . ( 6 ) ~.~S(A) x~[m,Ml Denote