Katalin Balla

A Unified Approach to Linear Differential Algebraic Equations and their Adjoints

Instead of a single matrix occurring in the standard setting, the leading term of the linear differential algebraic equation is composed of a pair of well matched matrices. An index notion is proposed for the equations. The coefficients are assumed to be continuous and only certain subspaces have to be continuously differentiable. The solvability of lower index problems is proved. The solution representations are based on the solutions of certain inherent regular ordinary differential equations that are uniquely determined by the problem data. The assumptions allow for a unified treatment of the original equation and its adjoint. Both equations have the same index and are solvable simultaneously. Their fundamental solution matrices satisfy a relation that generalizes the classical Lagrange identity.

Linear differential algebraic equations of index 1 and their adjoint equations

For linear differential algebraic equations of tractability index 1 the notion of the adjoint equation is analysed in full detail. The solvability of this adjoint equation is shown at the lowest possible smoothness. The fundamental matrices of both equations are defined and their relationships are characterized.

Transfer of Boundary Conditions for DAEs of Index 1

In this paper, the concept of Abramovs method for transferring boundary conditions posed for regular ordinary differential equations (ODEs) is applied to index-1 differential algebraic equations (DAEs). Having discussed the reduction of inhomogeneous problems to homogeneous ones and analyzed the underlying ideas of Abramovs method, we consider boundary value problems for index-1 linear DAEs both with constant and varying leading matrices. We describe the relations defining the subspaces of solutions satisfying the prescribed boundary conditions at one end of the interval. The index-1 DAEs which realize the transfer are given and their properties are studied. The results are reformulated for inhomogeneous index-1 DAEs as well.

Boundary conditions and their transfer for differential-algebraic equations of index 1

Abstract For the homogeneous linear index 1-tractable DAE we give the characterization of the linear subspace defined by a homogeneous linear relationship prescribed at an arbitrarily fixed point.

Linear subspaces for linear DAEs of index 1

Abstract In this paper, the representation of linear subspaces of solutions of index 1 DAEs A ( x ) y ′ + B ( x ) y = 0 is given. It is shown that the solutions of adjoint systems are strongly related to the subspaces.

On the computation of Bessel functions of first kind

A new algorithm is proposed for the evaluation of the Bessel function valuesJp(x) with non-negative indexp and real positive argumentx. The basic ideas are the transfer of boundary conditions and the modified Prüfer transformation applied to Bessel equation. Special attention is paid to the accuracy of the interval reduction and the stability of the auxiliary initial value problems.ZusammenfassungZur Berechnung der BesselfunktionswerteJp(x) mit nichtnegativem Indexp und reellem Argumentx wird ein neuer Algorithmus empfohlen. Die Methode besteht aus der modifizierten Prüfer-Transformation der Besselschen Gleichung mit Übertragung der Randbedingungen. Besondere Aufmerksamkeit ist der Genauigkeit der Intervallreduktion und der Stabilität der Hilfsanfangswertprobleme gewidmet.

Asymptotic behavior of certain Riccati difference equations

Abstract Matrix Riccati difference equations are investigated on the infinite index set. Under natural assumptions an existence and uniqueness theorem is proven. The existence of the asymptotic expansion of the solution and computability of its coefficients are shown, provided the coefficients of the equation have such an expansion.

On the reduction of Holt's problem to a finite interval

The reduction of Holts problem to a finite interval is studied. We give the error estimate for the reduction. The solution of the boundary value problem is computed by a variant of the “chasing” method.

The simultaneous computation of Bessel functions of first and second kind

Abstract Based on the qualitative properties of Bessels differential equation and its solutions, a method is proposed for the simultaneous evaluation of Bessel functions of first and second kind. Special attention is paid to the numerical properties of the method and to the errors of approximation.

Air Pollution Modeling in Action

In the paper we describe an air pollution model as applied to a composting plant. Two technologies of the composting process are compared. The model is based upon the Hungarian National Standards. The region is analyzed from meteorological and geomorphologic points of view. For illustration we add grayscale maps of relative concentration for three cases.