Value Problems For Differential-algebraic Equations Research Articles

Linear differential-algebraic boundary value problem with singular pulse influence

The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M. O. Bogolybov, A. D. Myshkis, A. M. Samoilenko, M. O. Perestyk and O. A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green’s operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action generalizes the results of S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.

Linear differential-algebraic boundary value problem with nonsingular pulse influence

The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N.N. Luzin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M.O. Bogolybov, A.D. Myshkis, A.M. Samoilenko, M.O. Perestyk and O.A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A.M. Samoilenko, M.O. Perestyuk and O.A Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green's operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with nonsingular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action neralizes the results of S. Campbell, A.M. Samoilenko, M.O. Perestyuk and O.A Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.

On sinc discretization method and block-tridiagonal preconditioning for second-order differential-algebraic equations

In this paper, we obtain a sinc discretization method for solving the boundary value problems of differential-algebraic equations (DAEs) and prove that the discrete solutions converge to the true solutions of the DAEs exponentially. In the process of presenting the solution, we find that the discrete solution is determined by a large and ill-conditioned linear system. To efficiently solve the linear system, we construct block-tridiagonal preconditioners for the iterative methods. Moreover, we derive the bounds for the eigenvalues of the preconditioned matrices. Numerical experiments are given to illustrate the effective performance and applicability of our method.

Numerical solution of differential-algebraic equations using the spline collocation-variation method

Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.

Two stage waveform relaxation method for the initial value problems of differential-algebraic equations

In this paper, we consider a two stage strategy for waveform relaxation (WR) iterations, applied to initial value problems for differential-algebraic equations (DAEs) in the form Aẏ(t)+By(t)=f(t). Outer iterations of TSWR are defined by MAẏ(k+1)(t)+M1y(k+1)(t)=N1y(k)(t)+NAẏ(k)(t)+f(t), where A=MA−NA, B=M1−N1, and each iteration y(k+1)(t) is computed using an inner iterative process, based on another splitting M1=M2−N2. Meanwhile, by the means of the Theta method, the discretized TSWR of DAEs is realized. Furthermore, when MA is an Hermitian positive semi-definite matrix with P-regular splittings, the convergence and the comparison theorems of TSWR are analyzed. Finally, the numerical experiments are presented.

The method of normal splines for linear DAEs on the number semi-axis

The method of normal spline-collocation (NSC), applicable to a wide class of ordinary linear singular differential and integral equations, is specified for the boundary value problems for differential-algebraic equations of second order on the number semi-axis. The method consists in minimization of a norm of the collocation systems' solutions in an appropriate Hilbert–Sobolev space. The NSC method does not use the notion of differentiation index and it is applicable to DAEs of any index as well as to equations not reducible to the normal form. The problems on the infinite interval can be solved in two ways. The first way is based on the use of the original space of functions defined on the semi-axis, and the second way is based on a singular transformation of the semi-axis into the unit segment. A new reproducing kernel, that provides the first way, is presented. An algorithm to create a non-uniform collocation grid is described.

On consistent initial conditions for circuit's DAEs with higher index

In this paper, we study efficient numerical methods for obtaining consistent initial conditions for systems of differential-algebraic equations (DAEs) with higher index arising e.g., from electronic circuits. We show that the class of Gear's backward differentiation formulas, unlike other multi-step techniques, are useful means for obtaining consistent initial conditions when carefully implemented. The new method does not employ burdensome techniques such as the QR factorization or singular value decomposition which require the order of n/sup s/ operations, where n is the size of the system. Therefore, these techniques are prohibitive for large circuits. The numerical experiments suggest that the method works reliably even for index-3 DAEs. Furthermore, the method is not only restricted to initial value problems for DAEs but can also be applied to solvers for boundary value problems based on shooting techniques.

Boundary value problems for differential-algebraic equations

This paper shows how certain regularization methods can be used to determine the solution structure of linear differential systems subject to linear constraints and boundary conditions. Attention is restricted to the relatively straightforward index-one problems and to certain index-two problems, where endpoint jumps are related to the underlying boundary layer structure of the corresponding singular perturbation problems.

A modified lobatto collocation for linear boundary value problems of differential-algebraic equations

The Lobatto collocation method is modified for efficiently solving linear boundary value problems of differential-algebraic equations with index 1. The stability and superconvergence of this method are established. Numerical implementations are discussed and a numerical example is given.