Solution of the Cauchy Problem for Two Differential-Algebraic Equations with a Fredholm Operator

2005 ◽  
Vol 41 (10) ◽  
pp. 1486-1489 ◽  
Author(s):  
S. P. Zubova
Keyword(s):  
Cauchy Problem ◽  
Fredholm Operator ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
The Cauchy Problem

Solving a second-order algebro-differential equation with respect to the derivative

2021 ◽  
pp. 414-420
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Rigid Body ◽  
Fredholm Operator ◽  
Arbitrary Dimension ◽  
Second Order ◽  
Index Zero ◽  
Second Order Differential Equations ◽  
The Cauchy Problem ◽  
One Step

We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.

scholarly journals Boundary value problems for systems of non-degenerate difference-algebraic equations

2019 ◽  
Keyword(s):  
Cauchy Problem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Linear Boundary ◽  
Algebraic Equations ◽  
Green Operator ◽  
Solvability Conditions ◽  
The Cauchy Problem ◽  
Generalized Green Operator

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 boundary value problems for difference equations, initiated in A.A. Markov, S.N. Bernstein, Ya.S. Besikovich, A.O. Gelfond, S.L. Sobolev, V.S. Ryaben'kii, V.B. Demidovich, A. Halanay, G.I. Marchuk, A.A. Samarskii, Yu.A. Mitropolsky, D.I. Martynyuk, G.M. Vayniko, A.M. Samoilenko, O.A. Boichuk and O.M. Stanzhitsky. Study of nonlinear singularly perturbed boundary value problems for difference equations in partial differences is devoted to the work of V.P. Anosov, L.S. Frank, P.E. Sobolevskii, A.L. Skubachevskii and A. Asheraliev. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A.M. Samoilenko and O.A. Boichuk on linear boundary value problems for difference-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 linear boundary value problem for a difference-algebraic equation. The solvability conditions are found in the paper, as well as the construction of a generalized Green operator for the Cauchy problem for a difference-algebraic system. The solvability conditions are found, as well as the construction of a generalized Green operator for a linear Noetherian difference-algebraic boundary value problem. An original classification of critical and noncritical cases for linear difference-algebraic boundary value problems is proposed.

Asymptotic solution of the Cauchy problem for the first-order equation with perturbed Fredholm operator

2020 ◽  
pp. 48-56
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Cauchy Problem ◽  
Small Parameter ◽  
Fredholm Operator ◽  
Gas Flow ◽  
First Order ◽  
Flexible Plates ◽  
The Cauchy Problem ◽  
Plates And Shells ◽  
Step Transition ◽  
The Right

We consider the Cauchy problem for a first-order differentialequation in a Banach space. The equation contains a small parameter in the highest derivative and a Fredholm operator perturbed by an operator addition on the right-hand side. Systems with small parameter in the highest derivative describe the motion of a viscous flow, the behavior of thin and flexible plates and shells, the process of a supersonic viscous gas flow around a blunt body, etc. The presence of a boundary layer phenomenon is revealed; in this case, even a small additive has a strong influence on the behavior of the solution. Asymptotic expansion of the solution in powers of small parameter is constructed by means of the Vasil’yeva- Vishik-Lyusternik method. Asymptotic property of the expansion is proved. To construct the regular part of the expansion, the equation decomposition method is used. It is consisted in a step-by-step transition to similar problems of decreasing dimensions.

LINEAR TRANSFORMATIONS AND FLOQUET THEOREM FOR LINEAR IMPLICIT DYNAMIC EQUATIONS ON TIME SCALES

2013 ◽  
Vol 06 (01) ◽  
pp. 1350004 ◽  
Author(s):  
Nguyen Huu Du ◽  
Nguyen Chi Liem
Keyword(s):  
Cauchy Problem ◽  
Time Scales ◽  
Differential Algebraic Equations ◽  
Dynamic Equations ◽  
Algebraic Equations ◽  
Linear Transformations ◽  
Difference Systems ◽  
Implicit Dynamic Equation ◽  
The Stability ◽  
Floquet Theorem

This paper is concerned with Cauchy problem, Lyapunov transformations, Floquet and Lyapunov theorems for linear implicit dynamic equation AtxΔ= Btx with index-1 on time scales. The stability of this equation under the act of these Lyapunov transformations is also considered. The results are a unification and generalization of previous results for differential-algebraic equations and implicit difference systems.

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