scholarly journals On Initial Value and Boundary Value Problem of Linear Differential Algebra System

2019 ◽  
Author(s):  
Shu-yan YAN
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Differential Algebra ◽  
Initial Value ◽  
Linear Differential

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

scholarly journals A Comparison between the fourth order linear differential equation with its boundary value problem

2020 ◽  
Vol 99 (3) ◽  
pp. 18-25
Author(s):  
Karwan H.F. Jwamer ◽  
◽  
Rando R.Q. Rasul ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Fourth Order ◽  
Upper Bound ◽  
Boundary Value ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential

In this paper, we study a fourth order linear differential equation. We found an upper bound for the solutions of this differential equation and also, we prove that all the solutions are in L4(0, ∞). By comparing these results we obtain that all the eigenfunction of the boundary value problem generated by this differential equation are bounded and in L4(0, ∞).

scholarly journals NONSINGULAR INTEGRO-DIFFERENTIAL BOUNDARY VALUE PROBLEM NOT SOLVED WITH RESPECT TO THE DERIVATIVE

2020 ◽  
Vol 8 (2) ◽  
pp. 127-138
Author(s):  
S. Chuiko ◽  
O. Chuiko ◽  
V. Kuzmina
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Systematic Study ◽  
Critical Case ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Linear Boundary ◽  
Actual Problem ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boi- chuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of linear boundary-value problems for ordinary di- fferential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the linear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear osci- llations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the linear boundary value problems for the integro-differential boundary value problem not solved with respect to the derivative, in parti- cular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear integro-differential boundary value problem not solved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian integro-differential boundary value problem not solved with respect to the derivative. The proposed scheme of the research of the nonlinear Noetherian integro-differential boundary value problem not solved with respect to the derivative in the critical case in this article can be transferred to the seminonlinear integro-differential boundary value problem not solved with respect to the derivative.

A Method for the Numerical Solution of a Boundary Value Problem for a Linear Differential Equation with Interval Parameters

2019 ◽  
Vol 16 (07) ◽  
pp. 1850115 ◽  
Author(s):  
Nizami A. Gasilov ◽  
Müjdat Kaya
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Numerical Method ◽  
Boundary Value ◽  
Real Life ◽  
Boundary Values ◽  
Right Hand ◽  
Linear Differential ◽  
The Right

In many real life applications, the behavior of the system is modeled by a boundary value problem (BVP) for a linear differential equation. If the uncertainties in the boundary values, the right-hand side function and the coefficient functions are to be taken into account, then in many cases an interval boundary value problem (IBVP) arises. In this study, for such an IBVP, we propose a different approach than the ones in common use. In the investigated IBVP, the boundary values are intervals. In addition, we model the right-hand side and coefficient functions as bunches of real functions. Then, we seek the solution of the problem as a bunch of functions. We interpret the IBVP as a set of classical BVPs. Such a classical BVP is constructed by taking a real number from each boundary interval, and a real function from each bunch. We define the bunch consisting of the solutions of all the classical BVPs to be the solution of the IBVP. In this context, we develop a numerical method to obtain the solution. We reduce the complexity of the method from [Formula: see text] to [Formula: see text] through our analysis. We demonstrate the effectiveness of the proposed approach and the numerical method by test examples.

Adjoint Interior-Point Boundary Conditions for Linear Differential Operators

1977 ◽  
Vol 20 (4) ◽  
pp. 447-450 ◽  
Author(s):  
Robert Neff Bryan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Interior Point ◽  
Differential Operators ◽  
Boundary Value ◽  
Point Boundary ◽  
Linear Differential Operators ◽  
Linear Differential

The investigations reported in this paper were prompted by a remark by A. M. Krall in [2] that certain functional which appear in the boundary conditions of the system adjoint to a given linear differential boundary value problem seem artificial in that setting.

Method of Forced Monotonicity for Conjugate type Boundary Value Problems for Ordinary Differential Equations

1988 ◽  
Vol 31 (1) ◽  
pp. 79-84
Author(s):  
P. W. Eloe ◽  
P. L. Saintignon
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Linear Differential Operator ◽  
Boundary Value ◽  
Linear Differential ◽  
Type Boundary ◽  
Sign Conditions ◽  
Conjugate Type

AbstractLet I = [a, b] ⊆ R and let L be an nth order linear differential operator defined on Cn(I). Let 2 ≦ k ≦ n and let a ≦ x1 < x2 < … < xn = b. A method of forced mono tonicity is used to construct monotone sequences that converge to solutions of the conjugate type boundary value problem (BVP) Ly = f(x, y),y(i-1) = rij where 1 ≦i ≦ mj, 1 ≦ j ≦ k, mj = n, and f : I X R → R is continuous. A comparison theorem is employed and the method requires that the Green's function of an associated BVP satisfies certain sign conditions.

STABILITY OF EQUILIBRIUM STATES OF NONLINEAR STRUCTURES AND CHAOS PHENOMENON.

1992 ◽  
Vol 02 (02) ◽  
pp. 271-283 ◽  
Author(s):  
D. SHILKRUT
Keyword(s):  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Geometrically Nonlinear ◽  
Geometrical Approach ◽  
Initial Value ◽  
Nonlinear Structure ◽  
Deterministic Systems ◽  
Classical Chaos ◽  
Problem Characteristic

The “classical” chaos of deterministic systems is characteristic for the motion of dynamical systems. Recently, some attempts were made to find static analogies of chaos [Thompson & Virgin, 1988; Naschie & Athel, 1989; Naschie, 1989]. However, this was considered for structures in specific artificial conditions (for example, infinitely long bars with sinusoidal geometric imperfections) transferring de facto the boundary value problem (which always describes static deformation of structures) into an initial value problem characteristic for problems of motion. In this article, chaotic (unpredictable) behavior is described for a usual (not special) nonlinear structure in statics, which is governed, naturally, by a boundary value problem in a finite interval of the argument. The behavior of this structure (geometrically nonlinear plate), which is an example of the class of static chaotic structures, is investigated by a new geometrical approach called the “deformation map.” The presented results are one of the first steps in the chapter of chaos in statics, and therefore the link between “classical” and static chaos needs further investigations.

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