scholarly journals ELASTIC BENDING OF A THREE-LAYER CIRCULAR PLATE WITH STEP-VARIABLE THICKNESS

2021 ◽  
Vol 1 (54) ◽  
pp. 25-29
Author(s):  
Denis V. LEONENKO ◽  
Keyword(s):  
Boundary Value Problem ◽  
Circular Plate ◽  
Variable Thickness ◽  
Boundary Value ◽  
Median Plane ◽  
Inhomogeneous System ◽  
Equilibrium Equations ◽  
Elastic Circular Plate ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

The bending of a three-layer elastic circular plate with step-variable thickness is considered. To describe kinematics of asymmetrical in thickness core pack, the broken line hypotheses are accepted. In thin bearing layers, Kirchhoff’s hypotheses are valid. In a relatively thick filler incompressible in thickness, Timoshenko’s hypothesis on the straightness and incompressibility of the deformed normal is fulfilled. The formulation of the corresponding boundary value problem is presented. Equilibrium equations are obtained by the variational Lagrange method. The solution of the boundary value problem is reduced to finding three required functions in each section, deflection, shear and radial displacement of the median plane of the filler. An inhomogeneous system of ordinary linear differential equations is obtained for these functions. The boundary conditions correspond to rigid pinching of the plate contour. A parametric analysis of the obtained solution is carried out.

scholarly journals NON-AXISYMMETRIC DEFORMATION OF A CIRCULAR THREE-LAYER PLATE IN ITS OWN PLANE

2021 ◽  
Vol 1 (54) ◽  
pp. 38-45
Author(s):  
Eduard I. STAROVOITOV ◽  
◽  
Alina V. NESTSIAROVICH ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Axisymmetric Deformation ◽  
Linear Differential Equations ◽  
Radial Coordinate ◽  
Inhomogeneous System ◽  
Equilibrium Equations ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

A statement is given for the boundary value problem of non-axisymmetric deformation of an elastic threelayer circular plate in its own plane. The plate contour is pinched. Physical equations of state in the plate layers are described using the linear theory of elasticity, taking into account temperature influence on the elastic characteristics of materials. Equilibrium equations are obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the boundary value problem is reduced to finding the radial and tangential displacements in the layers of the plate. These displacements satisfy an inhomogeneous system of ordinary linear differential equations. To solve it, the method of decomposition into trigonometric Fourier series is applied. After substituting the series into the original system of equilibrium equations and performing the corresponding transformations, a system of ordinary linear differential equations is obtained to determine the four radial functions in each term of the series. The analytical solution is written out in the final form in the case of cosine radial and sinusoidal circumferential loads that depend linearly on the radial coordinate. The load is applied in the middle plane of the filler. Numerical approbation of the solution is carried out. The dependence of radial and tangential displacements on polar coordinates and temperature is investigated. Graphs of changes in displacements along the radius of the plate for different values of the angular coordinate are given. The weak dependence of displacements on temperature is illustrated when the plate contour is fixed.

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.

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