Solutions Of Nonlinear Boundary Value Problems Research Articles

Deformations of Single-Crystal Silicon Circular Plate: Theory and Experiment

In this paper, the experimental methodology for the single-crystal circular plate deformation measurement and subsequent procedure for the quantitation of its mechanical properties are developed. The procedure is based on a new numerical-analytical solution of non-linear boundary-value problem for finite deformations of a circular anisotropic plate. Using the developed method, a study of the deformation of single-crystal circular plates formed on the basis of a silicon-on-insulator structure was carried out. The values of residual stresses are determined and it is shown that the presence of these stresses increases the flexural rigidity of the plate by several times.

On Approximate Solutions of Nonlinear Boundary-Value Problems by the Newton–Kantorovich Method

We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton–Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Lienard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Lienard equations, we use discrepancies in the original equations.

Fractal quintic spline method for nonlinear boundary-value problems

In this article, numerical solutions of nonlinear boundary-value problems are obtained using fractal quintic spline. Convergence analysis of the proposed method is also established. Proposed method has fourth-order convergence. Numerical examples are provided to show practical usefulness of the method and numerical results are compared with the existing numerical methods.

On Solutions of Nonlinear Boundary-Value Problems Whose Components Vanish at Certain Points

We show how an appropriate parametrization technique and successive approximations can help to investigate nonlinear boundary-value problems for systems of differential equations under the condition that the components of solutions vanish at certain unknown points. The technique can be applied to nonlinearities involving the signs of the absolute value and positive or negative parts of functions under boundary conditions of various types.

Variational Method for the Solution of Nonlinear Boundary-Value Problems of the Dynamics of Bounded Volumes of Liquid with Variable Boundaries

We study nonlinear boundary-value problems of the hydrodynamic type formulated for the domains whose boundaries are unknown in advance. Our investigations are based on the variational formulations of these problems with the help of specially introduced integral functionals with variable domains of integration. It is shown that the required solutions of the boundary-value problems are, in a certain sense, equivalent to finding stationary points of the analyzed functionals. These are pairs formed by the families of domains and functions defined in these domains. By analyzing an example of the problem of space motion of a vessel with elastic walls partially filled with an ideal incompressible liquid, we propose a method for the construction (in the analytic form) both of the required solutions and of the boundaries of domains (deformed walls and the shape of the perturbed free liquid surface).

Multiple Solutions of Nonlinear Boundary-value Problems for Ordinary Differential Equations

We consider a Hamilton system related to the Trott curve in Harnack’s theorem. This theorem says that the maximal number of ovals for the fourth-order curve is four. We consider the related Hamilton system which has more ovals than prescribed by Harnack’s theorem. We give an explanation and consider the Dirichlet boundary-value problem for the system. Precise estimation is given for the number of solutions to the Dirichlet problem.

Existence and multiplicity of symmetric positive solutions for nonlinear boundary-value problems with p-Laplacian operator

In this paper, we establish the existence and multiplicity of symmetric positive solutions for a class of p-Laplacian fourth-order differential equations with integral boundary conditions. Our proofs use the Leray-Schauder nonlinear alternative and Krasnoselkii’s fixed-point theorem in cones.

On the parametrization of boundary-value problems with two-point nonlinear boundary conditions

We obtain some results for the solutions of nonlinear boundary-value problems of a certain type with two-point nonlinear boundary conditions and reveal the efficiency of the procedure of reduction of the analyzed problem to a parametrized boundary-value problem with linear boundary conditions containing certain artificially introduced parameters. To study the transformed two-point problem, we propose a method based on the approximations of a special type constructed in the analytic form. It is shown that these approximations uniformly converge to a parametrized limit function and establish the relationship between this function and the exact solution. The proposed procedure leads to a certain system of algebraic equations whose solutions give numerical values of the parameters corresponding to the solution of the given two-point nonlinear boundary-value problem.

Analysis of bifurcation bending modes of arches and panels

This paper reports the results of numerical analysis of the bifurcation solutions of nonlinear boundary-value problems of plane bending of elastic arches and panels. Problems are formulated for a system of six nonlinear ordinary differential equations of the first order with independent fields of finite displacements and rotations. Two loading versions (by follower and conservative pressures) and two versions of boundary conditions (rigid clamping and pinning) are considered. In the case of clamped arches and panels, the set of solutions consists of symmetric and asymmetric bending modes which exist only for positive values of the load parameter. In the case of pinning, the set of solutions includes symmetric and asymmetrical modes which correspond to positive, negative, and zero values of the parameter. In both problems, the phase relations between the state parameter and the load parameter are bifurcated, ambiguous, have isolated branches, and admit a catastrophe — a finite jump from the fundamental equilibrium mode to a buckled mode.

Mixed projection-mesh scheme of the finite-element method to solve boundary-value problems describing the non-isothermal processes of elastoplastic deformation

A mixed projection-mesh scheme for solving a boundary-value problem of thermal plasticity is formulated in a quasi-static statement when the process of non-isothermal elastoplastic deformation of a body is a sequence of equilibrium states. In this case, the stress-strain state depends on the loading history, and the process of inelastic deformation is to be observed over the whole time interval under study. The correctness and convergence of the mixed approximations for stresses, strains and displacements are investigated as applied to the solution of nonlinear boundary-value problems that describe the non-isothermal processes of active loading taking into account the initial strains dependent on the history of deformation and heating. The properties of the projecting operators are studied in detail, and on this basis, the condition that ensures the existence, uniqueness and stability of solution is formulated. The results of the analysis of special formulas of the interpolation-type numerical integration are presented, the use of which considerably simplifies the computation procedure for solving equations of the mixed method. The convergence and accuracy estimations are based on the results of the theory of the generalized boundary-value problems and methods of the functional analysis. According to the estimations obtained, the accuracy of solution of a finite-dimensional problem at the initial stages of loading should be sufficient to avoid the effect of increase of the first coefficients in the expansion of the total error on the accuracy of solution of the elastoplastic problem at the subsequent stages of loading.

Spectral factorization in the disk algebra

Every strictly positive function f, given on the unit circle of the complex plane, defines an outer function. This article investigates the behavior of these outer functions on the boundary of the unit disk. It is shown that even if the given function f on the boundary is continuous, the corresponding outer function is generally not continuous on the closure of the unit disk. Moreover, any subset E∈ [-π ,π) of Lebesgue measure zero is a valid divergence set for outer functions of some continuous functions f. These results are applied to study the solutions of non-linear boundary-value problems and the factorization of spectral density functions.

Iteration Algorithms for Solving Boundary-Value Problems of the Theory of Small Elastic-Plastic Strains on the Basis of the Mixed Finite Element Method

A mixed projection-mesh scheme has been formulated for the solution of nonlinear boundary-value problems of the theory of small elastic-plastic strains in displacements, strains, and stresses. Iteration algorithms of the solution of nonlinear equations of the mixed method have been considered. Numerical results of the solution of two model problems are presented. The data obtained on the basis of the classical and mixed finite element methods are compared.

Mixed projection-mesh scheme of the finite-element method for the solution of the boundary-value problems of the theory of small elastic-plastic strains

A mixed projection-mesh scheme for the solution of nonlinear boundary-value problems of the theory of small elastic-plastic strains has been formulated. Correctness and convergence of the mixed approximations for stresses, strains, and displacements have been analyzed. The properties of projection operators are studied in detail, and on the basis of the results obtained, a condition has been formulated, which ensures the existence, uniqueness, and stability of the solution to a discrete problem. Application of the numerical integration has been analyzed and the obtained results are presented. The correctness and convergence estimates are based on the theory of generalized functions and the functional analysis method.

Interpolation Based Numerical Procedure for Solving Two-Point Nonlinear Boundary Value Problems

The present paper offers a simple, efficient and accurate straightforward numerical method for solving two-point nonlinear boundary value problems (NBVPs), which are frequently encountered in a broad discipline of sciences. The proposed technique is based on interpolation procedures and utilizes the capabilities of readily applicable functions, which are embedded in modern commercial mathematical software (Mathematica). The whole approach is computationally effortless, does not contain the drawbacks of other existing methods related to the solution of NBVPs and under the restriction of a sufficient guess of the interpolation domains is capable of tackling a variety of such problems, originated from different scientific areas. Several applications of the method produce results in excellent agreement with existing ones of the relevant literature, the latter being the product of far more sophisticated algorithms.

Elastoplastic state of nonshallow flexible ellipsoidal shells under concentric loading

The elastoplastic stress-strain state of flexible ellipsoidal shells under concentric loading is investigated by the solution of nonlinear boundary-value problems. Numerical results are presented for open shells in the case of a free (unsupported) edge and a shell with a central opening (the periphery of the opening is supported by a thin ring). Solutions of the problems in linear and nonlinear statements are analyzed. Numerical results for ellipsoidal and spherical shells are compared.

Some remarks concerning the convergence of the numerical-analytic method of successive approximations

We establish new improved estimates necessary for the justification of the numerical-analytic method for the investigation of the existence and construction of approximate solutions of nonlinear boundary-value problems for ordinary differential equations.

Analytic Solution of Nonlinear Boundary-Value Problems in Several Dimensions by Decomposition

Nonlinear partial differential equations are systematically solved by the decomposition method of G. Adomian for general boundary conditions described by boundary operator equations. Linear partial differential equations and linear or nonlinear ordinary differential equations become special cases all dealt with by a single procedure.

Enhancing transformations and parameter estimation for analytical signals

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Approximate Solutions of Non-linear Boundary-value Problems

On decrit une methode pour obtenir des solutions approchees de problemes aux valeurs limites non lineaires, basee sur une application du principe de maximum. Cas de l'equation de Liouville

Stability of solutions of nonlinear boundary-value problems for hyperbolic systems

Stability of solutions of nonlinear boundary-value problems for hyperbolic systems