Help Of Variational Method Research Articles

Multiple Normalized Solutions to a Choquard Equation Involving Fractional p-Laplacian in ℝN

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in RN. With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained for the above problem.

Kirchhoff equations with strong singularity in closed manifolds

In this paper, we study a class of Kirchhoff-type problems involving strong singular nonlinearities on compact Riemannian manifolds. With the help of variational method and Nehari method, we prove that this problem has a unique positive solution.

Existence results for fractional Dirichlet problems of p-Laplacian with instantaneous and non-instantaneous impulses on time scales

In this paper, by means of a relation between Riemann-Liouville fractional derivative and Caputo fractional derivative on time scales and with the help of variational methods and critical point theory, we obtain the existence results for a class of fractional Dirichlet problems of p-Laplacian with instantaneous and non-instantaneous impulses on time scales.

Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable s(x,·)-order fractional p1(x,·) and p2(x,·)-Laplacian. Assuming some reasonable conditions and with the help of variational methods, we reach a positive energy solution and a negative energy solution in an appropriate space of functions. The main difficulties and innovations are the Choquard nonlinearities and Kirchhoff functions with the presence of double Laplace operators involving two variable parameters.

Application of the variational method for studying stability of the 3D Lotka — Volterra system

The paper investigates the stability of the Lotka — Volterra dynamic system ≪predator — two preys≫ with the help of variational method. This method is applicable for studying equilibrium stability non-linear dynamical systems of any kind and in any dimension. It approves its effectiveness even in cases where the use of the Lyapunov function causes great difficulties (for example, when solving tasks of large dimension). The proposed method is based on a variation method for estimating the rate of phase-state metric change for the perturbed system when the variable t → ∞. The disadvantage of the variation method is the fact that this method provides only the necessary conditions of stability. However, its it forms the first step in studying the stability of non-linear dynamical systems of any kind and any dimension.

Free vibration and static analyses of metal-ceramic FG beams via high-order variational MFEM

There is not enough mixed finite element method (MFEM) model developed for static and dynamic analysis of functionally graded material (FGM) beams in the literature. The main purpose of this study is to develop a reliable and efficient computational modeling using an efficient functional in MFEM for free vibration and static analysis of FGM composite beams subject to high order shear deformation effects. The modeling of material properties was performed using mixture rule and Mori-Tanaka scheme which are more realistic determination techniques. This method based on the assumption that a two phase composite material consisting of matrix reinforced by spherical particles, randomly distributed in the beam. To explain the displacement components of the shear deformation effects, it was accepted that the shear deformation effects change sinusoidal. Partial differential field equations were obtained with the help of variational methods and then these equations were transformed into a novel functional for FGM beams with the help of Gâteaux differential derivative operator. Thanks to the Gâteaux differential method, the compatibility of the field equations was checked, and the field equations and boundary conditions were reflected to the function. A MFEM model was developed with a total of 10 degrees of freedom to apply the obtained functional. In the numerical applications section, free vibration and flexure problems solutions of FGM composite beams were compared with those predicted by other theories to show the effects of shear deformation, thickness changing and boundary conditions.

EXISTENCE OF WEAK SOLUTIONS FOR <i>Ψ</i>-CAPUTO FRACTIONAL BOUNDARY VALUE PROBLEM VIA VARIATIONAL METHODS

<abstract> This article is about a <italic>ψ</italic>-Caputo fractional boundary value problem which is investigated with the help of variational methods and critical point theory. A touchstone is obtained on the existence of the solution of the problem with the help of a functional. The problem is reduced into an equivalent form such that the solutions of the problem coincide with the critical points of a functional. Using aid from critical point theory, sufficient conditions are obtained for the existence of at least one solution. In the end, an example is also given to enrich our results. </abstract>

Experimental and theoretical investigation on flexure performance of pultruded GFRP composite beams with damage analyses

Fiber reinforced polymer (FRP) composite materials produced by pultrusion technique attract the attention of researchers due to their superior properties. Limited studies are available in the literature where experimental and theoretical investigations on the FRP composites are carried out together. In this study, the flexure performances of pultruded glass FRP (P-GFRP) composite beams were investigated experimentally and theoretically. Theoretical approaches for flexural analysis of P-GFRP composite beams were developed with the help of variational methods. Kinematic relations of composite beams were defined based on high order shear deformation beam theory. Effective material properties of composite beams were obtained by using mixture rule model. The differential field equations were transformed to functional with Gâteaux differential method. The element matrix with a total of 10 degrees of freedom was obtained by using the mixed finite element method (MFEM). Moreover, classical finite element modeling (FEM) was performed with the help of ABAQUS program. Various beam specimens with different fiber orientations were extracted from the P-GFRP box profile. Tensile and burnout tests were performed to determine the mechanical properties of the obtained P-GFRP composite beam specimens. Three-point bending test was utilized to investigate flexure performance. Kinetic, macro and micro mechanical damage analyzes were performed to better understand the behavior of the P-GFRP composite beams. Experimental results, theoretical solution results and FEM simulation results were compared and numerical values ​​were found to be very close to each other.

Ground state sign-changing solutions for the Schrödinger–Kirchhoff equation in [formula omitted

In this paper, with the help of variational methods in association with the deformation lemma and Miranda's theorem, we investigate the existence of a least energy sign-changing solution which has precisely two nodal domains for the following Schrödinger–Kirchhoff equation in R3:{−(a+b∫R3|∇u|2dx)Δu+V(x)u=f(u)in R3,u∈H1(R3), where a,b>0 and the potential V:R3→R+ is locally Hölder continuous and not necessarily radially symmetric.

Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids

This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary-value problems in (Math. Meth. Appl. Sci. 2006; 29:2225–2244) with the help of variational methods based on duality theory from convex analysis. In the present paper, it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn- and Friedrichs-type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate that contains only one constant coming from the following assertion: the L2 norm of a vector-valued function from H1(Ω) in the factor space generated by the equivalence with respect to rigid motions is bounded by the L2 norm of the symmetric part of the gradient tensor. As for some ‘simple’ domains such as squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains. Copyright © 2009 John Wiley & Sons, Ltd.

Influence of Built-In Internal Electric Field on Ferroelectric Film Properties and Phase Diagram

The influence of film-substrate mismatch effect, surface piezoelectric effect caused by broken symmetry on the film surface, surface tension and depolarization field on the ferroelectric film properties was considered in the phenomenological Ginzburg-Landau theory framework. The Euler-Lagrange equations for three polarization components were solved with the help of variational method proposed by us earlier. This approach leads to the free energy in the form of algebraic expression for different powers of polarization components with the coefficients dependent on film thickness, misfit strain, temperature etc. The odd powers of polarization related to built-in electric field normal to the surface originated from misfit strain and piezoelectricity in vicinity of surface appeared in the free energy expansion. The obtained free energy opens the way for the calculation of the ferroelectric film properties by conventional procedure of minimization. The calculations had shown that the built-in field drastically influences all the properties. In particular the smearing of polarization and dielectric permittivity, shift of its maximum temperature, the asymmetry of hysteresis loop were obtained. The latter speaks in favour of statement that the films self-polarization phenomenon originated from built-in field. The calculation of phase diagrams of PZT 50/50 compressive- or tensile-strained films was performed. The temperature behaviour of pyroelectric coefficient and dielectric permittivity lead to the forecast of the electret-like polar state, i.e. existence of pyroelectricity below the critical thickness of ferroelectric-paraelectric phase transitions. The possibility of ferroelectric phase conservation up to monolayer film on substrate for compressive strain was demonstrated.

Ferroelectric thin films phase diagrams with self-polarized phase and electret state

We calculated the three components of polarization in phenomenological theory framework by consideration of three Euler-Lagrange equations allowing for mismatch effect and influence of misfit dislocations, surface piezoelectric effect caused by broken symmetry on the film surface, surface tension, and depolarization field. The equations were solved with the help of variational method proposed earlier [M. D. Glinchuk et al., Physica B 332, 356 (2002)]. This approach leads to the free energy in the form of algebraic expression for different powers of polarization components with the coefficients dependent on film thickness, misfit strain, temperature, etc. The odd powers of polarization related to built-in electric field normal to the surface originated from misfit strain and piezoelectricity in the vicinity of the surface which appeared in the free energy expansion. The obtained free energy opens the way for the calculation of the ferroelectric film properties by conventional procedure of minimization. As an example we calculated phase diagrams of PbZr0.5Ti0.5O3 [PZT(50∕50)] compressive- or tensile-strained films. The temperature behavior of pyroelectric coefficient and dielectric permittivity lead to the forecast of the electretlike polar state, i.e., existence of pyroelectricity below the critical thickness of ferroelectric-paraelectric phase transitions. The mismatch-induced electric field was shown to be the physical nature of the self-polarization phenomenon in ferroelectric thin films.

Two-dimensional Hierarchical Models for Prismatic Shells with Thickness Vanishing at the Boundary

We construct variational hierarchical two-dimensional models for elastic, prismatic shells of variable thickness vanishing at boundary. With the help of variational methods, existence and uniqueness theorems for the corresponding two-dimensional boundary value problems are proved in appropriate weighted functional spaces. By means of the solutions of these two-dimensional boundary value problems, a sequence of approximate solutions in the corresponding three-dimensional region is constructed. We establish that this sequence converges in the Sobolev space H1 to the solution of the original three-dimensional boundary value problem.

Single-particle states and the connection between Jastrow and Brueckner theory

Starting from Jastrow theory we derive with the help of variational methods the first orders of the hole-line expansion for the ground-state energy and for the associated wave function of a finite Fermi system. The implications of a variation of the energy expansion with respect to the single-particle basis are discussed both for Jastrow theory and for Brueckner theory. We elucidate the mechanism which leads to the invariance of the complete hole-line expansion with respect to a change of the single-particle basis. As a consequence, variation of the complete expansion in either theory gives a non-trivial result only if that result was built into the expansion from the outset. Nonetheless, variation of the lowest-order terms of the expansion leads to definite results and yields the necessary conditions for natural orbitals.