Variational Perturbation Method Research Articles

Existence and uniqueness of solution for a singular elliptic differential equation

Abstract In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: − Δ u − 1 2 ( x ⋅ ∇ u ) = μ h ( x ) u q − 1 + λ u − u p , x ∈ R N , u ( x ) → 0 , as ∣ x ∣ → + ∞ , \left\{\begin{array}{l}-\Delta u-\frac{1}{2}\left(x\cdot \nabla u)=\mu h\left(x){u}^{q-1}+\lambda u-{u}^{p},\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| x| \to +\infty ,\end{array}\right. where N ⩾ 3 N\geqslant 3 , 0 < q < 1 0\lt q\lt 1 , λ > 0 \lambda \gt 0 , p > 1 p\gt 1 , μ > 0 \mu \gt 0 is a parameter and the function h ( x ) h\left(x) satisfies certain conditions. To start with, based on the variational argument and perturbation method, we obtain the existence and uniqueness of the positive solution for the aforementioned singular elliptic differential equation as λ > N 2 \lambda \gt \frac{N}{2} . In addition, there is no solution as λ ⩽ N 2 \lambda \leqslant \frac{N}{2} . Later, from an experimental point of view, we give the numerical solution of the aforementioned singular elliptic differential equation by means of a neural network in some special cases, which enrich the theoretical results. Our conclusions partially extend the results corresponding to the nonsingular case.

Existence of nontrivial distributional solutions for a class of Schrödinger-Poisson system with Sobolev critical nonlinearity and zero mass

This paper studies a class of Schrödinger-Poisson system in R3 with Sobolev critical nonlinearity and zero mass −Δu+ϕu=λf(u)+u5,x∈R3;−Δϕ=u2,x∈R3, where λ is a positive parameter and f is a continuous function defined on R with some suitable assumptions. One nontrivial distributional solution is obtained by a combination of variational methods and perturbation method.

On the Numerical Solution of 1D and 2D KdV Equations Using Variational Homotopy Perturbation and Finite Difference Methods

The KdV equation has special significance as it describes various physical phenomena. In this paper, we use two methods, namely, a variational homotopy perturbation method and a classical finite-difference method, to solve 1D and 2D KdV equations with homogeneous and non-homogeneous source terms by considering five numerical experiments with initial and boundary conditions. The variational homotopy perturbation method is a semi-analytic technique for handling linear as well as non-linear problems. We derive classical finite difference methods to solve the five numerical experiments. We compare the performance of the two classes of methods for these numerical experiments by computing absolute and relative errors at some spatial nodes for short, medium and long time propagation. The logarithm of maximum error vs. time from the numerical methods is also obtained for the experiments undertaken. The stability and consistency of the finite difference scheme is obtained. To the best of our knowledge, a comparison between the variational homotopy perturbation method and the classical finite difference method to solve these five numerical experiments has not been undertaken before. The ideal extension of this work would be an application of the employed methods for fractional and stochastic KdV type equations and their variants.

Parameter-dependent multiplicity results of sign-changing solutions for quasilinear elliptic equations

Existence of sign-changing solutions to quasilinear elliptic equations of the form [Formula: see text] under the Dirichlet boundary condition, where [Formula: see text] ([Formula: see text]) is a bounded domain with smooth boundary and [Formula: see text] is a parameter, is studied. In particular, we examine how the number of sign-changing solutions depends on the parameter [Formula: see text]. In the case considered here, there exists no nontrivial solution for [Formula: see text] sufficiently small. We prove that, as [Formula: see text] becomes large, there exist both arbitrarily many sign-changing solutions with negative energy and arbitrarily many sign-changing solutions with positive energy. The results are proved via a variational perturbation method. We construct new invariant sets of descending flow so that sign-changing solutions to the perturbed equations outside of these sets are obtained, and then we take limits to obtain sign-changing solutions to the original equation.

Approximate Solution of Generalized Modified b-Equation by Optimal Auxiliary Function Method

In this study, the implantation of a new semi-analytical method called the optimal auxiliary function method (OAFM) has been extended to partial differential equations. The adopted method was tested upon for approximate solution of generalized modified b-equation. The first-order numerical solution obtained by OAFM has been compared with the variational homotopy perturbation method (VHPM). The method possesses the auxiliary function and control parameters which can be easily handled during simulation of the nonlinear problem. From the numerical and graphical results, we concluded the method is very effective and easy to implement for the nonlinear PDEs.

Flexoelectricity and the entropic force between fluctuating fluid membranes

Biological membranes undergo noticeable thermal fluctuations at physiological temperatures. When two membranes approach each other, they hinder the out-of-plane fluctuations of the other. This hindrance leads to an entropic repulsive force between membranes which, in an interplay with attractive and repulsive forces owing to other sources, affects a range of biological functions: cell adhesion, membrane fusion, self-assembly, binding–unbinding transition among others. In this work, we take cognizance of the fact that biological membranes are not purely mechanical entities and, owing to the phenomenon of flexoelectricity, exhibit a coupling between deformation and electric polarization. The ensuing coupled mechanics–electrostatics–statistical mechanics problem is analytically intractable. We use a variational perturbation method to analyze, in closed form, the contribution of flexoelectricity to the entropic force between two fluctuating membranes and discuss its possible physical implications. We find that flexoelectricity leads to a correction that switches from an enhanced attraction at close membrane separations and an enhanced repulsion when the membranes are further apart.

Analysis of Nonlinear Active Noise Behavior of Fuzzy Controller Using Non-Perturbation Methods

In this paper, a Fuzzy controller model has been converted into a time-dependent nonlinear model and then quadratic Riccati differential equation was analyzed to satisfy the solution of the nonlinear active noise behavior of Fuzzy controller. Further, the approximate solutions of this equation using non-perturbation methods i.e., adomian decomposition method (ADM), variational iterational method (VIM) and homotopy perturbation method (HPM) were investigated. A comparison of these methods has also been given with tabular and graphical presentations. Our results reveal that VIM provides the closest approximate solution and fast convergence for the proposed model as compare to ADM and HPM.

Buckling and Post-buckling Analysis of Flatness Deflects in High-strength automotive sheet

Shape defect is a common problem in the production of automobile plate. But, the wave type function used by previous scholars can not reflect the actual situation of the site well. To solve this problem, high-order trigonometric functions are used to describe wave patterns, it expresses the fact that the wave pattern does not cover the whole width. Based on the buckling theory of plates, the shape defect was investigated in this paper. Using S.Timoshenko principle of least work, a new mechanical model of longitudinal buckling was established. And the mathematic mode of critical buckling was achieved by Galerkin’s principle of virtual displacement, the path generation of post-buckling was found with Perturbation-Variational Method, and the mode of post-bucking was achieved. The rolling experiment was carried out in test rolling mill. In this experiment, variety of different shape defect model were obtained by designing different roll-shaped curves for work roll, and the mode was good in according to the analytical and numerical results.

Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations

In the present research work, a newly developed technique which is known as variational homotopy perturbation transform method is implemented to solve fractional-order acoustic wave equations. The basic idea behind the present research work is to extend the variational homotopy perturbation method to variational homotopy perturbation transform method. The proposed scheme has confirmed, that it is an accurate and straightforward technique to solve fractional-order partial differential equations. The validity of the method is verified with the help of some illustrative examples. The obtained solutions have shown close contact with the exact solutions. Furthermore, the highest degree of accuracy has been achieved by the suggested method. In fact, the present method can be considered as one of the best analytical techniques compared to other analytical techniques to solve non-linear fractional partial differential equations.

Localized nodal solutions for quasilinear Schrödinger equations

In this paper, we study the existence of localized nodal solutions for a class of semiclassical quasilinear Schrödinger equations including, as a special case, the Modified Nonlinear Schrödinger Equation (MNLS)ε2(Δv+12vΔv2)−V(x)v+|v|q−2v=0,inRN,v(x)→0as|x|→∞. We establish for small ε the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function, by developing new variational perturbation method to treat this class of non-smooth variational problems. The new method allows the perturbed variational functionals to share critical points with the original functional. This method allows us to avoid any limiting process from the perturbed problems to the original problem, and it is effective in dealing with multiple existence of solutions.

Variational homotopy perturbation method for solving Lane-Emden initial value problems

Variational homotopy perturbation method for solving Lane-Emden initial value problems

Sharp-interface approach for simulating solid-state dewetting in two dimensions: A Cahn–Hoffman [formula omitted]-vector formulation

By using a Cahn–Hoffman ξ-vector formulation, we propose a sharp-interface approach for solving solid-state dewetting problems in two dimensions. First, based on the thermodynamic variation and smooth vector-field perturbation method, we rigorously derive a sharp-interface model with weakly anisotropic surface energies, and this model describes the interface evolution which occurs through surface diffusion flow and contact line migration. Second, a parametric finite element method in terms of the ξ-vector formulation is proposed for numerically solving the sharp-interface model. By performing numerical simulations, we examine several specific evolution processes for solid-state dewetting of thin films, e.g., the evolution of small islands, pinch-off of large islands and power-law retraction dynamics of semi-infinite step films, and these simulation results are consistent with experimental observations. Finally, we also include the strong surface energy anisotropy into the sharp-interface model and design its corresponding numerical scheme via the ξ-vector formulation.

New analytical solution for stream\u2013aquifer interaction under constant replenishment

In the present paper, approximate analytical solutions for flow in unconfined aquifers with streams are obtained assuming one-dimensional horizontal groundwater flow in homogeneous and isotropic aquifer with recharge effect using variational homotopy perturbation method which is a combination of variational iteration method and homotopy perturbation method. The governing nonlinear Boussinesq equation is obtained by using a basic principle of conservation of mass followed by Dupuit’s assumptions. Sensitivity of the parameters has been analyzed. The obtained approximate analytical solutions are numerically validated. The present analytical method is reliable and can be equally well applied to other nonlinear equations arising in the stream–aquifer interaction problems.

A comparative analysis for the solution of nonlinear Burgers' equation.

In this work, a comparative study of seven well-known mathematical techniques for the coupled Burgers' equations is reported. The techniques involve in this comparison are as follows: Laplace transform Adomian decomposition method, Laplace transform homotopy perturbation method, Variational iteration method, Variational iteration decomposition method, Variational iteration homotopy perturbation method, the optimal homotopy asymptotic method, and OHAM with Daftardar-Jafari polynomial. Here we considered a practical example which consists of coupled Burgers' equations with the kinematic viscosity ε=1. Convergence and stability analysis is a major part of this analysis. After a careful observation, it is found that the variational iteration method has faster convergence than all the remaining methods. Adomian decomposition method and Homotopy perturbation method show weaker stability in comparison with other involved techniques.

A diagrammatic analysis of the variational perturbation method for classical fluids.

The statistical mechanics of classical fluids can be approached from the particle perspective, where the focus is on the various positions and interactions of the particles, and from the field perspective, where the focus is on the form of the interaction fields generated by the particles. In this work, we combine these two perspectives by examining the variational perturbation method for classical fluids, which has been widely used to describe nonuniform electrolyte systems. Most of this work has been for low orders of the approximation, it has been limited to cases where the electrostatic interactions are weak. We present an exact diagrammatic representation of the method, which greatly facilitates the enumeration and evaluation of higher order corrections to the free energy functional. This framework is able to encapsulate several different approximate theories. Performing a cumulant expansion, leads to the Debye-Hückel and higher order corrections. Including the contribution of chain diagrams leads to a theory closely related to the splitting theory [Hatlo and Lue, Europhys. Lett., 2010, 89, 25002], which has been shown to be accurate from the weak through to the strong coupling limits. Including all chain and ring diagrams leads to the hypernetted chain approximation; this is a more direct route to the conventional derivation, which also requires a renormalization of the Mayer f-bonds to the total correlation functions. These approximations to the variational perturbation method are applied to the classical one-component plasma in order to assess their relative accuracy and understand their relationship to each other. Strategies for developing improved approximations are discussed.

Dispersion managed dissipative solitons with higher order nonlinearity and frequency-selective feedback

We present the generation and interaction dynamics of dispersion managed soliton in a dissipative cubic-quintic nonlinear optical fiber coupled with frequency-selective feedback. Analytically, perturbative variational method is used to obtain evolution equations for different soliton parameters that are subsequently solved to study the propagation characteristics of the dispersion managed dissipative solitons (DMDS). The DMDSs show breather like characteristics and are found to be robust against certain level of initial noise. Both the standard Gaussian and more generic super-Gaussian pulse profile are shown to yield stable DMDS. Interaction of two DMDSs of same phase may lead to a bound state. By varying the temporal separation and phase difference between the DMDSs the bound state can be switched between low to high speed regime.

Analysis of nano droplet dynamics with various sphericities using efficient computational techniques

Motion of a vertically falling nano droplet in incompressible Newtonian media with initial velocity is investigated. The instantaneous velocity and acceleration are carried out by using the variational iteration method (VIM) and homotopy perturbation method (HPM), which are analytical solution techniques. The obtained results are compared with Runge–Kutta method in order to verify the accuracy of the proposed methods. The results show that, the analytical solutions are in good agreement with each other and with the numerical solution. Also, the effects of sphericity (φ) on the velocity and acceleration profiles of the nano droplet are explained. Moreover, the results demonstrate that the VIM-Pade and HPM-Pade are very effective in generating analytical solutions for even highly nonlinear problems.

Thermal fluctuations and effective bending stiffness of elastic thin sheets and graphene: A nonlinear analysis

The study of statistical mechanics of thermal fluctuations of graphene—the prototypical two-dimensional material—is rendered rather complicated due to the necessity of accounting for geometric deformation nonlinearity. Unlike fluid membranes such as lipid bilayers, coupling of stretching and flexural modes in solid membranes like graphene leads to a highly anharmonic elastic Hamiltonian. Existing treatments draw heavily on analogies in the high-energy physics literature and are hard to extend or modify in the typical contexts that permeate materials, mechanics and some of the condensed matter physics literature. In this study, using a variational perturbation method, we present a “mechanics-oriented” treatment of the thermal fluctuations of elastic sheets such as graphene and evaluate their effect on the effective bending stiffness at finite temperatures. In particular, we explore the size, pre-strain and temperature dependency of the out-of-plane fluctuations, and demonstrate how an elastic sheet becomes effectively stiffer at larger sizes. Our derivations provide a transparent approach that can be extended to include multi-field couplings and anisotropy for other 2D materials. To reconcile our analytical results with atomistic considerations, we also perform molecular dynamics simulations on graphene and contrast the obtained results and physical insights with those in the literature.

Numerical Solutions of a Quadratic Integral Equations by Using Variational Iteration and Homotopy Perturbation Methods

In this paper, the approximate solutions for quadratic integral equations (QIEs) are given by the variational iteration method(VIM) and homotopy perturbation method (HPM). These methods produce the solutions in terms of convergent series without needing to restrictive assumptions, to illustrate the ability and credibility of the methods, we deal with some examples that show simplicity and effectiveness.

Variational Homotopy Perturbation Method for Solving Riccati Type Differential Problems

In this paper, a Variational homotopy perturbation method is proposed to solve nonlinear Riccati differential equation. By combining the Variational Iteration Method and the Homotopy Perturbation Method, this technique possesses a fast convergence rate with high accuracy. The results reveal that the proposed method is very effective and simple.