Wide Class Of Nonlinear Problems Research Articles

On the nonlinear perturbations of self-adjoint operators

Abstract Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation T x = N ( x ) Tx=N\left(x) , where T T is a self-adjoint operator in a real Hilbert space ℋ {\mathcal{ {\mathcal H} }} and N N is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.

On some results of third-grade non-Newtonian fluid flow between two parallel plates

In this paper, the main focus is to discuss the non-Newtonian third-grade fluid flow between two parallel plates. The mathematical model is consequent to the continuity and momentum equations. Three cases have been discussed such as the plane Couette flow, the plane Poiseuille flow and the last one is a plane Couette-Poiseuille flow. The modeled differential equations are converted into dimensionless form by suitable non-dimensional parameters, then they are solved by the variational iteration method (VIM). It is noticed that the variational iteration method is convenient to apply and is very helpful for finding solutions of a wide class of nonlinear problems.

A New Homotopy Perturbation Scheme for Solving Singular Boundary Value Problems Arising in Various Physical Models

Abstract The classical homotopy perturbation method proposed by J. H. He, Comput. Methods Appl. Mech. Eng. 178, 257 (1999) is useful for obtaining the approximate solutions for a wide class of nonlinear problems in terms of series with easily calculable components. However, in some cases, it has been found that this method results in slowly convergent series. To overcome the shortcoming, we present a new reliable algorithm called the domain decomposition homotopy perturbation method (DDHPM) to solve a class of singular two-point boundary value problems with Neumann and Robin-type boundary conditions arising in various physical models. Five numerical examples are presented to demonstrate the accuracy and applicability of our method, including thermal explosion, oxygen-diffusion in a spherical cell and heat conduction through a solid with heat generation. A comparison is made between the proposed technique and other existing seminumerical or numerical techniques. Numerical results reveal that only two or three iterations lead to high accuracy of the solution and this newly improved technique introduces a powerful improvement for solving nonlinear singular boundary value problems (SBVPs).

On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics

In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

Linearization methods for optimizing the low thrust spacecraft trajectory: Theoretical aspects

The theoretical aspects of the modified linearization method, which makes it possible to solve a wide class of nonlinear problems on optimizing low-thrust spacecraft trajectories (V. V. Efanov et al., 2009; V. V. Khartov et al., 2010) are examined. The main modifications of the linearization method are connected with its refinement for optimizing the main dynamic systems and design parameters of the spacecraft.

Solution of System of Higher Order Integro-Differential Equations by Perturbed Variational Iteration Method

Variational Iteration Method (VIM) is a numerical method for solving a wide class of non-linear problems, first envisioned by the Chinese Mathematician He (1998). In this paper, higher order integro differential equations are reduced to a system of integral equations. The reduced system is then perturbed by using Chebyshev Polynomial and solved by Variational Iteration method. The results obtained for some illustrative examples showed that the perturbed variational iteration method is efficient and reliable. Examples are given to illustrate the efficiency and implementation of the method.

Some Relatively New Techniques for Nonlinear Problems

This paper outlines a detailed study of some relatively new techniques which are originated by He for solving diversified nonlinear problems of physical nature. In particular, we will focus on the variational iteration method (VIM) and its modifications, the homotopy perturbation method (HPM), the parameter expansion method, and exp‐function method. These relatively new but very reliable techniques proved useful for solving a wide class of nonlinear problems and are capable to cope with the versatility of the physical problems. Several examples are given to reconfirm the efficiency of these algorithms. Some open problems are also suggested for future research work.

Equation-free implementation of statistical moment closures

We present a general numerical scheme for the practical implementation of statistical moment closures suitable for modeling complex, large-scale, nonlinear systems. Building on recently developed equation-free methods, this approach numerically integrates the closure dynamics, the equations of which may not even be available in closed form. Although closure dynamics introduce statistical assumptions of unknown validity, they can have significant computational advantages as they typically have fewer degrees of freedom and may be much less stiff than the original detailed model. The numerical closure approach can in principle be applied to a wide class of nonlinear problems, including strongly coupled systems (either deterministic or stochastic) for which there may be no scale separation. We demonstrate the equation-free approach for implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic partial differential equation.

Towards a Unified Approach to Nonexistence of Solutions for a Class of Differential Inequalities

In this paper we deal with an approach for obtaining a priori estimates for solutions of nonlinear partial differential equations and inequalities. The goal is to apply these estimates to find necessary conditions for solvability. This approach, which is simply based of test functions, covers a wide class of nonlinear problems for which we address the question of nonexistence of solutions. In recent years, nonexistence theorems for equations and systems of partial differential equations, have received considerable attention. There is a rather extensive bibliography on this subject. For quasilinear parabolic problems we refer the interested reader to [35],[41], and for more recent results to [20], [7],

Symmetry approach to reveal hidden coherent structures in quantum optics. General outlook and examples

A general invariant-algebraic approach to reveal hidden coherent structures (closed complexes and configurations of states) is developed in quantum-optical models due to symmetry of their HamiltonianH. This approach is based on using the mathematical concept of dual algebraic pairs, incorporating action of both invariance groups and dynamic-symmetry algebras. Its general features are demonstrated on some recent examples: (i)G-invariant biphotons and related invariant treatment of unpolarized light and (ii) quasispin clusters and optical atoms in nonlinear models of quantum optics. The first example leads to a positive solution (within the framework of composite field theory) of the old problem put in polarization optics by Fresnel—the existence of elementary waves of unpolarized light. The second example yields a new mathematical technique for analyzing a wide class of nonlinear problems in quantum optics and laser physics that, in turn, enables one to reveal new coherent effects and phenomena in the process of time evolution of the models under study.

Analysis of a Nonlinear First-Order System with a White Noise Input

A general technique for computing the power spectrum of nonlinear systems is presented, and the method is applied to the Brownian motion of a particle with idealized Coulomb damping. In this method the transition probability is obtained by solving the appropriate Fokker-Planck equation. Then the autocorrelation function is calculated by an appropriate integration, and finally the power spectrum is found by means of the Wiener-Khintchine relation. Comparison is made with the method of equivalent linearization, which is applicable to a wide class of nonlinear problems.