**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**3463

# Search results for: Non-linear quasi-static solution.

##### 3463 In-situ Quasistatic Compression and Microstructural Characterization of Aluminum Foams of Different Cell Topology

**Authors:**
M. A. Islam,
P. J. Hazell,
J. P. Escobedo,
M. Saadatfar

**Abstract:**

Metallic foams have good potential for lightweight structures for impact and blast mitigation. Therefore it is important to find out the optimized foam structure (i.e. cell size, shape, relative density, and distribution) to maximise energy absorption. In this paper, quasistatic compression and microstructural characterization of closed-cell aluminium foams of different pore size and cell distributions have been carried out. We present results for two different aluminium metal foams of density 0.49-0.51 g/cc and 0.31- 0.34 g/cc respectively that have been tested in quasi-static compression. The influence of cell geometry and cell topology on quasistatic compression behaviour has been investigated using optical microscope and computed tomography (micro-CT) analysis. It is shown that the deformation is not uniform in the structure and collapse begins at the weakest point.

**Keywords:**
Metal foams,
micro-CT,
cell topology,
quasistatic
compression.

##### 3462 On the Approximate Solution of a Nonlinear Singular Integral Equation

**Authors:**
Nizami Mustafa,
C. Ardil

**Abstract:**

In this study, the existence and uniqueness of the solution of a nonlinear singular integral equation that is defined on a region in the complex plane is proven and a method is given for finding the solution.

**Keywords:**
Approximate solution,
Fixed-point principle,
Nonlinear singular integral equations,
Vekua integral operator

##### 3461 On the Positive Definite Solutions of Nonlinear Matrix Equation

**Authors:**
Tian Baoguang,
Liang Chunyan,
Chen Nan

**Abstract:**

In this paper, the nonlinear matrix equation is investigated. Based on the fixed-point theory, the boundary and the existence of the solution with the case r>-δ_{i} are discussed. An algorithm that avoids matrix inversion with the case -1<-δ_{i}<0 is proposed.

**Keywords:**
Nonlinear matrix equation,
Positive definite solution,
The maximal-minimal solution,
Iterative method,
Free-inversion

##### 3460 Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method

**Authors:**
Gülnihal Meral

**Abstract:**

**Keywords:**
Density Dependent Nonlinear Reaction-Diffusion Equation,
Differential Quadrature Method,
Implicit Euler Method.

##### 3459 Application of He-s Amplitude Frequency Formulation for a Nonlinear Oscillator with Fractional Potential

**Abstract:**

In this paper, He-s amplitude frequency formulation is used to obtain a periodic solution for a nonlinear oscillator with fractional potential. By calculation and computer simulations, compared with the exact solution shows that the result obtained is of high accuracy.

**Keywords:**
He's amplitude frequency formulation,
Periodic solution,
Nonlinear oscillator,
Fractional potential.

##### 3458 Solution of Two-Point Nonlinear Boundary Problems Using Taylor Series Approximation and the Ying Buzu Shu Algorithm

**Authors:**
U. C. Amadi,
N. A. Udoh

**Abstract:**

One of the major challenges faced in solving initial and boundary problems is how to find approximate solutions with minimal deviation from the exact solution without so much rigor and complications. The Taylor series method provides a simple way of obtaining an infinite series which converges to the exact solution for initial value problems and this method of solution is somewhat limited for a two point boundary problem since the infinite series has to be truncated to include the boundary conditions. In this paper, the Ying Buzu Shu algorithm is used to solve a two point boundary nonlinear diffusion problem for the fourth and sixth order solution and compare their relative error and rate of convergence to the exact solution.

**Keywords:**
Ying Buzu Shu,
nonlinear boundary problem,
Taylor series algorithm,
infinite series.

##### 3457 Solution of Nonlinear Second-Order Pantograph Equations via Differential Transformation Method

**Authors:**
Nemat Abazari,
Reza Abazari

**Abstract:**

In this work, we successfully extended one-dimensional differential transform method (DTM), by presenting and proving some theorems, to solving nonlinear high-order multi-pantograph equations. This technique provides a sequence of functions which converges to the exact solution of the problem. Some examples are given to demonstrate the validity and applicability of the present method and a comparison is made with existing results.

**Keywords:**
Nonlinear multi-pantograph equation,
delay differential equation,
differential transformation method,
proportional delay conditions,
closed form solution.

##### 3456 On the Strong Solutions of the Nonlinear Viscous Rotating Stratified Fluid

**Authors:**
A. Giniatoulline

**Abstract:**

**Keywords:**
Galerkin method,
Navier-Stokes equations,
nonlinear partial differential equations,
Sobolev spaces,
stratified fluid.

##### 3455 Power Series Solution to Sliding Velocity in Three-Dimensional Multibody Systems with Impact and Friction

**Authors:**
Hesham A. Elkaranshawy,
Amr M. Abdelrazek,
Hosam M. Ezzat

**Abstract:**

**Keywords:**
Impact with friction,
nonlinear ordinary differential
equations,
power series solutions,
rough collision.

##### 3454 A Study on Metal Hexagonal Honeycomb Crushing Under Quasi-Static Loading

**Authors:**
M. Zarei Mahmoudabadi,
M. Sadighi

**Abstract:**

In the study of honeycomb crushing under quasistatic loading, two parameters are important, the mean crushing stress and the wavelength of the folding mode. The previous theoretical models did not consider the true cylindrical curvature effects and the flow stress in the folding mode of honeycomb material. The present paper introduces a modification on Wierzbicki-s model based on considering two above mentioned parameters in estimating the mean crushing stress and the wavelength through implementation of the energy method. Comparison of the results obtained by the new model and Wierzbicki-s model with existing experimental data shows better prediction by the model presented in this paper.

**Keywords:**
Crush strength,
Flow stress,
Honeycomb,
Quasistatic
load.

##### 3453 Quasilinearization–Barycentric Approach for Numerical Investigation of the Boundary Value Fin Problem

**Authors:**
Alireza Rezaei,
Fatemeh Baharifard,
Kourosh Parand

**Abstract:**

In this paper we improve the quasilinearization method by barycentric Lagrange interpolation because of its numerical stability and computation speed to achieve a stable semi analytical solution. Then we applied the improved method for solving the Fin problem which is a nonlinear equation that occurs in the heat transferring. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The modified QLM is iterative but not perturbative and gives stable semi analytical solutions to nonlinear problems without depending on the existence of a smallness parameter. Comparison with some numerical solutions shows that the present solution is applicable.

**Keywords:**
Quasilinearization method,
Barycentric lagrange interpolation,
nonlinear ODE,
fin problem,
heat transfer.

##### 3452 Existence of Solution of Nonlinear Second Order Neutral Stochastic Differential Inclusions with Infinite Delay

**Authors:**
Yong Li

**Abstract:**

The paper is concerned with the existence of solution of nonlinear second order neutral stochastic differential inclusions with infinite delay in a Hilbert Space. Sufficient conditions for the existence are obtained by using a fixed point theorem for condensing maps.

**Keywords:**
Mild solution,
Convex multivalued map,
Neutral
stochastic differential inclusions.

##### 3451 Analytical Based Truncation Principle of Higher-Order Solution for a x1/3 Force Nonlinear Oscillator

**Authors:**
Md. Alal Hosen

**Abstract:**

In this paper, a modified harmonic balance method based an analytical technique has been developed to determine higher-order approximate periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x^{1/3}. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. In this article, different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. We find a modified harmonic balance method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the x^{1/3} force nonlinear oscillator but it is also useful for many other nonlinear problems.

**Keywords:**
Approximate solutions,
Harmonic balance method,
Nonlinear oscillator,
Perturbation.

##### 3450 Two-Dimensional Solitary Wave Solution to the Quadratic Nonlinear Schrdinger Equation

**Authors:**
Sarun Phibanchon

**Abstract:**

The solitary wave solution of the quadratic nonlinear Schrdinger equation is determined by the iterative method called Petviashvili method. This solution is also used for the initial condition for the time evolution to study the stability analysis. The spectral method is applied for the time evolution.

**Keywords:**
soliton,
iterative method,
spectral method,
plasma

##### 3449 Lagrangian Method for Solving Unsteady Gas Equation

**Authors:**
Amir Taghavi,
kourosh Parand,
Hosein Fani

**Abstract:**

In this paper we propose, a Lagrangian method to solve unsteady gas equation which is a nonlinear ordinary differential equation on semi-infnite interval. This approach is based on Modified generalized Laguerre functions. This method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare this work with some other numerical results. The findings show that the present solution is highly accurate.

**Keywords:**
Unsteady gas equation,
Generalized Laguerre functions,
Lagrangian method,
Nonlinear ODE.

##### 3448 A First Course in Numerical Methods with “Mathematica“

**Authors:**
Andrei A. Kolyshkin

**Abstract:**

**Keywords:**
Numerical methods,
"Mathematica",
e-learning.

##### 3447 Axisymmetric Nonlinear Analysis of Point Supported Shallow Spherical Shells

**Authors:**
M. Altekin,
R. F. Yükseler

**Abstract:**

Geometrically nonlinear axisymmetric bending of a shallow spherical shell with a point support at the apex under linearly varying axisymmetric load was investigated numerically. The edge of the shell was assumed to be simply supported or clamped. The solution was obtained by the finite difference and the Newton-Raphson methods. The thickness of the shell was considered to be uniform and the material was assumed to be homogeneous and isotropic. Sensitivity analysis was made for two geometrical parameters. The accuracy of the algorithm was checked by comparing the deflection with the solution of point supported circular plates and good agreement was obtained.

**Keywords:**
Bending,
nonlinear,
plate,
point support,
shell.

##### 3446 Unique Positive Solution of Nonlinear Fractional Differential Equation Boundary Value Problem

**Authors:**
Fengxia Zheng

**Abstract:**

By using two new fixed point theorems for mixed monotone operators, the positive solution of nonlinear fractional differential equation boundary value problem is studied. Its existence and uniqueness is proved, and an iterative scheme is constructed to approximate it.

**Keywords:**
Fractional differential equation,
boundary value problem,
positive solution,
existence and uniqueness,
fixed point theorem,
mixed monotone operator.

##### 3445 The Finite Difference Scheme for the Suspended String Equation with the Nonlinear Damping Term

**Authors:**
Jaipong Kasemsuwan

**Abstract:**

**Keywords:**
Finite-difference method,
the nonlinear damped
equation,
the numerical simulation,
the suspended string equation

##### 3444 Nonlinear Dynamic Analysis of Base-Isolated Structures Using a Partitioned Solution Approach and an Exponential Model

**Authors:**
Nicolò Vaiana,
Filip C. Filippou,
Giorgio Serino

**Abstract:**

The solution of the nonlinear dynamic equilibrium equations of base-isolated structures adopting a conventional monolithic solution approach, i.e. an implicit single-step time integration method employed with an iteration procedure, and the use of existing nonlinear analytical models, such as differential equation models, to simulate the dynamic behavior of seismic isolators can require a significant computational effort. In order to reduce numerical computations, a partitioned solution method and a one dimensional nonlinear analytical model are presented in this paper. A partitioned solution approach can be easily applied to base-isolated structures in which the base isolation system is much more flexible than the superstructure. Thus, in this work, the explicit conditionally stable central difference method is used to evaluate the base isolation system nonlinear response and the implicit unconditionally stable Newmark’s constant average acceleration method is adopted to predict the superstructure linear response with the benefit in avoiding iterations in each time step of a nonlinear dynamic analysis. The proposed mathematical model is able to simulate the dynamic behavior of seismic isolators without requiring the solution of a nonlinear differential equation, as in the case of widely used differential equation model. The proposed mixed explicit-implicit time integration method and nonlinear exponential model are adopted to analyze a three dimensional seismically isolated structure with a lead rubber bearing system subjected to earthquake excitation. The numerical results show the good accuracy and the significant computational efficiency of the proposed solution approach and analytical model compared to the conventional solution method and mathematical model adopted in this work. Furthermore, the low stiffness value of the base isolation system with lead rubber bearings allows to have a critical time step considerably larger than the imposed ground acceleration time step, thus avoiding stability problems in the proposed mixed method.

**Keywords:**
Base-isolated structures,
earthquake engineering,
mixed time integration,
nonlinear exponential model.

##### 3443 Spectral Investigation for Boundary Layer Flow over a Permeable Wall in the Presence of Transverse Magnetic Field

**Authors:**
Saeed Sarabadan,
Mehran Nikarya,
Kouroah Parand

**Abstract:**

**Keywords:**
MHD Falkner-Skan,
nonlinear ODE,
spectral
collocation method,
Bessel functions,
skin friction,
velocity.

##### 3442 Automatic Iterative Methods for the Multivariate Solution of Nonlinear Algebraic Equations

**Authors:**
Rafat Alshorman,
Safwan Al-Shara',
I. Obeidat

**Abstract:**

**Keywords:**
Nonlinear Algebraic Equations,
Iterative Methods,
Homotopy
Analysis Method.

##### 3441 Approximate Solution to Non-Linear Schrödinger Equation with Harmonic Oscillator by Elzaki Decomposition Method

**Authors:**
Emad K. Jaradat,
Ala’a Al-Faqih

**Abstract:**

Nonlinear Schrödinger equations are regularly experienced in numerous parts of science and designing. Varieties of analytical methods have been proposed for solving these equations. In this work, we construct an approximate solution for the nonlinear Schrodinger equations, with harmonic oscillator potential, by Elzaki Decomposition Method (EDM). To illustrate the effects of harmonic oscillator on the behavior wave function, nonlinear Schrodinger equation in one and two dimensions is provided. The results show that, it is more perfectly convenient and easy to apply the EDM in one- and two-dimensional Schrodinger equation.

**Keywords:**
Non-linear Schrodinger equation,
Elzaki decomposition method,
harmonic oscillator,
one and two- dimensional Schrodinger equation.

##### 3440 Instability of a Nonlinear Differential Equation of Fifth Order with Variable Delay

**Authors:**
Cemil Tunc

**Abstract:**

In this paper, we study the instability of the zero solution to a nonlinear differential equation with variable delay. By using the Lyapunov functional approach, some sufficient conditions for instability of the zero solution are obtained.

**Keywords:**
Instability,
Lyapunov-Krasovskii functional,
delay differential equation,
fifth order.

##### 3439 A Sum Operator Method for Unique Positive Solution to a Class of Boundary Value Problem of Nonlinear Fractional Differential Equation

**Authors:**
Fengxia Zheng,
Chuanyun Gu

**Abstract:**

**Keywords:**
Fractional differential equation,
Boundary value
problem,
Positive solution,
Existence and uniqueness,
Fixed point
theorem of a sum operator.

##### 3438 The Existence and Uniqueness of Positive Solution for Nonlinear Fractional Differential Equation Boundary Value Problem

**Authors:**
Chuanyun Gu,
Shouming Zhong

**Abstract:**

In this paper, the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problem is concerned by a fixed point theorem of a sum operator. Our results can not only guarantee the existence and uniqueness of positive solution, but also be applied to construct an iterative scheme for approximating it. Finally, the example is given to illustrate the main result.

**Keywords:**
Fractional differential equation,
Boundary value problem,
Positive solution,
Existence and uniqueness,
Fixed point theorem of a sum operator

##### 3437 Periodic Solutions for Some Strongly Nonlinear Oscillators by He's Energy Balance Method

**Abstract:**

In this paper, applying He-s energy balance method to determine frequency formulation relations of nonlinear oscillators with discontinuous term or fractional potential. By calculation and computer simulations, compared with the exact solutions show that the results obtained are of high accuracy.

**Keywords:**
He's energy balance method,
periodic solution,
nonlinear oscillator,
discontinuous,
fractional potential.

##### 3436 An Asymptotic Solution for the Free Boundary Parabolic Equations

**Authors:**
Hsuan-Ku Liu,
Ming Long Liu

**Abstract:**

In this paper, we investigate the solution of a two dimensional parabolic free boundary problem. The free boundary of this problem is modelled as a nonlinear integral equation (IE). For this integral equation, we propose an asymptotic solution as time is near to maturity and develop an integral iterative method. The computational results reveal that our asymptotic solution is very close to the numerical solution as time is near to maturity.

**Keywords:**
Integral equation,
asymptotic solution,
free boundary problem,
American exchange option.

##### 3435 Existence and Uniqueness of Positive Solution for Nonlinear Fractional Differential Equation with Integral Boundary Conditions

**Authors:**
Chuanyun Gu

**Abstract:**

**Keywords:**
Fractional differential equation,
positive solution,
existence and uniqueness,
fixed point theorem,
generalized concave
and convex operator,
integral boundary conditions.

##### 3434 A New Nonlinear PID Controller and its Parameter Design

**Authors:**
Yongping Ren,
Zongli Li,
Fan Zhang

**Abstract:**

**Keywords:**
Nonlinear PID controller,
stability,
gain equivalence,
dissipative,
T-Passivity.