**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**42

# Search results for: Approximate solution

##### 42 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.

##### 41 A Study of Hamilton-Jacobi-Bellman Equation Systems Arising in Differential Game Models of Changing Society

**Authors:**
Weihua Ruan,
Kuan-Chou Chen

**Abstract:**

**Keywords:**
Differential games,
Hamilton-Jacobi-Bellman
equations,
infinite horizon,
political-economy models.

##### 40 Non-Linear Vibration and Stability Analysis of an Axially Moving Beam with Rotating-Prismatic Joint

**Authors:**
M. Najafi,
F. Rahimi Dehgolan

**Abstract:**

In this paper, the dynamic modeling of a single-link flexible beam with a tip mass is given by using Hamilton's principle. The link has been rotational and translational motion and it was assumed that the beam is moving with a harmonic velocity about a constant mean velocity. Non-linearity has been introduced by including the non-linear strain to the analysis. Dynamic model is obtained by Euler-Bernoulli beam assumption and modal expansion method. Also, the effects of rotary inertia, axial force, and associated boundary conditions of the dynamic model were analyzed. Since the complex boundary value problem cannot be solved analytically, the multiple scale method is utilized to obtain an approximate solution. Finally, the effects of several conditions on the differences among the behavior of the non-linear term, mean velocity on natural frequencies and the system stability are discussed.

**Keywords:**
Non-linear vibration,
stability,
axially moving beam,
bifurcation,
multiple scales method.

##### 39 Two-Dimensional Observation of Oil Displacement by Water in a Petroleum Reservoir through Numerical Simulation and Application to a Petroleum Reservoir

**Authors:**
Ahmad Fahim Nasiry,
Shigeo Honma

**Abstract:**

**Keywords:**
Numerical simulation,
immiscible,
finite difference,
IADI,
IADE,
waterflooding.

##### 38 APPLE: Providing Absolute and Proportional Throughput Guarantees in Wireless LANs

**Authors:**
Zhijie Ma,
Qinglin Zhao,
Hongning Dai,
Huan Zhang

**Abstract:**

**Keywords:**
IEEE 802.11e,
throughput guarantee,
priority.

##### 37 Recovering the Boundary Data in the Two Dimensional Inverse Heat Conduction Problem Using the Ritz-Galerkin Method

**Authors:**
Saeed Sarabadan,
Kamal Rashedi

**Abstract:**

**Keywords:**
Inverse problem,
parabolic equations,
heat equation,
Ritz-Galerkin method,
Landweber iterations.

##### 36 Flow of a Second Order Fluid through Constricted Tube with Slip Velocity at Wall Using Integral Method

**Authors:**
Nosheen Zareen Khan,
Abdul Majeed Siddiqui,
Muhammad Afzal Rana

**Abstract:**

**Keywords:**
Approximate solution,
constricted tube,
non-Newtonian fluids,
Reynolds number.

##### 35 Cubic Trigonometric B-spline Approach to Numerical Solution of Wave Equation

**Authors:**
Shazalina Mat Zin,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail,
Muhammad Abbas

**Abstract:**

The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature.

**Keywords:**
Collocation method,
Cubic trigonometric B-spline,
Finite difference,
Wave equation.

##### 34 Stabilization of the Bernoulli-Euler Plate Equation: Numerical Analysis

**Authors:**
Carla E. O. de Moraes,
Gladson O. Antunes,
Mauro A. Rincon

**Abstract:**

The aim of this paper is to study the internal stabilization of the Bernoulli-Euler equation numerically. For this, we consider a square plate subjected to a feedback/damping force distributed only in a subdomain. An algorithm for obtaining an approximate solution to this problem was proposed and implemented. The numerical method used was the Finite Difference Method. Numerical simulations were performed and showed the behavior of the solution, confirming the theoretical results that have already been proved in the literature. In addition, we studied the validation of the numerical scheme proposed, followed by an analysis of the numerical error; and we conducted a study on the decay of the energy associated.

**Keywords:**
Bernoulli-Euler Plate Equation,
Numerical Simulations,
Stability,
Energy Decay,
Finite Difference Method.

##### 33 Simulation of Immiscibility Regions in Sodium Borosilicate Glasses

**Authors:**
D. Aboutaleb,
B. Safi

**Abstract:**

In this paper, sodium borosilicates glasses were prepared by melting in air. These heat-resistant transparent glasses have subjected subsequently isothermal treatments at different times, which have transformed them at opaque glass (milky white color). Such changes indicate that these glasses showed clearly phase separation (immiscibility). The immiscibility region in a sodium borosilicate ternary system was investigated in this work, i.e. to determine the regions from which some compositions can show phase separation. For this we went through the conditions of thermodynamic equilibrium, which were translated later by mathematical equations to find an approximate solution. The latter has been translated in a simulation which was established thereafter to find the immiscibility regions in this type of special glasses.

**Keywords:**
Sodium borosilicate,
heat-resistant,
isothermal treatments,
immiscibility,
thermodynamics four.

##### 32 Application of Double Side Approach Method on Super Elliptical Winkler Plate

**Authors:**
Hsiang-Wen Tang,
Cheng-Ying Lo

**Abstract:**

In this study, the static behavior of super elliptical Winkler plate is analyzed by applying the double side approach method. The lack of information about super elliptical Winkler plates is the motivation of this study and we use the double side approach method to solve this problem because of its superior ability on efficiently treating problems with complex boundary shape. The double side approach method has the advantages of high accuracy, easy calculation procedure and less calculation load required. Most important of all, it can give the error bound of the approximate solution. The numerical results not only show that the double side approach method works well on this problem but also provide us the knowledge of static behavior of super elliptical Winkler plate in practical use.

**Keywords:**
Super elliptical Winkler Plate,
double side approach method,
error bound.

##### 31 Approximate Solution of Some Mixed Boundary Value Problems of the Generalized Theory of Couple-Stress Thermo-Elasticity

**Authors:**
M. Chumburidze,
D. Lekveishvili

**Abstract:**

We have considered the harmonic oscillations and general dynamic (pseudo oscillations) systems of theory generalized Green-Lindsay of couple-stress thermo-elasticity for isotropic, homogeneous elastic media. Approximate solution of some mixed boundary value problems for finite domain, bounded by the some closed surface are constructed.

**Keywords:**
The couple-stress thermo-elasticity,
boundary value problems.

##### 30 MHD Unsteady Free Convection of Heat and Mass Transfer Flow through Porous Medium with Time Dependent Suction and Constant Heat Source/Sink

**Authors:**
Praveen Saraswat,
Rudraman Singh

**Abstract:**

In this paper, we have investigated the free convection MHD flow due to heat and mass transfer through porous medium bounded by an infinite vertical non-conducting porous plate with time dependent suction under the influence of uniform transverse magnetic field of strength H_{0}. When Temperature (T) and Concentration (C) at the plate is oscillatory with time about a constant non-zero mean. The velocity distribution, the temperature distribution, co-efficient of skin friction and role of heat transfer is investigated. Here the partial differential equations are involved. Exact solution is not possible so approximate solution is obtained and various graphs are plotted.

**Keywords:**
Time Dependent Suction,
Convection,
MHD,
Porous.

##### 29 On the Approximate Solution of Continuous Coefficients for Solving Third Order Ordinary Differential Equations

**Authors:**
A. M. Sagir

**Abstract:**

This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y''' = f(x, y, y', y''), y(α)=y_{0}, y'(α)=β, y''(α)=η with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non – stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution.

**Keywords:**
Block Method,
Hybrid,
Linear Multistep,
Self starting,
Third Order Ordinary Differential Equations.

##### 28 Constant Order Predictor Corrector Method for the Solution of Modeled Problems of First Order IVPs of ODEs

**Authors:**
A. A. James,
A. O. Adesanya,
M. R. Odekunle,
D. G. Yakubu

**Abstract:**

This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compete favorably with the existing methods.

**Keywords:**
Interpolation,
Approximate Solution,
Collocation,
Differential system,
Half step,
Converges,
Block method,
Efficiency.

##### 27 Laplace Decomposition Approximation Solution for a System of Multi-Pantograph Equations

**Authors:**
M. A. Koroma,
C. Zhan,
A. F. Kamara,
A. B. Sesay

**Abstract:**

In this work we adopt a combination of Laplace transform and the decomposition method to find numerical solutions of a system of multi-pantograph equations. The procedure leads to a rapid convergence of the series to the exact solution after computing a few terms. The effectiveness of the method is demonstrated in some examples by obtaining the exact solution and in others by computing the absolute error which decreases as the number of terms of the series increases.

**Keywords:**
Laplace decomposition,
pantograph equations,
exact
solution,
numerical solution,
approximate solution.

##### 26 An Application of the Sinc-Collocation Method to a Three-Dimensional Oceanography Model

**Authors:**
Y. Mohseniahouei,
K. Abdella,
M. Pollanen

**Abstract:**

**Keywords:**
Boundary Value Problems,
Differential Equations,
Sinc Numerical Methods,
Wind-Driven Currents

##### 25 Reliability Approximation through the Discretization of Random Variables using Reversed Hazard Rate Function

**Authors:**
Tirthankar Ghosh,
Dilip Roy,
Nimai Kumar Chandra

**Abstract:**

Sometime it is difficult to determine the exact reliability for complex systems in analytical procedures. Approximate solution of this problem can be provided through discretization of random variables. In this paper we describe the usefulness of discretization of a random variable using the reversed hazard rate function of its continuous version. Discretization of the exponential distribution has been demonstrated. Applications of this approach have also been cited. Numerical calculations indicate that the proposed approach gives very good approximation of reliability of complex systems under stress-strength set-up. The performance of the proposed approach is better than the existing discrete concentration method of discretization. This approach is conceptually simple, handles analytic intractability and reduces computational time. The approach can be applied in manufacturing industries for producing high-reliable items.

**Keywords:**
Discretization,
Reversed Hazard Rate,
Exponential
distribution,
reliability approximation,
engineering item.

##### 24 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.

##### 23 Exact Solutions of the Helmholtz equation via the Nikiforov-Uvarov Method

**Authors:**
Said Laachir,
Aziz Laaribi

**Abstract:**

The Helmholtz equation often arises in the study of physical problems involving partial differential equation. Many researchers have proposed numerous methods to find the analytic or approximate solutions for the proposed problems. In this work, the exact analytical solutions of the Helmholtz equation in spherical polar coordinates are presented using the Nikiforov-Uvarov (NU) method. It is found that the solution of the angular eigenfunction can be expressed by the associated-Legendre polynomial and radial eigenfunctions are obtained in terms of the Laguerre polynomials. The special case for k=0, which corresponds to the Laplace equation is also presented.

**Keywords:**
Helmholtz equation,
Nikiforov-Uvarov method,
exact solutions,
eigenfunctions.

##### 22 Approximate Solutions to Large Stein Matrix Equations

**Authors:**
Khalide Jbilou

**Abstract:**

In the present paper, we propose numerical methods for solving the Stein equation AXC - X - D = 0 where the matrix A is large and sparse. Such problems appear in discrete-time control problems, filtering and image restoration. We consider the case where the matrix D is of full rank and the case where D is factored as a product of two matrices. The proposed methods are Krylov subspace methods based on the block Arnoldi algorithm. We give theoretical results and we report some numerical experiments.

**Keywords:**
IEEEtran,
journal,
LATEX,
paper,
template.

##### 21 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

##### 20 Mechanical Quadrature Methods and Their Extrapolations for Solving First Kind Boundary Integral Equations of Anisotropic Darcy-s Equation

**Authors:**
Xin Luo,
Jin Huang,
Chuan-Long Wang

**Abstract:**

The mechanical quadrature methods for solving the boundary integral equations of the anisotropic Darcy-s equations with Dirichlet conditions in smooth domains are presented. By applying the collectively compact theory, we prove the convergence and stability of approximate solutions. The asymptotic expansions for the error show that the methods converge with the order O (h3), where h is the mesh size. Based on these analysis, extrapolation methods can be introduced to achieve a higher convergence rate O (h5). An a posterior asymptotic error representation is derived in order to construct self-adaptive algorithms. Finally, the numerical experiments show the efficiency of our methods.

**Keywords:**
Darcy's equation,
anisotropic,
mechanical quadrature methods,
extrapolation methods,
a posteriori error estimate.

##### 19 New Newton's Method with Third-order Convergence for Solving Nonlinear Equations

**Authors:**
Osama Yusuf Ababneh

**Abstract:**

For the last years, the variants of the Newton-s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we present a new Newton-s method based on contra harmonic mean with cubically convergent. Numerical examples show that the new method can compete with the classical Newton's method.

**Keywords:**
Third-order convergence,
non-linear equations,
root finding,
iterative method.

##### 18 Ordinary Differential Equations with Inverted Functions

**Authors:**
Thomas Kampke

**Abstract:**

Equations with differentials relating to the inverse of an unknown function rather than to the unknown function itself are solved exactly for some special cases and numerically for the general case. Invertibility combined with differentiability over connected domains forces solutions always to be monotone. Numerical function inversion is key to all solution algorithms which either are of a forward type or a fixed point type considering whole approximate solution functions in each iteration. The given considerations are restricted to ordinary differential equations with inverted functions (ODEIs) of first order. Forward type computations, if applicable, admit consistency of order one and, under an additional accuracy condition, convergence of order one.

**Keywords:**
Euler method,
fixed points,
golden section,
multi-step procedures,
Runge Kutta methods.

##### 17 On a Way for Constructing Numerical Methods on the Joint of Multistep and Hybrid Methods

**Authors:**
G.Mehdiyeva,
M.Imanova,
V.Ibrahimov

**Abstract:**

**Keywords:**
Multistep and hybrid methods,
initial value problem,
degree and stability of hybrid methods

##### 16 An Approximate Solution of the Classical Van der Pol Oscillator Coupled Gyroscopically to a Linear Oscillator Using Parameter-Expansion Method

**Authors:**
Mohammad Taghi Darvishi,
Samad Kheybari

**Abstract:**

In this article, we are dealing with a model consisting of a classical Van der Pol oscillator coupled gyroscopically to a linear oscillator. The major problem is analyzed. The regular dynamics of the system is considered using analytical methods. In this case, we provide an approximate solution for this system using parameter-expansion method. Also, we find approximate values for frequencies of the system. In parameter-expansion method the solution and unknown frequency of oscillation are expanded in a series by a bookkeeping parameter. By imposing the non-secularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation. One iteration step provides an approximate solution which is valid for the whole solution domain.

**Keywords:**
Parameter-expansion method,
classical Van der Pol oscillator.

##### 15 A Projection Method Based on Extended Krylov Subspaces for Solving Sylvester Equations

**Authors:**
Yiqin Lin,
Liang Bao,
Yimin Wei

**Abstract:**

In this paper we study numerical methods for solving Sylvester matrix equations of the form AX +XBT +CDT = 0. A new projection method is proposed. The union of Krylov subspaces in A and its inverse and the union of Krylov subspaces in B and its inverse are used as the right and left projection subspaces, respectively. The Arnoldi-like process for constructing the orthonormal basis of the projection subspaces is outlined. We show that the approximate solution is an exact solution of a perturbed Sylvester matrix equation. Moreover, exact expression for the norm of residual is derived and results on finite termination and convergence are presented. Some numerical examples are presented to illustrate the effectiveness of the proposed method.

**Keywords:**
Arnoldi process,
Krylov subspace,
Iterative method,
Sylvester equation,
Dissipative matrix.

##### 14 The Homotopy Analysis Method for Solving Discontinued Problems Arising in Nanotechnology

**Authors:**
Hassan Saberi-Nik,
Mahin Golchaman

**Abstract:**

This paper applies the homotopy analysis method method to a nonlinear differential-difference equation arising in nanotechnology. Continuum hypothesis on nanoscales is invalid, and a differential-difference model is considered as an alternative approach to describing discontinued problems. Comparison of the approximate solution with the exact one reveals that the method is very effective.

**Keywords:**
Homotopy analysis method,
differential-difference,
nanotechnology.

##### 13 A New Approach to the Approximate Solutions of Hamilton-Jacobi Equations

**Authors:**
Joe Imae,
Kenjiro Shinagawa,
Tomoaki Kobayashi,
Guisheng Zhai

**Abstract:**

We propose a new approach on how to obtain the approximate solutions of Hamilton-Jacobi (HJ) equations. The process of the approximation consists of two steps. The first step is to transform the HJ equations into the virtual time based HJ equations (VT-HJ) by introducing a new idea of ‘virtual-time’. The second step is to construct the approximate solutions of the HJ equations through a computationally iterative procedure based on the VT-HJ equations. It should be noted that the approximate feedback solutions evolve by themselves as the virtual-time goes by. Finally, we demonstrate the effectiveness of our approximation approach by means of simulations with linear and nonlinear control problems.

**Keywords:**
Nonlinear Control,
Optimal Control,
Hamilton-Jacobi Equation,
Virtual-Time