**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**2787

# Search results for: Exact solution

##### 2787 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.

##### 2786 Exact Solution of the Ising Model on the 15 X 15 Square Lattice with Free Boundary Conditions

**Authors:**
Seung-Yeon Kim

**Abstract:**

**Keywords:**
Phase transition,
Ising magnet,
Square lattice,
Freeboundary conditions,
Exact solution.

##### 2785 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.

##### 2784 Exploring Solutions in Extended Horava-Lifshitz Gravity

**Authors:**
Aziza Altaibayeva,
Ertan Gudekli,
Ratbay Myrzakulov

**Abstract:**

In this letter, we explore exact solutions for the Horava-Lifshitz gravity. We use of an extension of this theory with first order dynamical lapse function. The equations of motion have been derived in a fully consistent scenario. We assume that there are some spherically symmetric families of exact solutions of this extended theory of gravity. We obtain exact solutions and investigate the singularity structures of these solutions. Specially, an exact solution with the regular horizon is found.

**Keywords:**
Quantum gravity,
Horava-Lifshitz gravity,
black hole,
spherically symmetric space times.

##### 2783 An Exact Solution of Axi-symmetric Conductive Heat Transfer in Cylindrical Composite Laminate under the General Boundary Condition

**Authors:**
M.kayhani,
M.Nourouzi,
A. Amiri Delooei

**Abstract:**

**Keywords:**
exact solution,
composite laminate,
heat conduction,
cylinder,
Fourier transformation.

##### 2782 A Note on the Numerical Solution of Singular Integral Equations of Cauchy Type

**Authors:**
M. Abdulkawi,
Z. K. Eshkuvatov,
N. M. A. Nik Long

**Abstract:**

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebyshev polynomials of the first kind. It is shown that the numerical solution of characteristic singular integral equation is identical with the exact solution, when the force function is a cubic function. Moreover, it also shown that this numerical method gives exact solution for other singular integral equations with degenerate kernels.

**Keywords:**
Singular integral equations,
Cauchy kernel,
Chebyshev polynomials,
interpolation.

##### 2781 An Exact Solution to Support Vector Mixture

**Authors:**
Monjed Ezzeddinne,
Nicolas Lefebvre,
Régis Lengellé

**Abstract:**

**Keywords:**
Identification,
Learning systems,
Mixture ofExperts,
Support Vector Machines.

##### 2780 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.

##### 2779 Some Exact Solutions of the (2+1)-Dimensional Breaking Soliton Equation using the Three-wave Method

**Authors:**
Mohammad Taghi Darvishi,
Mohammad Najafi

**Abstract:**

This paper considers the (2+1)-dimensional breaking soliton equation in its bilinear form. Some exact solutions to this equation are explicitly derived by the idea of three-wave solution method with the assistance of Maple. We can see that the new idea is very simple and straightforward.

**Keywords:**
Soliton solution,
computerized symbolic computation,
painleve analysis,
(2+1)-dimensional breaking soliton equation,
Hirota's bilinear form.

##### 2778 Simulink Approach to Solve Fuzzy Differential Equation under Generalized Differentiability

**Authors:**
N. Kumaresan ,
J. Kavikumar,
Kuru Ratnavelu

**Abstract:**

**Keywords:**
Fuzzy differential equation,
Generalized differentiability,
H-difference and Simulink.

##### 2777 Solution of Fuzzy Differential Equation under Generalized Differentiability by Genetic Programming

**Authors:**
N. Kumaresan,
J. Kavikumar,
M. Kumudthaa,
Kuru Ratnavelu

**Abstract:**

**Keywords:**
Fuzzy differential equation,
Generalized differentiability,
Genetic programming and H-difference.

##### 2776 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.

##### 2775 An Approximation Method for Exact Boundary Controllability of Euler-Bernoulli System

**Authors:**
Abdelaziz Khernane,
Naceur Khelil,
Leila Djerou

**Abstract:**

**Keywords:**
Boundary control,
exact controllability,
finite
difference methods,
functional optimization.

##### 2774 Extension of a Smart Piezoelectric Ceramic Rod

**Authors:**
Ali Reza Pouladkhan,
Jalil Emadi,
Hamed Habibolahiyan

**Abstract:**

This paper presents an exact solution and a finite element method (FEM) for a Piezoceramic Rod under static load. The cylindrical rod is made from polarized ceramics (piezoceramics) with axial poling. The lateral surface of the rod is traction-free and is unelectroded. The two end faces are under a uniform normal traction. Electrically, the two end faces are electroded with a circuit between the electrodes, which can be switched on or off. Two cases of open and shorted electrodes (short circuit and open circuit) will be considered. Finally, a finite element model will be used to compare the results with an exact solution. The study uses ABAQUS (v.6.7) software to derive the finite element model of the ceramic rod.

**Keywords:**
Finite element method,
Ceramic rod; Axial
poling,
Normal traction,
Short circuit,
Open circuit.

##### 2773 Approximated Solutions of Two-Point Nonlinear Boundary Problem by a Combination of Taylor Series Expansion and Newton Raphson Method

**Authors:**
Chinwendu. B. Eleje,
Udechukwu P. Egbuhuzor

**Abstract:**

One of the difficulties encountered in solving nonlinear Boundary Value Problems (BVP) by many researchers is finding approximated solutions with minimum deviations from the exact solutions without so much rigor and complications. In this paper, we propose an approach to solve a two point BVP which involves a combination of Taylor series expansion method and Newton Raphson method. Furthermore, the fourth and sixth order approximated solutions are obtained and we compare their relative error and rate of convergence to the exact solution. Finally, some numerical simulations are presented to show the behavior of the solution and its derivatives.

**Keywords:**
Newton Raphson method,
non-linear boundary value problem,
Taylor series approximation,
Michaelis-Menten equation.

##### 2772 Exact Solution of Some Helical Flows of Newtonian Fluids

**Authors:**
Imran Siddique

**Abstract:**

**Keywords:**
Newtonian fluids,
Velocity field,
Exact solutions,
Shear stress,
Cylindrical domains.

##### 2771 Traveling Wave Solutions for the (3+1)-Dimensional Breaking Soliton Equation by (G'/G)- Expansion Method and Modified F-Expansion Method

**Authors:**
Mohammad Taghi Darvishi,
Maliheh Najafi,
Mohammad Najafi

**Abstract:**

In this paper, using (G/G )-expansion method and modified F-expansion method, we give some explicit formulas of exact traveling wave solutions for the (3+1)-dimensional breaking soliton equation. A modified F-expansion method is proposed by taking full advantages of F-expansion method and Riccati equation in seeking exact solutions of the equation.

**Keywords:**
Exact solution,
The (3+1)-dimensional breaking soliton equation,
( G G )-expansion method,
Riccati equation,
Modified Fexpansion method.

##### 2770 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.

##### 2769 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.

##### 2768 Solving Inhomogeneous Wave Equation Cauchy Problems using Homotopy Perturbation Method

**Authors:**
Mohamed M. Mousa,
Aidarkhan Kaltayev

**Abstract:**

In this paper, He-s homotopy perturbation method (HPM) is applied to spatial one and three spatial dimensional inhomogeneous wave equation Cauchy problems for obtaining exact solutions. HPM is used for analytic handling of these equations. The results reveal that the HPM is a very effective, convenient and quite accurate to such types of partial differential equations (PDEs).

**Keywords:**
Homotopy perturbation method,
Exact solution,
Cauchy problem,
inhomogeneous wave equation

##### 2767 Numerical Solution of Infinite Boundary Integral Equation by Using Galerkin Method with Laguerre Polynomials

**Authors:**
N. M. A. Nik Long,
Z. K. Eshkuvatov,
M. Yaghobifar,
M. Hasan

**Abstract:**

In this paper the exact solution of infinite boundary integral equation (IBIE) of the second kind with degenerate kernel is presented. Moreover Galerkin method with Laguerre polynomial is applied to get the approximate solution of IBIE. Numerical examples are given to show the validity of the method presented.

**Keywords:**
Approximation,
Galerkin method,
Integral
equations,
Laguerre polynomial.

##### 2766 HPM Solution of Momentum Equation for Darcy-Brinkman Model in a Parallel Plates Channel Subjected to Lorentz Force

**Authors:**
Asghar Shirazpour,
Seyed Moein Rassoulinejad Mousavi,
Hamid Reza Seyf

**Abstract:**

In this paper an analytical solution is presented for fully developed flow in a parallel plates channel under the action of Lorentz force, by use of Homotopy Perturbation Method (HPM). The analytical results are compared with exact solution and an excellent agreement has been observed between them for both Couette and Poiseuille flows. Moreover, the effects of key parameters have been studied on the dimensionless velocity profile.

**Keywords:**
Lorentz Force,
Porous Media,
Homotopy Perturbation method

##### 2765 Control of the Thermal Evaporation of Organic Semiconductors via Exact Linearization

**Authors:**
Martin Steinberger,
Martin Horn

**Abstract:**

In this article, a high vacuum system for the evaporation of organic semiconductors is introduced and a mathematical model is given. Based on the exact input output linearization a deposition rate controller is designed and tested with different evaporation materials.

**Keywords:**
Effusion cell,
organic semiconductors,
deposition rate,
exact linearization.

##### 2764 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.

##### 2763 A Comparison of Exact and Heuristic Approaches to Capital Budgeting

**Authors:**
Jindřiška Šedová,
Miloš Šeda

**Abstract:**

**Keywords:**
Capital budgeting,
knapsack problem,
GAMS,
heuristic method,
genetic algorithm.

##### 2762 Constructing Approximate and Exact Solutions for Boussinesq Equations using Homotopy Perturbation Padé Technique

**Authors:**
Mohamed M. Mousa,
Aidarkhan Kaltayev

**Abstract:**

**Keywords:**
Homotopy perturbation method,
Padé approximants,
cubic Boussinesq equation,
modified Boussinesq equation.

##### 2761 Transient Population Dynamics of Phase Singularities in 2D Beeler-Reuter Model

**Authors:**
Hidetoshi Konno,
Akio Suzuki

**Abstract:**

The paper presented a transient population dynamics of phase singularities in 2D Beeler-Reuter model. Two stochastic modelings are examined: (i) the Master equation approach with the transition rate (i.e., λ(n, t) = λ(t)n and μ(n, t) = μ(t)n) and (ii) the nonlinear Langevin equation approach with a multiplicative noise. The exact general solution of the Master equation with arbitrary time-dependent transition rate is given. Then, the exact solution of the mean field equation for the nonlinear Langevin equation is also given. It is demonstrated that transient population dynamics is successfully identified by the generalized Logistic equation with fractional higher order nonlinear term. It is also demonstrated the necessity of introducing time-dependent transition rate in the master equation approach to incorporate the effect of nonlinearity.

**Keywords:**
Transient population dynamics,
Phase singularity,
Birth-death process,
Non-stationary Master equation,
nonlinear Langevin equation,
generalized Logistic equation.

##### 2760 Exact Analysis of Resonance Frequencies of Simply Supported Cylindrical Shells

**Authors:**
A. Farshidianfar,
P. Oliazadeh,
M. H. Farshidianfar

**Abstract:**

In order to study the free vibration of simply supported circular cylindrical shells; an analytical procedure is developed and discussed in detail. To identify its’ validity, the exact technique was applied to four different shell theories 1) Soedel, 2) Flugge, 3) Morley-Koiter, and 4) Donnell. The exact procedure was compared favorably with experimental results and those obtained using the numerical finite element method. A literature review reveals that beam functions are used extensively as an approximation for simply supported boundary conditions. The effects of this approximate method were also investigated on the natural frequencies by comparing results with those of the exact analysis.

**Keywords:**
Circular Cylindrical Shell,
Free Vibration,
Natural Frequency.

##### 2759 New Exact Three-Wave Solutions for the (2+1)-Dimensional Asymmetric Nizhnik-Novikov-Veselov System

**Authors:**
Fadi Awawdeh,
O. Alsayyed

**Abstract:**

New exact three-wave solutions including periodic two-solitary solutions and doubly periodic solitary solutions for the (2+1)-dimensional asymmetric Nizhnik-Novikov- Veselov (ANNV) system are obtained using Hirota's bilinear form and generalized three-wave type of ansatz approach. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an e¤ective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics.

**Keywords:**
Soliton Solution,
Hirota Bilinear Method,
ANNV System.

##### 2758 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.