**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1067

# Search results for: Cauchy equation

##### 1067 Existence of Iterative Cauchy Fractional Differential Equation

**Authors:**
Rabha W. Ibrahim

**Abstract:**

Our main aim in this paper is to use the technique of non expansive operators to more general iterative and non iterative fractional differential equations (Cauchy type ). The non integer case is taken in sense of Riemann-Liouville fractional operators. Applications are illustrated.

**Keywords:**
Fractional calculus,
fractional differential equation,
Cauchy equation,
Riemann-Liouville fractional operators,
Volterra
integral equation,
non-expansive mapping,
iterative differential equation.

##### 1066 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

##### 1065 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.

##### 1064 Stability of Fractional Differential Equation

**Authors:**
Rabha W. Ibrahim

**Abstract:**

We study a Dirichlet boundary value problem for Lane-Emden equation involving two fractional orders. Lane-Emden equation has been widely used to describe a variety of phenomena in physics and astrophysics, including aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres,and thermionic currents. However, ordinary Lane-Emden equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractalmedium, numerous generalizations of Lane-Emden equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Lane-Emden equation. This gives rise to the fractional Lane-Emden equation with a single index. Recently, a new type of Lane-Emden equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskiis fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space. Ulam-Hyers stability for iterative Cauchy fractional differential equation is defined and studied.

**Keywords:**
Fractional calculus,
fractional differential equation,
Lane-Emden equation,
Riemann-Liouville fractional operators,
Volterra integral equation.

##### 1063 On Method of Fundamental Solution for Nondestructive Testing

**Abstract:**

**Keywords:**
ill-posed,
TSVD,
Laplace's equation,
inverse problem,
L-curve,
Generalized Cross Validation.

##### 1062 Study on Optimal Control Strategy of PM2.5 in Wuhan, China

**Authors:**
Qiuling Xie,
Shanliang Zhu,
Zongdi Sun

**Abstract:**

In this paper, we analyzed the correlation relationship among PM2.5 from other five Air Quality Indices (AQIs) based on the grey relational degree, and built a multivariate nonlinear regression equation model of PM2.5 and the five monitoring indexes. For the optimal control problem of PM2.5, we took the partial large Cauchy distribution of membership equation as satisfaction function. We established a nonlinear programming model with the goal of maximum performance to price ratio. And the optimal control scheme is given.

**Keywords:**
Grey relational degree,
multiple linear regression,
membership function,
nonlinear programming.

##### 1061 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.

##### 1060 Learning an Overcomplete Dictionary using a Cauchy Mixture Model for Sparse Decay

**Authors:**
E. S. Gower,
M. O. J. Hawksford

**Abstract:**

**Keywords:**
expectation-maximization,
Pitman estimator,
sparsedecomposition

##### 1059 Integral Operators Related to Problems of Interface Dynamics

**Authors:**
Pa Pa Lin

**Abstract:**

This research work is concerned with the eigenvalue problem for the integral operators which are obtained by linearization of a nonlocal evolution equation. The purpose of section II.A is to describe the nature of the problem and the objective of the project. The problem is related to the “stable solution" of the evolution equation which is the so-called “instanton" that describe the interface between two stable phases. The analysis of the instanton and its asymptotic behavior are described in section II.C by imposing the Green function and making use of a probability kernel. As a result , a classical Theorem which is important for an instanton is proved. Section III devoted to a study of the integral operators related to interface dynamics which concern the analysis of the Cauchy problem for the evolution equation with initial data close to different phases and different regions of space.

**Keywords:**
Evolution,
Green function,
instanton,
integral operators.

##### 1058 Analytical Solution of Time-Harmonic Torsional Vibration of a Cylindrical Cavity in a Half-Space

**Authors:**
M.Eskandari-Ghadi,
M.Mahmoodian

**Abstract:**

**Keywords:**
Cosine transform,
Half space,
Isotropic,
Singular
integral equation,
Torsion

##### 1057 On the Integer Solutions of the Pell Equation x2 - dy2 = 2t

**Authors:**
Ahmet Tekcan,
Betül Gezer,
Osman Bizim

**Abstract:**

Let k ≥ 1 and t ≥ 0 be two integers and let d = k2 + k be a positive non-square integer. In this paper, we consider the integer solutions of Pell equation x2 - dy2 = 2t. Further we derive a recurrence relation on the solutions of this equation.

**Keywords:**
Pell equation,
Diophantine equation.

##### 1056 On the Flow of a Third Grade Viscoelastic Fluid in an Orthogonal Rheometer

**Authors:**
Carmen D. Pricinâ,
E. Corina Cipu,
Victor Ţigoiu

**Abstract:**

The flow of a third grade fluid in an orthogonal rheometer is studied. We employ the admissible velocity field proposed in [5]. We solve the problem and obtain the velocity field as well as the components for the Cauchy tensor. We compare the results with those from [9]. Some diagrams concerning the velocity and Cauchy stress components profiles are presented for different values of material constants and compared with the corresponding values for a linear viscous fluid.

**Keywords:**
Non newtonian fluid flow,
orthogonal rheometer,
third grade fluid.

##### 1055 The Proof of Two Conjectures Related to Pell-s Equation x2 −Dy2 = ± 4

**Authors:**
Armend Sh. Shabani

**Abstract:**

**Keywords:**
Pell's equation,
solutions of Pell's equation.

##### 1054 An Analytical Method for Solving General Riccati Equation

**Authors:**
Y. Pala,
M. O. Ertas

**Abstract:**

In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

**Keywords:**
Riccati Equation,
ordinary differential equation,
nonlinear differential equation,
analytical solution,
proper solution.

##### 1053 The Pell Equation x2 − Py2 = Q

**Authors:**
Ahmet Tekcan,
Arzu Özkoç,
Canan Kocapınar,
Hatice Alkan

**Abstract:**

**Keywords:**
Pell equation,
solutions of Pell equation.

##### 1052 The Diophantine Equation y2 − 2yx − 3 = 0 and Corresponding Curves over Fp

**Authors:**
Ahmet Tekcan,
Arzu Özkoç,
Hatice Alkan

**Abstract:**

**Keywords:**
Diophantine equation,
Pell equation,
quadratic form.

##### 1051 Solution of The KdV Equation with Asymptotic Degeneracy

**Authors:**
Tapas Kumar Sinha,
Joseph Mathew

**Abstract:**

Recently T. C. Au-Yeung, C.Au, and P. C. W. Fung [2] have given the solution of the KdV equation [1] to the boundary condition , where b is a constant. We have further extended the method of [2] to find the solution of the KdV equation with asymptotic degeneracy. Via simulations we find both bright and dark Solitons (i.e. Solitons with opposite phases).

**Keywords:**
KdV equation,
Asymptotic Degeneracy,
Solitons,
Inverse Scattering

##### 1050 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.

##### 1049 Study of Cahn-Hilliard Equation to Simulate Phase Separation

**Authors:**
Nara Guimarães,
Marcelo Aquino Martorano,
Douglas Gouvêa

**Abstract:**

An investigation into Cahn-Hilliard equation was carried out through numerical simulation to identify a possible phase separation for one and two dimensional domains. It was observed that this equation can reproduce important mass fluxes necessary for phase separation within the miscibility gap and for coalescence of particles.

**Keywords:**
Cahn-Hilliard equation,
miscibility gap,
phase
separation.

##### 1048 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.

##### 1047 Traveling Wave Solutions for the Sawada-Kotera-Kadomtsev-Petviashivili Equation and the Bogoyavlensky-Konoplechenko Equation by (G'/G)- Expansion Method

**Authors:**
Nisha Goyal,
R.K. Gupta

**Abstract:**

This paper presents a new function expansion method for finding traveling wave solutions of a nonlinear equations and calls it the G G -expansion method, given by Wang et al recently. As an application of this new method, we study the well-known Sawada-Kotera-Kadomtsev-Petviashivili equation and Bogoyavlensky-Konoplechenko equation. With two new expansions, general types of soliton solutions and periodic solutions for these two equations are obtained.

**Keywords:**
Sawada-Kotera-Kadomtsev-Petviashivili equation,
Bogoyavlensky-Konoplechenko equation,

##### 1046 Traveling Wave Solutions for Shallow Water Wave Equation by (G'/G)-Expansion Method

**Authors:**
Anjali Verma,
Ram Jiwari,
Jitender Kumar

**Abstract:**

This paper presents a new function expansion method for finding traveling wave solution of a non-linear equation and calls it the (G'/G)-expansion method. The shallow water wave equation is reduced to a non linear ordinary differential equation by using a simple transformation. As a result the traveling wave solutions of shallow water wave equation are expressed in three forms: hyperbolic solutions, trigonometric solutions and rational solutions.

**Keywords:**
Shallow water wave equation,
Exact solutions,
(G'/G) expansion method.

##### 1045 Robust Coherent Noise Suppression by Point Estimation of the Cauchy Location Parameter

**Authors:**
Ephraim Gower,
Thato Tsalaile,
Monageng Kgwadi,
Malcolm Hawksford.

**Abstract:**

This paper introduces a new point estimation algorithm, with particular focus on coherent noise suppression, given several measurements of the device under test where it is assumed that 1) the noise is first-order stationery and 2) the device under test is linear and time-invariant. The algorithm exploits the robustness of the Pitman estimator of the Cauchy location parameter through the initial scaling of the test signal by a centred Gaussian variable of predetermined variance. It is illustrated through mathematical derivations and simulation results that the proposed algorithm is more accurate and consistently robust to outliers for different tailed density functions than the conventional methods of sample mean (coherent averaging technique) and sample median search.

**Keywords:**
Central limit theorem,
Fisher-Cramer Rao,
gamma
function,
Pitman estimator.

##### 1044 Ginzburg-Landau Model for Curved Two-Phase Shallow Mixing Layers

**Authors:**
Irina Eglite,
Andrei A. Kolyshkin

**Abstract:**

**Keywords:**
Shallow water equations,
mixing layer,
weakly
nonlinear analysis,
Ginzburg-Landau equation

##### 1043 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.

##### 1042 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.

##### 1041 Extend Three-wave Method for the (3+1)-Dimensional Soliton Equation

**Authors:**
Somayeh Arbabi Mohammad-Abadi,
Maliheh Najafi

**Abstract:**

In this paper, we study (3+1)-dimensional Soliton equation. We employ the Hirota-s bilinear method to obtain the bilinear form of (3+1)-dimensional Soliton equation. Then by the idea of extended three-wave method, some exact soliton solutions including breather type solutions are presented.

**Keywords:**
Three-wave method,
(3+1)-dimensional Soliton equation,
Hirota's bilinear form.

##### 1040 Parallel Algorithm for Numerical Solution of Three-Dimensional Poisson Equation

**Authors:**
Alibek Issakhov

**Abstract:**

**Keywords:**
MPI,
OpenMP,
three dimensional Poisson equation

##### 1039 Iterative solutions to the linear matrix equation AXB + CXTD = E

**Authors:**
Yongxin Yuan,
Jiashang Jiang

**Abstract:**

**Keywords:**
matrix equation,
iterative algorithm,
parameter estimation,
minimum norm solution.

##### 1038 Mechanical Equation of State in an Al-Li Alloy

**Authors:**
Jung-Ho Moon,
Tae Kwon Ha

**Abstract:**

Existence of plastic equation of state has been investigated by performing a series of load relaxation tests at various temperatures using an Al-Li alloy. A plastic equation of state is first developed from a simple kinetics consideration for a mechanical activation process of a leading dislocation piled up against grain boundaries. A series of load relaxation test has been conducted at temperatures ranging from 200 to 530^{o}C to obtain the stress-strain rate curves. A plastic equation of state has been derived from a simple consideration of dislocation kinetics and confirmed by experimental results.

**Keywords:**
Plastic equation of state,
Dislocation kinetics,
Load relaxation test,
Al-Li alloy,
Microstructure.