**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1434

# Search results for: Integral equation

##### 1434 On One Application of Hybrid Methods For Solving Volterra Integral Equations

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

**Abstract:**

**Keywords:**
Volterra integral equation,
hybrid methods,
stability
and degree,
methods of quadrature

##### 1433 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

##### 1432 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.

##### 1431 On the Existence and Global Attractivity of Solutions of a Functional Integral Equation

**Authors:**
Asadollah Aghajani,
Yaghoub Jalilian

**Abstract:**

Using the concept of measure of noncompactness, we present some results concerning the existence, uniform local attractivity and global attractivity of solutions for a functional integral equation. Our results improve and extend some previous known results and based on weaker conditions. Some examples which show that our results are applicable when the previous results are inapplicable are also included.

**Keywords:**
Functional integral equation,
fixed-point,
measure of noncompactness,
attractive solution,
asymptotic stability.

##### 1430 Solution of First kind Fredholm Integral Equation by Sinc Function

**Authors:**
Khosrow Maleknejad,
Reza Mollapourasl,
Parvin Torabi,
Mahdiyeh Alizadeh,

**Abstract:**

**Keywords:**
Integral equation,
Fredholm type,
Collocation method,
Sinc approximation.

##### 1429 Approximate Solution of Nonlinear Fredholm Integral Equations of the First Kind via Converting to Optimization Problems

**Authors:**
Akbar H. Borzabadi,
Omid S. Fard

**Abstract:**

**Keywords:**
Fredholm integral equation,
Optimization method,
Optimal control,
Nonlinear and linear programming

##### 1428 Advanced Gronwall-Bellman-Type Integral Inequalities and Their Applications

**Authors:**
Zixin Liu,
Shu Lü,
Shouming Zhong,
Mao Ye

**Abstract:**

**Keywords:**
Gronwall-Bellman-Type integral inequalities,
integrodifferential equation,
p-exponentially stable,
mixed delays.

##### 1427 Existence of Solutions for a Nonlinear Fractional Differential Equation with Integral Boundary Condition

**Abstract:**

This paper deals with a nonlinear fractional differential equation with integral boundary condition of the following form: Dαt x(t) = f(t, x(t),Dβ t x(t)), t ∈ (0, 1), x(0) = 0, x(1) = 1 0 g(s)x(s)ds, where 1 < α ≤ 2, 0 < β < 1. Our results are based on the Schauder fixed point theorem and the Banach contraction principle.

**Keywords:**
Fractional differential equation,
Integral boundary condition,
Schauder fixed point theorem,
Banach contraction principle.

##### 1426 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

##### 1425 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.

##### 1424 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.

##### 1423 Optimal Relaxation Parameters for Obtaining Efficient Iterative Methods for the Solution of Electromagnetic Scattering Problems

**Authors:**
Nadaniela Egidi,
Pierluigi Maponi

**Abstract:**

The approximate solution of a time-harmonic electromagnetic scattering problem for inhomogeneous media is required in several application contexts and its two-dimensional formulation is a Fredholm integral equation of second kind. This integral equation provides a formulation for the direct scattering problem but has to be solved several times in the numerical solution of the corresponding inverse scattering problem. The discretization of this Fredholm equation produces large and dense linear systems that are usually solved by iterative methods. To improve the efficiency of these iterative methods, we use the Symmetric SOR preconditioning and propose an algorithm to evaluate the associated relaxation parameter. We show the efficiency of the proposed algorithm by several numerical experiments, where we use two Krylov subspace methods, i.e. Bi-CGSTAB and GMRES.

**Keywords:**
Fredholm integral equation,
iterative method,
preconditioning,
scattering problem.

##### 1422 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.

##### 1421 The Application of Hybrid Orthonomal Bernstein and Block-Pulse Functions in Finding Numerical Solution of Fredholm Fuzzy Integral Equations

**Authors:**
Mahmoud Zarrini,
Sanaz Torkaman

**Abstract:**

In this paper, we have proposed a numerical method for solving fuzzy Fredholm integral equation of the second kind. In this method a combination of orthonormal Bernstein and Block-Pulse functions are used. In most cases, the proposed method leads to the exact solution. The advantages of this method are shown by an example and calculate the error analysis.

**Keywords:**
Fuzzy Fredholm Integral Equation,
Bernstein,
Block-Pulse,
Orthonormal.

##### 1420 Power Series Form for Solving Linear Fredholm Integral Equations of Second Order via Banach Fixed Point Theorem

**Authors:**
Adil AL-Rammahi

**Abstract:**

In this paper, a new method for solution of second order linear Fredholm integral equation in power series form was studied. The result is obtained by using Banach fixed point theorem.

**Keywords:**
Fredholm integral equation,
power series,
Banach fixed point theorem,
Linear Systems.

##### 1419 The Particle Swarm Optimization Against the Runge’s Phenomenon: Application to the Generalized Integral Quadrature Method

**Authors:**
A. Zerarka,
A. Soukeur,
N. Khelil

**Abstract:**

In the present work, we introduce the particle swarm optimization called (PSO in short) to avoid the Runge-s phenomenon occurring in many numerical problems. This new approach is tested with some numerical examples including the generalized integral quadrature method in order to solve the Volterra-s integral equations

**Keywords:**
Integral equation,
particle swarm optimization,
Runge's phenomenon.

##### 1418 Numerical Solution for Elliptical Crack with Developing Cusps Subject to Shear Loading

**Authors:**
Nik Mohd Asri Nik Long,
Koo Lee Feng,
Zainidin K. Eshkuvatov,
A. A. Khaldjigitov

**Abstract:**

This paper study the behavior of the solution at the crack edges for an elliptical crack with developing cusps, Ω in the plane elasticity subjected to shear loading. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over Ω and it is then transformed into a similar equation over a circular region, D, using conformal mapping. An appropriate collocation points are chosen on the region D to reduce the hypersingular integral equation into a system of linear equations with (2N+1)(N+1) unknown coefficients, which will later be used in the determination of shear stress intensity factors and maximum shear stress intensity. Numerical solution for the considered problem are compared with the existing asymptotic solution, and displayed graphically. Our results give a very good agreement to the existing asymptotic solutions.

**Keywords:**
Elliptical crack,
stress intensity factors,
hyper singular integral equation,
shear loading,
conformal mapping.

##### 1417 Basket Option Pricing under Jump Diffusion Models

**Authors:**
Ali Safdari-Vaighani

**Abstract:**

**Keywords:**
Radial basis function,
basket option,
jump diffusion,
RBF-PUM.

##### 1416 An Asymptotic Formula for Pricing an American Exchange Option

**Authors:**
Hsuan-Ku Liu

**Abstract:**

In this paper, the American exchange option (AEO) valuation problem is modelled as a free boundary problem. The critical stock price for an AEO is satisfied an integral equation implicitly. When the remaining time is large enough, an asymptotic formula is provided for pricing an AEO. The numerical results reveal that our asymptotic pricing formula is robust and accurate for the long-term AEO.

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

##### 1415 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.

##### 1414 Contact Problem for an Elastic Layered Composite Resting on Rigid Flat Supports

**Authors:**
T. S. Ozsahin,
V. Kahya,
A. Birinci,
A. O. Cakiroglu

**Abstract:**

**Keywords:**
Frictionless contact,
Layered composite,
Singularintegral equation,
The theory of elasticity.

##### 1413 High Accuracy Eigensolutions in Elasticity for Boundary Integral Equations by Nyström Method

**Authors:**
Pan Cheng,
Jin Huang,
Guang Zeng

**Abstract:**

**Keywords:**
boundary integral equation,
extrapolation algorithm,
aposteriori error estimate,
elasticity.

##### 1412 Displacement Solution for a Static Vertical Rigid Movement of an Interior Circular Disc in a Transversely Isotropic Tri-Material Full-Space

**Authors:**
D. Mehdizadeh,
M. Rahimian,
M. Eskandari-Ghadi

**Abstract:**

This article is concerned with the determination of the static interaction of a vertically loaded rigid circular disc embedded at the interface of a horizontal layer sandwiched in between two different transversely isotropic half-spaces called as tri-material full-space. The axes of symmetry of different regions are assumed to be normal to the horizontal interfaces and parallel to the movement direction. With the use of a potential function method, and by implementing Hankel integral transforms in the radial direction, the government partial differential equation for the solely scalar potential function is transformed to an ordinary 4th order differential equation, and the mixed boundary conditions are transformed into a pair of integral equations called dual integral equations, which can be reduced to a Fredholm integral equation of the second kind, which is solved analytically. Then, the displacements and stresses are given in the form of improper line integrals, which is due to inverse Hankel integral transforms. It is shown that the present solutions are in exact agreement with the existing solutions for a homogeneous full-space with transversely isotropic material. To confirm the accuracy of the numerical evaluation of the integrals involved, the numerical results are compared with the solutions exists for the homogeneous full-space. Then, some different cases with different degrees of material anisotropy are compared to portray the effect of degree of anisotropy.

**Keywords:**
Transversely isotropic,
rigid disc,
elasticity,
dual integral equations,
tri-material full-space.

##### 1411 Hybrid Function Method for Solving Nonlinear Fredholm Integral Equations of the Second Kind

**Authors:**
jianhua Hou,
Changqing Yang,
and Beibo Qin

**Abstract:**

A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm type equations which have many applications in mathematical physics are then considered. The method is based on hybrid function approximations. The properties of hybrid of block-pulse functions and Chebyshev polynomials are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

**Keywords:**
Hybrid functions,
Fredholm integral equation,
Blockpulse,
Chebyshev polynomials,
product operational matrix.

##### 1410 Order Reduction of Linear Dynamic Systems using Stability Equation Method and GA

**Authors:**
G. Parmar,
R. Prasad,
S. Mukherjee

**Abstract:**

The authors present an algorithm for order reduction of linear dynamic systems using the combined advantages of stability equation method and the error minimization by Genetic algorithm. The denominator of the reduced order model is obtained by the stability equation method and the numerator terms of the lower order transfer function are determined by minimizing the integral square error between the transient responses of original and reduced order models using Genetic algorithm. The reduction procedure is simple and computer oriented. It is shown that the algorithm has several advantages, e.g. the reduced order models retain the steady-state value and stability of the original system. The proposed algorithm has also been extended for the order reduction of linear multivariable systems. Two numerical examples are solved to illustrate the superiority of the algorithm over some existing ones including one example of multivariable system.

**Keywords:**
Genetic algorithm,
Integral square error,
Orderreduction,
Stability equation method.

##### 1409 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.

##### 1408 Flutter Analysis of Slender Beams with Variable Cross Sections Based on Integral Equation Formulation

**Authors:**
Z. El Felsoufi,
L. Azrar

**Abstract:**

This paper studies a mathematical model based on the integral equations for dynamic analyzes numerical investigations of a non-uniform or multi-material composite beam. The beam is subjected to a sub-tangential follower force and elastic foundation. The boundary conditions are represented by generalized parameterized fixations by the linear and rotary springs. A mathematical formula based on Euler-Bernoulli beam theory is presented for beams with variable cross-sections. The non-uniform section introduces non-uniformity in the rigidity and inertia of beams and consequently, more complicated equilibrium who governs the equation. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebro-differential system related to internal and boundary unknowns. A generalized formula for the deflection, the slope, the moment and the shear force are presented. The free vibration of non-uniform loaded beams is formulated in a compact matrix form and all needed matrices are explicitly given. The dynamic stability analysis of slender beam is illustrated numerically based on the coalescence criterion. A realistic case related to an industrial chimney is investigated.

**Keywords:**
Chimney,
BEM and integral equation formulation,
non uniform cross section,
vibration and Flutter.

##### 1407 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.

##### 1406 On Fourier Type Integral Transform for a Class of Generalized Quotients

**Authors:**
A. S. Issa,
S. K. Q. AL-Omari

**Abstract:**

**Keywords:**
Fourier type integral,
Fourier integral,
generalized
quotient,
Boehmian,
distribution.

##### 1405 Frictionless Contact Problem Between Two Orthotropic Elastic Layers

**Authors:**
V. Kahya,
A. Birinci,
R. Erdol

**Abstract:**

**Keywords:**
Frictionless contact,
Initial separation,
Orthotropicmaterial,
Singular integral equation.