**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1442

# Search results for: Volterra integral equation

##### 1442 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

##### 1441 Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations

**Authors:**
Fuziyah Ishak,
Siti Norazura Ahmad

**Abstract:**

Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

**Keywords:**
Accuracy,
extended trapezoidal method,
numerical solution,
Volterra integro-differential equations.

##### 1440 Research of a Multistep Method Applied to Numerical Solution of Volterra Integro-Differential Equation

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

**Abstract:**

**Keywords:**
Volterra integro-differential equations,
multistepmethods,
finite-difference methods,
initial value problem

##### 1439 Spline Collocation for Solving System of Fredholm and Volterra Integral Equations

**Authors:**
N. Ebrahimi,
J. Rashidinia

**Abstract:**

In this paper, numerical solution of system of Fredholm and Volterra integral equations by means of the Spline collocation method is considered. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The solution is collocated by cubic B-spline and the integrand is approximated by the Newton-Cotes formula. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods to illustrate the accuracy and the implementation of our method.

**Keywords:**
Convergence analysis,
Cubic B-spline,
Newton-
Cotes formula,
System of Fredholm and Volterra integral equations.

##### 1438 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.

##### 1437 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.

##### 1436 Optimal Control of Volterra Integro-Differential Systems Based On Legendre Wavelets and Collocation Method

**Authors:**
Khosrow Maleknejad,
Asyieh Ebrahimzadeh

**Abstract:**

In this paper, the numerical solution of optimal control problem (OCP) for systems governed by Volterra integro-differential (VID) equation is considered. The method is developed by means of the Legendre wavelet approximation and collocation method. The properties of Legendre wavelet together with Gaussian integration method are utilized to reduce the problem to the solution of nonlinear programming one. Some numerical examples are given to confirm the accuracy and ease of implementation of the method.

**Keywords:**
Collocation method,
Legendre wavelet,
optimal control,
Volterra integro-differential equation.

##### 1435 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

##### 1434 The Dividend Payments for General Claim Size Distributions under Interest Rate

**Authors:**
Li-Li Li,
Jinghai Feng,
Lixin Song

**Abstract:**

**Keywords:**
Dividend payout,
Integro-differential equation,
Jumpdiffusion model,
Volterra equation

##### 1433 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.

##### 1432 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.

##### 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 Volterra Filtering Techniques for Removal of Gaussian and Mixed Gaussian-Impulse Noise

**Authors:**
M. B. Meenavathi,
K. Rajesh

**Abstract:**

In this paper, we propose a new class of Volterra series based filters for image enhancement and restoration. Generally the linear filters reduce the noise and cause blurring at the edges. Some nonlinear filters based on median operator or rank operator deal with only impulse noise and fail to cancel the most common Gaussian distributed noise. A class of second order Volterra filters is proposed to optimize the trade-off between noise removal and edge preservation. In this paper, we consider both the Gaussian and mixed Gaussian-impulse noise to test the robustness of the filter. Image enhancement and restoration results using the proposed Volterra filter are found to be superior to those obtained with standard linear and nonlinear filters.

**Keywords:**
Gaussian noise,
Image enhancement,
Imagerestoration,
Linear filters,
Nonlinear filters,
Volterra series.

##### 1420 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.

##### 1419 Volterra Filter for Color Image Segmentation

**Authors:**
M. B. Meenavathi,
K. Rajesh

**Abstract:**

**Keywords:**
Color image segmentation,
HSI space,
K–means
clustering,
Volterra filter.

##### 1418 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.

##### 1417 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.

##### 1416 Application of the Hybrid Methods to Solving Volterra Integro-Differential Equations

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

**Abstract:**

**Keywords:**
Integro-differential equations,
initial value
problem,
hybrid methods,
predictor-corrector method

##### 1415 2n Positive Periodic Solutions to n Species Non-autonomous Lotka-Volterra Competition Systems with Harvesting Terms

**Authors:**
Yongkun Li,
Kaihong Zhao

**Abstract:**

By using Mawhin-s continuation theorem of coincidence degree theory, we establish the existence of 2n positive periodic solutions for n species non-autonomous Lotka-Volterra competition systems with harvesting terms. An example is given to illustrate the effectiveness of our results.

**Keywords:**
Positive periodic solutions,
Lotka-Volterra competition system,
coincidence degree,
harvesting term.

##### 1414 Nonlinear Acoustic Echo Cancellation Using Volterra Filtering with a Variable Step-Size GS-PAP Algorithm

**Authors:**
J. B. Seo,
K. J. Kim,
S. W. Nam

**Abstract:**

**Keywords:**
Acoustic echo cancellation (AEC),
Volterra filtering,
variable step-size,
GS-PAP.

##### 1413 Basket Option Pricing under Jump Diffusion Models

**Authors:**
Ali Safdari-Vaighani

**Abstract:**

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