**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**56

# Search results for: integral equation

##### 56 Peridynamic Modeling of an Isotropic Plate under Tensile and Flexural Loading

**Authors:**
Eda Gök

**Abstract:**

Peridynamics is a new modeling concept of non-local interactions for solid structures. The formulations of Peridynamic (PD) theory are based on integral equations rather than differential equations. Through, undefined equations of associated problems are avoided. PD theory might be defined as continuum version of molecular dynamics. The medium is usually modeled with mass particles bonded together. Particles interact with each other directly across finite distances through central forces named as bonds. The main assumption of this theory is that the body is composed of material points which interact with other material points within a finite distance. Although, PD theory developed for discontinuities, it gives good results for structures which have no discontinuities. In this paper, displacement control of the isotropic plate under the effect of tensile and bending loading has been investigated by means of PD theory. A MATLAB code is generated to create PD bonds and corresponding surface correction factors. Using generated MATLAB code the geometry of the specimen is generated, and the code is implemented in Finite Element Software. The results obtained from non-local continuum theory are compared with the Finite Element Analysis results and analytical solution. The results show good agreement.

**Keywords:**
Flexural loading,
non-local continuum mechanics,
Peridynamic theory,
solid structures,
tensile loading.

##### 55 Reduction in Population Growth under Various Contraceptive Strategies in Uttar Pradesh, India

**Authors:**
Prashant Verma,
K. K. Singh,
Anjali Singh,
Ujjaval Srivastava

**Abstract:**

Contraceptive policies have been derived to achieve desired reductions in the growth rate and also, applied to the data of Uttar-Pradesh, India for illustration. Using the Lotka’s integral equation for the stable population, expressions for the proportion of contraceptive users at different ages have been obtained. At the age of 20 years, 42% of contraceptive users is imperative to reduce the present annual growth rate of 0.036 to 0.02, assuming that 40% of the contraceptive users discontinue at the age of 25 years and 30% again continue contraceptive use at age 30 years. Further, presuming that 75% of women start using contraceptives at the age of 23 years, and 50% of the remaining women start using contraceptives at the age of 28 years, while the rest of them start using it at the age of 32 years. If we set a minimum age of marriage as 20 years, a reduction of 0.019 in growth rate will be obtained. This study describes how the level of contraceptive use at different age groups of women reduces the growth rate in the state of Uttar Pradesh. The article also promotes delayed marriage in the region.

**Keywords:**
Child bearing,
contraceptive devices,
contraceptive policies,
population growth,
stable population.

##### 54 Generation of Numerical Data for the Facilitation of the Personalized Hyperthermic Treatment of Cancer with An Interstital Antenna Array Using the Method of Symmetrical Components

**Authors:**
Prodromos E. Atlamazoglou

**Abstract:**

**Keywords:**
Hyperthermia,
integral equations,
insulated antennas,
method of symmetrical components.

##### 53 Semilocal Convergence of a Three Step Fifth Order Iterative Method under Höolder Continuity Condition in Banach Spaces

**Authors:**
Ramandeep Behl,
Prashanth Maroju,
S. S. Motsa

**Abstract:**

**Keywords:**
Hölder continuity condition,
Fréchet derivative,
fifth
order convergence,
recurrence relations.

##### 52 Non-Local Behavior of a Mixed-Mode Crack in a Functionally Graded Piezoelectric Medium

**Authors:**
Nidhal Jamia,
Sami El-Borgi

**Abstract:**

In this paper, the problem of a mixed-Mode crack embedded in an infinite medium made of a functionally graded piezoelectric material (FGPM) with crack surfaces subjected to electro-mechanical loadings is investigated. Eringen’s non-local theory of elasticity is adopted to formulate the governing electro-elastic equations. The properties of the piezoelectric material are assumed to vary exponentially along a perpendicular plane to the crack. Using Fourier transform, three integral equations are obtained in which the unknown variables are the jumps of mechanical displacements and electric potentials across the crack surfaces. To solve the integral equations, the unknowns are directly expanded as a series of Jacobi polynomials, and the resulting equations solved using the Schmidt method. In contrast to the classical solutions based on the local theory, it is found that no mechanical stress and electric displacement singularities are present at the crack tips when nonlocal theory is employed to investigate the problem. A direct benefit is the ability to use the calculated maximum stress as a fracture criterion. The primary objective of this study is to investigate the effects of crack length, material gradient parameter describing FGPMs, and lattice parameter on the mechanical stress and electric displacement field near crack tips.

**Keywords:**
Functionally graded piezoelectric material,
mixed-mode crack,
non-local theory,
Schmidt method.

##### 51 Integral Methods in the Determination of Temperature Fields of Cooled Blades of Gas Turbines

**Authors:**
C. Ardil

**Abstract:**

**Keywords:**
Integral methods,
determination of temperature fields,
cooled blades,
gas turbines.

##### 50 Numerical Modeling of Temperature Fields in Aviation Gas Turbine Elements

**Authors:**
A. M. Pashaev,
R. A. Sadihov,
A. S. Samedov,
C. Ardil

**Abstract:**

A mathematical model and a numerical method for computing the temperature field of the profile part of convectionally cooled blades are developed. The theoretical substantiation of the method is proved by corresponding theorems. To this end, convergent quadrature processes were developed and error estimates were obtained in terms of the Zygmund continuity moduli. The boundary conditions for heat exchange are determined from the solution of the corresponding integral equations and empirical relations. The reliability of the developed methods is confirmed by calculation and experimental studies of the thermohydraulic characteristics of the nozzle apparatus of the first stage of the gas turbine.

**Keywords:**
Aviation gas turbine,
temperature field,
cooled blades,
numerical modeling.

##### 49 Nonlinear Slow Shear Alfven Waves in Electron- Positron-Ion Plasma Including Full Ion Dynamics

**Authors:**
B. Ghosh,
H. Sahoo,
K. K. Mondal

**Abstract:**

**Keywords:**
Alfven waves,
Sagdeev potential,
Solitary waves.

##### 48 The Solution of the Direct Problem of Electrical Prospecting with Direct Current under Conditions of Ground Surface Relief

**Authors:**
Balgaisha Mukanova,
Tolkyn Mirgalikyzy

**Abstract:**

Theory of interpretation of electromagnetic fields studied in the electrical prospecting with direct current is mainly developed for the case of a horizontal surface observation. However in practice we often have to work in difficult terrain surface. Conducting interpretation without the influence of topography can cause non-existent anomalies on sections. This raises the problem of studying the impact of different shapes of ground surface relief on the results of electrical prospecting's research. This research examines the numerical solutions of the direct problem of electrical prospecting for two-dimensional and three-dimensional media, taking into account the terrain. The problem is solved using the method of integral equations. The density of secondary currents on the relief surface is obtained.

**Keywords:**
Ground surface relief,
method of integral equations,
numerical method.

##### 47 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.

##### 46 Modeling of Temperature Fields of Gas Turbine Blades by Considering Heat Flow and Specified Temperature

**Authors:**
C. Ardil

**Abstract:**

**Keywords:**
Modeling of temperature fields,
gas turbine blades,
integral methods,
cooled blades,
gas turbines.

##### 45 Optimal Design for SARMA(P,Q)L Process of EWMA Control Chart

**Authors:**
Y. Areepong

**Abstract:**

The main goal of this paper is to study Statistical Process Control (SPC) with Exponentially Weighted Moving Average (EWMA) control chart when observations are serially-correlated. The characteristic of control chart is Average Run Length (ARL) which is the average number of samples taken before an action signal is given. Ideally, an acceptable ARL of in-control process should be enough large, so-called (ARL_{0}). Otherwise it should be small when the process is out-of-control, so-called Average of Delay Time (ARL_{1}) or a mean of true alarm. We find explicit formulas of ARL for EWMA control chart for Seasonal Autoregressive and Moving Average processes (SARMA) with Exponential white noise. The results of ARL obtained from explicit formula and Integral equation are in good agreement. In particular, this formulas for evaluating (ARL_{0}) and (ARL_{1}) be able to get a set of optimal parameters which depend on smoothing parameter (λ) and width of control limit (H) for designing EWMA chart with minimum of (ARL_{1}).

**Keywords:**
Average Run Length1,
Optimal parameters,
Exponentially Weighted Moving Average (EWMA) control chart.

##### 44 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.

##### 43 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.

##### 42 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.

##### 41 Continuous Adaptive Robust Control for Nonlinear Uncertain Systems

**Authors:**
Dong Sang Yoo

**Abstract:**

We consider nonlinear uncertain systems such that a priori information of the uncertainties is not available. For such systems, we assume that the upper bound of the uncertainties is represented as a Fredholm integral equation of the first kind and we propose an adaptation law that is capable of estimating the upper bound and design a continuous robust control which renders nonlinear uncertain systems ultimately bounded.

**Keywords:**
Adaptive Control,
Estimation,
Fredholm Integral,
Uncertain System.

##### 40 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.

##### 39 Mechanical Quadrature Methods for Solving First Kind Boundary Integral Equations of Stationary Stokes Problem

**Authors:**
Xin Luo,
Jin Huang,
Pan Cheng

**Abstract:**

By means of Sidi-Israeli’s quadrature rules, mechanical quadrature methods (MQMs) for solving the first kind boundary integral equations (BIEs) of steady state Stokes problem are presented. The convergence of numerical solutions by MQMs is proved based on Anselone’s collective compact and asymptotical compact theory, and the asymptotic expansions with the odd powers of the errors are provided, which implies that the accuracy of the approximations by MQMs possesses high accuracy order O (h3). Finally, the numerical examples show the efficiency of our methods.

**Keywords:**
Stokes problem,
boundary integral equation,
mechanical
quadrature methods,
asymptotic expansions.

##### 38 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.

##### 37 Numerical Solution of Hammerstein Integral Equations by Using Quasi-Interpolation

**Authors:**
M. Zarebnia,
S. Khani

**Abstract:**

In this paper first, a numerical method based on quasiinterpolation for solving nonlinear Fredholm integral equations of the Hammerstein-type is presented. Then, we approximate the solution of Hammerstein integral equations by Nystrom’s method. Also, we compare the methods with some numerical examples.

**Keywords:**
Hammerstein integral equations,
quasi-interpolation,
Nystrom’s method.

##### 36 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.

##### 35 Complexity Reduction Approach with Jacobi Iterative Method for Solving Composite Trapezoidal Algebraic Equations

**Authors:**
Mohana Sundaram Muthuvalu,
Jumat Sulaiman

**Abstract:**

In this paper, application of the complexity reduction approach based on half- and quarter-sweep iteration concepts with Jacobi iterative method for solving composite trapezoidal (CT) algebraic equations is discussed. The performances of the methods for CT algebraic equations are comparatively studied by their application in solving linear Fredholm integral equations of the second kind. Furthermore, computational complexity analysis and numerical results for three test problems are also included in order to verify performance of the methods.

**Keywords:**
Complexity reduction approach,
Composite trapezoidal
scheme,
Jacobi method,
Linear Fredholm integral equations

##### 34 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.

##### 33 Numerical Approximation to the Performance of CUSUM Charts for EMA (1) Process

**Authors:**
K. Petcharat,
Y. Areepong,
S. Sukparungsri,
G. Mititelu

**Abstract:**

**Keywords:**
Cumulative Sum Chart,
Moving Average
Observation,
Average Run Length,
Numerical Approximations.

##### 32 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

##### 31 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.

##### 30 On Modified Numerical Schemes in Vortex Element Method for 2D Flow Simulation Around Airfoils

**Authors:**
Ilia Marchevsky,
Victoriya Moreva

**Abstract:**

The problem of incompressible steady flow simulation around an airfoil is discussed. For some simplest airfoils (circular, elliptical, Zhukovsky airfoils) the exact solution is known from complex analysis. It allows to compute the intensity of vortex layer which simulates the airfoil. Some modifications of the vortex element method are proposed and test computations are carried out. It-s shown that the these approaches are much more effective in comparison with the classical numerical scheme.

**Keywords:**
Vortex element method,
vortex layer,
integral equation,
ill-conditioned matrix.

##### 29 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

##### 28 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

##### 27 Perfect Plastic Deformation of a Circular Thin Bronze Plate due to the Growth and Collapse of a Vapour Bubble

**Authors:**
M.T. Shervani-Tabar,
M. Rezaee,
E. Madadi Kandjani

**Abstract:**

Dynamics of a vapour bubble generated due to a high local energy input near a circular thin bronze plate in the absence of the buoyancy forces is numerically investigated in this paper. The bubble is generated near a thin bronze plate and during the growth and collapse of the bubble, it deforms the nearby plate. The Boundary Integral Equation Method is employed for numerical simulation of the problem. The fluid is assumed to be incompressible, irrotational and inviscid and the surface tension on the bubble boundary is neglected. Therefore the fluid flow around the vapour bubble can be assumed as a potential flow. Furthermore, the thin bronze plate is assumed to have perfectly plastic behaviour. Results show that the displacement of the circular thin bronze plate has considerable effect on the dynamics of its nearby vapour bubble. It is found that by decreasing the thickness of the thin bronze plate, the growth and collapse rate of the bubble becomes higher and consequently the lifetime of the bubble becomes shorter.

**Keywords:**
Vapour Bubble,
Thin Bronze Plate,
Boundary Integral
Equation Method.