Search results for: method of integral equations
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 8871

Search results for: method of integral equations

8871 The Approximate Solution of Linear Fuzzy Fredholm Integral Equations of the Second Kind by Using Iterative Interpolation

Authors: N. Parandin, M. A. Fariborzi Araghi

Abstract:

in this paper, we propose a numerical method for the approximate solution of fuzzy Fredholm functional integral equations of the second kind by using an iterative interpolation. For this purpose, we convert the linear fuzzy Fredholm integral equations to a crisp linear system of integral equations. The proposed method is illustrated by some fuzzy integral equations in numerical examples.

Keywords: Fuzzy function integral equations, Iterative method, Linear systems, Parametric form of fuzzy number.

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8870 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.

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8869 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.

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8868 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.

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8867 Integral Image-Based Differential Filters

Authors: Kohei Inoue, Kenji Hara, Kiichi Urahama

Abstract:

We describe a relationship between integral images and differential images. First, we derive a simple difference filter from conventional integral image. In the derivation, we show that an integral image and the corresponding differential image are related to each other by simultaneous linear equations, where the numbers of unknowns and equations are the same, and therefore, we can execute the integration and differentiation by solving the simultaneous equations. We applied the relationship to an image fusion problem, and experimentally verified the effectiveness of the proposed method.

Keywords: Integral images, differential images, differential filters, image fusion.

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8866 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.

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8865 High Accuracy Eigensolutions in Elasticity for Boundary Integral Equations by Nyström Method

Authors: Pan Cheng, Jin Huang, Guang Zeng

Abstract:

Elastic boundary eigensolution problems are converted into boundary integral equations by potential theory. The kernels of the boundary integral equations have both the logarithmic and Hilbert singularity simultaneously. We present the mechanical quadrature methods for solving eigensolutions of the boundary integral equations by dealing with two kinds of singularities at the same time. The methods possess high accuracy O(h3) and low computing complexity. The convergence and stability are proved based on Anselone-s collective compact theory. Bases on the asymptotic error expansion with odd powers, we can greatly improve the accuracy of the approximation, and also derive a posteriori error estimate which can be used for constructing self-adaptive algorithms. The efficiency of the algorithms are illustrated by numerical examples.

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

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8864 Solution of First kind Fredholm Integral Equation by Sinc Function

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

Abstract:

Sinc-collocation scheme is one of the new techniques used in solving numerical problems involving integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this paper, some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some algebraic equations. Then convergence with exponential rate is proved by a theorem to guarantee applicability of numerical technique. Finally, numerical examples are included to demonstrate the validity and applicability of the technique.

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

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8863 Research of a Multistep Method Applied to Numerical Solution of Volterra Integro-Differential Equation

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

Abstract:

Solution of some practical problems is reduced to the solution of the integro-differential equations. But for the numerical solution of such equations basically quadrature methods or its combination with multistep or one-step methods are used. The quadrature methods basically is applied to calculation of the integral participating in right hand side of integro-differential equations. As this integral is of Volterra type, it is obvious that at replacement with its integrated sum the upper limit of the sum depends on a current point in which values of the integral are defined. Thus we receive the integrated sum with variable boundary, to work with is hardly. Therefore multistep method with the constant coefficients, which is free from noted lack and gives the way for finding it-s coefficients is present.

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

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8862 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

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8861 Approximate Solution of Nonlinear Fredholm Integral Equations of the First Kind via Converting to Optimization Problems

Authors: Akbar H. Borzabadi, Omid S. Fard

Abstract:

In this paper we introduce an approach via optimization methods to find approximate solutions for nonlinear Fredholm integral equations of the first kind. To this purpose, we consider two stages of approximation. First we convert the integral equation to a moment problem and then we modify the new problem to two classes of optimization problems, non-constraint optimization problems and optimal control problems. Finally numerical examples is proposed.

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

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8860 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

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8859 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.

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8858 On One Application of Hybrid Methods For Solving Volterra Integral Equations

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

Abstract:

As is known, one of the priority directions of research works of natural sciences is introduction of applied section of contemporary mathematics as approximate and numerical methods to solving integral equation into practice. We fare with the solving of integral equation while studying many phenomena of nature to whose numerically solving by the methods of quadrature are mainly applied. Taking into account some deficiency of methods of quadrature for finding the solution of integral equation some sciences suggested of the multistep methods with constant coefficients. Unlike these papers, here we consider application of hybrid methods to the numerical solution of Volterra integral equation. The efficiency of the suggested method is proved and a concrete method with accuracy order p = 4 is constructed. This method in more precise than the corresponding known methods.

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

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8857 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.

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8856 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.

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8855 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.

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8854 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.

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8853 Integral Methods in the Determination of Temperature Fields of Cooled Blades of Gas Turbines

Authors: C. Ardil

Abstract:

A mathematical model and an effective numerical method for calculating the temperature field of the profile part of convection cooled blades have been 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 the calculation-experimental studies of the thermohydraulic characteristics of the nozzle apparatus of the first stage of a gas turbine.

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

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8852 Modeling of Temperature Fields of Gas Turbine Blades by Considering Heat Flow and Specified Temperature

Authors: C. Ardil

Abstract:

A new mathematical model for calculating the temperature field of the profile part of the cooled blades of gas turbines is developed. The theoretical substantiation of the method is based on the application of the method of potential theory (the method of boundary integral equations). The effectiveness of the implementation of the developed mathematical model is confirmed on the basis of a computational experiment.

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

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8851 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.

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8850 An Efficient Collocation Method for Solving the Variable-Order Time-Fractional Partial Differential Equations Arising from the Physical Phenomenon

Authors: Haniye Dehestani, Yadollah Ordokhani

Abstract:

In this work, we present an efficient approach for solving variable-order time-fractional partial differential equations, which are based on Legendre and Laguerre polynomials. First, we introduced the pseudo-operational matrices of integer and variable fractional order of integration by use of some properties of Riemann-Liouville fractional integral. Then, applied together with collocation method and Legendre-Laguerre functions for solving variable-order time-fractional partial differential equations. Also, an estimation of the error is presented. At last, we investigate numerical examples which arise in physics to demonstrate the accuracy of the present method. In comparison results obtained by the present method with the exact solution and the other methods reveals that the method is very effective.

Keywords: Collocation method, fractional partial differential equations, Legendre-Laguerre functions, pseudo-operational matrix of integration.

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8849 Fixed Point Equations Related to Motion Integrals in Renormalization Hopf Algebra

Authors: Ali Shojaei-Fard

Abstract:

In this paper we consider quantum motion integrals depended on the algebraic reconstruction of BPHZ method for perturbative renormalization in two different procedures. Then based on Bogoliubov character and Baker-Campbell-Hausdorff (BCH) formula, we show that how motion integral condition on components of Birkhoff factorization of a Feynman rules character on Connes- Kreimer Hopf algebra of rooted trees can determine a family of fixed point equations.

Keywords: Birkhoff Factorization, Connes-Kreimer Hopf Algebra of Rooted Trees, Integral Renormalization, Lax Pair Equation, Rota- Baxter Algebras.

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8848 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.

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8847 Application of the Hybrid Methods to Solving Volterra Integro-Differential Equations

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

Abstract:

Beginning from the creator of integro-differential equations Volterra, many scientists have investigated these equations. Classic method for solving integro-differential equations is the quadratures method that is successfully applied up today. Unlike these methods, Makroglou applied hybrid methods that are modified and generalized in this paper and applied to the numerical solution of Volterra integro-differential equations. The way for defining the coefficients of the suggested method is also given.

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

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8846 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.

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8845 Active Tendons for Seismic Control of Buildings

Authors: S. M. Nigdeli, M. H. Boduroglu

Abstract:

In this study, active tendons with Proportional Integral Derivation type controllers were applied to a SDOF and a MDOF building model. Physical models of buildings were constituted with virtual springs, dampers and rigid masses. After that, equations of motion of all degrees of freedoms were obtained. Matlab Simulink was utilized to obtain the block diagrams for these equations of motion. Parameters for controller actions were found by using a trial method. After earthquake acceleration data were applied to the systems, building characteristics such as displacements, velocities, accelerations and transfer functions were analyzed for all degrees of freedoms. Comparisons on displacement vs. time, velocity vs. time, acceleration vs. time and transfer function (Db) vs. frequency (Hz) were made for uncontrolled and controlled buildings. The results show that the method seems feasible.

Keywords: Active Tendons, Proportional Integral DerivationType Controllers, SDOF, MDOF, Earthquake, Building.

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8844 MP-SMC-I Method for Slip Suppression of Electric Vehicles under Braking

Authors: Tohru Kawabe

Abstract:

In this paper, a new SMC (Sliding Mode Control) method with MP (Model Predictive Control) integral action for the slip suppression of EV (Electric Vehicle) under braking is proposed. The proposed method introduce the integral term with standard SMC gain , where the integral gain is optimized for each control period by the MPC algorithms. The aim of this method is to improve the safety and the stability of EVs under braking by controlling the wheel slip ratio. There also include numerical simulation results to demonstrate the effectiveness of the method.

Keywords: Sliding Mode Control, Model Predictive Control, Integral Action, Electric Vehicle, Slip suppression.

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8843 Automatic Iterative Methods for the Multivariate Solution of Nonlinear Algebraic Equations

Authors: Rafat Alshorman, Safwan Al-Shara', I. Obeidat

Abstract:

Most real world systems express themselves formally as a set of nonlinear algebraic equations. As applications grow, the size and complexity of these equations also increase. In this work, we highlight the key concepts in using the homotopy analysis method as a methodology used to construct efficient iteration formulas for nonlinear equations solving. The proposed method is experimentally characterized according to a set of determined parameters which affect the systems. The experimental results show the potential and limitations of the new method and imply directions for future work.

Keywords: Nonlinear Algebraic Equations, Iterative Methods, Homotopy Analysis Method.

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8842 A Modification on Newton's Method for Solving Systems of Nonlinear Equations

Authors: Jafar Biazar, Behzad Ghanbari

Abstract:

In this paper, we are concerned with the further study for system of nonlinear equations. Since systems with inaccurate function values or problems with high computational cost arise frequently in science and engineering, recently such systems have attracted researcher-s interest. In this work we present a new method which is independent of function evolutions and has a quadratic convergence. This method can be viewed as a extension of some recent methods for solving mentioned systems of nonlinear equations. Numerical results of applying this method to some test problems show the efficiently and reliability of method.

Keywords: System of nonlinear equations.

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