My thesis presents the application of fuzzy integrals as tool for criteria aggregation in the decision problems. Naji qatanani abstract integral equations, in general, play a very important role in engineering and technology due to their wide range of applications. Greens functions as the kernel of the integral equations are introduced using simple practical problems. Volume 108, issue 2, 1 december 1999, pages 193200. Section 5 and 6 describe an experiment pertaining to syllableproximity evaluation using the fuzzy integrationbased aggregation. Inthispaper,weintroducetwodimensionalfuzzy integrals and propose some generalized quadrature rules and their dependent theorems for mappings of bounded variation. Some practical problems are solved in this chapter. Theory and numerical solution of volterra functional integral. Numerical solution of twodimensional linear fuzzy fredholm integral equations by the fuzzy lagrange interpolation article pdf available in advances in fuzzy systems 20181.
Method for solving fuzzy integrodifferential equation by. Existence and uniqueness theorem for fuzzy integral equation. In 12 nonlinear integral equation of volterra type are considered as follows. Fuzzy calculus is the study of theory and applications of integrals and derivatives of uncertain functions. Linearity versus nonlinearity of integral equations 1 4.
On the convergence of twodimensional fuzzy volterra. Numerical solution of linear fredholm fuzzy integral equations of the second kind by. This fuzzy procedure avoids the time consuming process of defuzzification required in a mamdani fuzzy model chengshion shieh 2014. Application of fuzzy laplace transforms for solving fuzzy. Solving fuzzy volterra integral equations via fuzzy sumudu. Solving a system of fuzzy integral equations by an analytic method. Numerical solution of fredholm fuzzy integral equations of the second kind via direct method using triangular functions.
Existence of solutions of general nonlinear fuzzy volterra. If f is nonzero, it is called an inhomogeneous integral equation. Numerical solution of nonlinear urisohnvolterra fuzzy functional. Also, the fuzzy integral equations have been studied by several authors, 14, 15. The fuzzy twodimensional differential transform method of fixed grid size is used to find approximate solutions of fpdes. We introduce a definition of the integral of a fuzzyvalued function that is only slightly different from the usual one, yet that is more intuitive and that can be applied to a larger class of functions. Finally in chapter 3, approximate methods for solving integral equations are discussed. Stochastic fuzzy differential equations with an application 125 where kk denotes a norm in ird. Pdf solving fuzzy integral equations of the second kind.
As an application, we propose an iterative numerical method in order to approximate the solution of nonlinear fuzzy fredholm integral equations in two variables, the fuzzy cubature rule being used in the construction of the numerical method. Liao employed the basic idea of the homotopy in topology to propose a general analytic method for nonlinear problems, namely ham see the monograph 15, and the papers 1618. Theory and applications of fuzzy volterra integral equations. Fuzzy volterra integral equations the integral equations which are discussed in this section are the volterra integral equations of the second kind. Pdf study of the approximate solution of fuzzy volterra.
There are plenty of solved examples in the text to illustrate the methods, along with problems to solve. A numerical method for solving fuzzy volterrafredholm integral equation of the second kind will is introduced. Hou,ad fuzzy integral equations john mordeson and william newman department of mathematicscomputer science, creighton unilersity, omaha, nebraska 68178 abstract we introduce a definition of the integral of a fuzzy valued function that is only slightly different from the usual one, yet that is more intuitive and that can be applied to a larger class of functions. Mt5802 integral equations introduction integral equations occur in a variety of applications, often being obtained from a differential equation.
A computational method based on hybrid of bernstein and blockpulse functions for solving linear fuzzy fredholm integral equations system journal of taibah university for science, volume 9, issue 2, 2015, pp. Theory and numerical solution of volterra functional integral equations hermann brunner department of mathematics and statistics memorial university of newfoundland st. Fuzzy logic is a very broad concept which includes fuzzy set theory, fuzzy measure, fuzzy integral, fuzzy control theory, fuzzy decision theory etc. So, instead of using deterministic models, we provide fuzzy integral equations of both linear and nonlinear forms. Integral equations occur in a variety of applications, often being obtained from a differential equation. The application of fuzzy integrals in multicriteria. Single and multidimensional integral equations david keffer department of chemical engineering university of tennessee, knoxville august 1999 table of contents 1. Recently, bede introduced a strongly generalized di. Also, in 2018, nouriani and ezzati 27 solved twodimensional linear fuzzy integral equation by using the fuzzy lagrange interpola. Abstract using fuzzy laplace transform method, the solution of fuzzy convolution volterra integral equation fcvie of the second kind with convolution fuzzy and crisp kernel is investigated. In this paper, we introduce twodimensional fuzzy integrals and propose some generalized quadrature rules and their depended theorems for henstock integrable, bounded mappings. Fuzzy number, fuzzy linear system, fuzzy integral equations 1 introduction the concept of integration of fuzzy functions was. Section 7 provides a brief summary of the conclusions. Solving linear fredholm fuzzy integral equations system by taylor expansion method a.
Fuzzy transform has been proposed as a pilot fuzzy approximation technique with the aim of being applied in up to now unusual application. The next covers fuzzy numbers and explains zadehs extension principle. Tiraie, afshin, a numerical method for solving double integral equations 2005. Analytical and numerical methods for solving linear fuzzy. Numerical solution of interval and fuzzy system of linear. Definition and background a fuzzy number is a fuzzy subset of the real line r i. A direct method for numerically solving integral equations system using. The functions of the equations are supposed to be discontinuous with respect to some variables and satisfy nonabsolute fuzzy integrability. This will be a useful resource book for those studying integral equations.
Study materials integral equations mathematics mit. We convert a nonlinear fuzzy volterrafredholm integral equation to a nonlinear system of volterrafredholm integral equation in crisp. In this paper, we introduce fuzzy quadratic integral equation of fractional. The main purpose of this paper is to approximate the solution of linear twodimensional fuzzy fredholm integral equations of the second kind 2dffie2. The study of fuzzy integral equations fie from both theoretical and numerical points of view has been developed in recent years after a distinct study of the existence of a unique solution for fuzzy fredholm integral equations had been carried out in. In fact, obtaining the exact solutions of such fuzzy integral equations is not possible in all cases because of the inherited restrictions form application of fuzzy concepts in these problems. The reason for doing this is that it may make solution of the problem easier or, sometimes, enable us to prove fundamental results on the existence and uniqueness of the solution. Pdf numerical solution of two dimensional nonlinear. In this paper, a reduction formula to reduce fuzzy linear differential equations to fuzzy linear integrodifferential equations of volterra type is produced. A numerical solution of fredholm fuzzy integral equations of the second kind. Corrigenda to numerical solution of fredholm fuzzy integral. Formulations of fuzzy integral equations in terms of the aumann integral do not reflect the behavior of corresponding crisp models.
Application of fuzzy differential transform method for solving fuzzy. Request pdf application of fuzzy differential transform method for solving fuzzy. Solving fuzzy integral equations of the second kind by fuzzy. Pdf existence and uniqueness of solutions for fuzzy quadratic. The topics of numerical methods for solving fuzzy integral equations have been rapidly growing in recent years and have been studies by authors of 6. Numerical solution of fuzzy volterra integral equations is considered in,15, 16. Now we are coming to the proper definition of riemann sums and the integral. Solving a system of fuzzy integral equations by an analytic.
Abstract in this paper, we use new parametic formof fuzzy numbers and convert a system of fuzzy integral equations to two system of integral equations in crisp. The topic of fuzzy integral equations which has attracted growing interest for some time, in particular in relation to fuzzy control, has been developed in recent years. Bounded solutions for fuzzy differential and integral equations. Method of successive approximations for fredholm ie s e i r e s n n a m u e n 2. Combining the error estimates 5 and 25 we obtain a practical stopping criterion of the. It is worth noting that integral equations often do not have an analytical solution, and must be solved numerically.
To facilitate the benefits of this proposal, an algorithmic form of the hpm is also designed to handle the same. The interest in fuzzy fredholm integral equations is based primarily on its applications in fuzzy. Solving the first kind fuzzy integral equations using a. In this paper, a numerical procedure for solving fuzzy fredholm integral equations of the second kind fies with arbitrary kernels have been investigated and residual minimization method is given. Fuzzy integral equations and strong fuzzy henstock integrals. Fuzzy integral equations advances in difference equations. Park and jeong 8 proved the existence and uniqueness of solutions of fuzzy. Solving fuzzy integral equations of the second kind by.
After this a lot research papers have appeared proposing solutions of various types of fuzzy equations viz. Finally, an algorithm is presented to solve the fuzzy integral equation by using the trapezoidal rule. Application of fuzzy bicubic splines interpolation for. Fuzzy set theoryand its applications, fourth edition. It is known that kird is a complete and separable metric space with respect to dh. Porter 1 introduction the integral equation problem is to nd the solution to.
In this paper we use fuzzy bunch functions to define every equation, and pay attention by finding a general formula of reduction to reduce fuzzy differential equations, and fuzzy volterra linear integral equations to fuzzy volterra linear. In this paper existence theorems for certain volterra integral equations and fredholm integral equation for the fuzzy set valued mappings are obtained. Pdf the fractional quadratic integral equations have wide applications in various. Then, by solving the linear system, unknowns are determined. It provides a complete treatment of numerous transform techniques fourier, laplace, mellin, hankel, hilbert. Since these equations usually can not be solved explicitly, so it is required to obtain approximate solutions. We show that our definition is equivalent to the extension principle for functions of this class. In all of the above, if the known function f is identically zero, the equation is called a homogeneous integral equation. Some authors discussed the solution of fuzzy integrodifferential equation by fuzzy differential transform method in their research paper. Each rule has a crisp output and the overall output is obtained via weighted average. Find materials for this course in the pages linked along the left. The fuzzy logic is applicable to many areas where human decision making plays an important. So, fuzzy convolution operator is proposed and related.
On the other hand, such techniques, including the fuzzy transform, are not excluded from the other techniques involved in fuzzy systems. The fredholm integral equation of the second kind hochstadt 1973 is given by where. In wu 2000 investigated the fuzzy riemann in tegral and its numerical integration. Chapter 8 deals with the applied problems of advanced nature such as arising in ocean waves, seismic response, transverse oscillations and flows of heat. Subrahmanyam and sudarsanam studied fuzzy volterra integral equations. In this paper, a twodimensional differential transform method to solve fuzzy partial differential equations fpdes is proposed. Abstract fuzzy integral equations fies topic is an important branch in fuzzy mathematics. Operator theory and integral equations 802660s lecturenotes secondprinting valery serov university of oulu 2012 edited by markus harju. Ganesan department of mathematics, srm university, ramapuram, chennai 600 089. Then by approximating bernstein polynomials, the obtained second kind fuzzy integral equation is solved. Despite many studies conducted to nd algorithms for selecting the optimum values of c, the optimal choice of shape parameter is an open problem which is still under intensive. Preliminaries let conv r n be a set of all nonempty convex compact subsets of the.
Much research has been undertaken on analyzing existence and uniqueness of solution and developing numerical algorithms for solving onedimensional integral equations. Fredholm integral equations by using fuzzy interpolation via iterative method such as. Fuzzy trapezoidal cubature rule and application to two. It is important to develop convergence and unique analysis for fuzzy integral equations. In this paper, we construct the fuzzy trapezoidal cubature rule providing its remainder estimate for the case of lipschitzian functions.
Consequently, they are illadapted to describe physical phenomena, even when vagueness and uncertainty are present. Numerical solution of twodimensional fuzzy fredholm. Application of fuzzy measure and fuzzy integral in. The theory of integral equations ie is exposed in the framework of hilbert spaces. Integral equation fuzzy number classical solution fredholm integral equation interval arithmetic these keywords were added by machine and not by the authors. Saburi department of mathematics science and research branch islamic azad university, tehran, iran abstract in this paper a numerical method for solving fuzzy partial di. Problems and exercises in integral equations internet archive. Solving linear fredholm fuzzy integral equations system by. Banifazel department of mathematics, urmia branch islamic azad university, urmia, iran abstract in this paper we intend to o. In this method, one parameter was created in the second kind equation. Solving fuzzy partial differential equations by fuzzy two. A method for solving fuzzy fredholm integral equations of the. In this paper, we deal with the existence problems of discontinuous fuzzy integral equations involving the strong fuzzy henstock integral in fuzzy number space.
The applications of fuzzy logic found in many domains. The first one starts by laying the groundwork of fuzzy intuitionistic fuzzy sets, fuzzy hedges, and fuzzy relations. Biacino and lettieri 6 have proposed different methods for solving the fuzzy equations. In this article, we focus on linear and nonlinear fuzzy volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method hpm to obtain fuzzy approximate solutions to them. This method is based on the concept of creating function series. Numerical solution of two dimensional nonlinear fuzzy fredholm integral equations of second kind using hybrid of blockpulse functions and bernstein polynomials article pdf available in filomat. Research article quadrature rules and iterative method for. The method of successive approximations for fredholms integral equation.
Introduction integral equations appears in most applied areas and are as important as differential equations. Pdf numerical solution of twodimensional linear fuzzy. Advanced analytical techniques for the solution of single. So, in this special issue, we intend to consider the numerical methods to solve fuzzy integral equations and the related topics with real applications. Recently the setvalued and fuzzy integral equations and inclusions began to be considered 614. An adaptive grid size mechanism based on the fixed grid size technique is also proposed. Solution of fuzzy volterra integral equations in a bernstein.
Analytical and numerical methods for solving linear fuzzy volterra integral equation of the second kind by jihan tahsin abdel rahim hamaydi supervised prof. On the integrals,series and integral equations of fuzzy setvalued functions, j. In fact, as we will see, many problems can be formulated equivalently as either a differential or an integral equation. For such integral equations the convergence technique bas been examined in considerable detail for the linear case by erdelyi 3, 4, and 5, and in some detail for the nonlinear case by erdelyi 6. A similar situation for fuzzy odes has been obviated by interpretation in terms of families of differential inclusions. Balachandran and dauer 1 established the existence of solutions of perturbed fuzzy integral equations. Numerical solution of two dimensional nonlinear fuzzy.
The book is mainly oriented towards the theory of compact integral operators, partial differential operators and boundary value problems. A numerical method for solving double integral equations. In this paper, first the regularization method applied to convert the first kind fuzzy integral equation into the second kind fuzzy integral equation. Solving fuzzy volterra integral equations via fuzzy sumudu transform norazrizal aswad abdul rahman1 and muhammad zaini ahmad1, a 1institute of engineering mathematics, pauh putra main campus, universiti malaysia perlis, 02600 arau, perlis. We use fuzzy twodimensional triangular functions 2dtfs to reduce the 2deffiee2 to a system of linear fredholm integral equations of the second kind with three variables in crisp. This process is experimental and the keywords may be updated as the learning algorithm improves. In this paper the substantiation of the averaging method for fuzzy integral equation using the second approach is considered. This branch of mathematical analysis, extensively investigated in recent years, has emerged as an effective and powerful tool for the mathematical modeling of several engineering and scientific phenomena. The reason for doing this is that it may make solution of the problem easier or, sometimes, enable us to prove fundamental results on. Johns, nl canada department of mathematics hong kong baptist university hong kong sar p. Fuzzy fredholm integral equation of the second kind annajah. In this chapter we will allow gx to be a fuzzy function andor.
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