Fredholm integral equations in a fredholm integral equation the limits of integration are. Volume 108, issue 2, 1 december 1999, pages 193200. A numerical method for fuzzy differential equations and. This unique work provides a new direction for the reader in the use of basic concepts of fuzzy differential equations, solutions and its applications. A method for solving fuzzy fredholm integral equations of the. Method of successive approximations for fredholm ie s e i r e s n n a m u e n 2. The classical fractional euler method has also been extended in the fuzzy setting in order to approximate the solutions of linear and nonlinear fuzzy fractional differential equations. Using the ham, it is possible to find the exact solution or an approximate solution of the problem. Accordingly various researchersbuckley 4, wasowski 5, biacino and lettieri 6 have proposed different methods for solving the fuzzy equations. Fuzzy integral equations and strong fuzzy henstock integrals shao, yabin and zhang, huanhuan, abstract and applied analysis, 20 existence of solutions of the wave equation involving the distributional henstockkurzweil integral liu, wei, lu, yueping, wang, ying, and ye, guoju, differential and integral equations, 2011. The classical fractional euler method has also been extended in the fuzzy setting in order to approximate the solutions of linear and. Solving linear fredholm fuzzy integral equations of the second kind.
Apr 17, 2019 explains fuzzy matrix operations and features examples. Solving fuzzy fractional differential equations using. In this paper, an application of the bernstein polynomials expansion method is applied to solve linear second kind fredholm and volterra integral equations systems. 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 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.
Pdf in this paper, a numerical procedure for solving fuzzy fredholm integral equations of the second kind fies with arbitrary kernels have been. Preliminaries let conv r n be a set of all nonempty convex compact subsets of the. Fuzzy number, fuzzy linear system, fuzzy integral equations 1 introduction the concept of integration of fuzzy functions was. Comparison of accurate solutions of nonlinear hammerstein fuzzy. A fuzzy linear system axb where a is a crisp matrix and b is a fuzzy number vector has been studied by friedman et al. Solving fuzzy nonlinear volterrafredholm integral equations by. In this chapter we will allow gx to be a fuzzy function andor. Solution of fuzzy volterra integral equations in a bernstein. Introduction integral equations appears in most applied areas and are as important as differential equations. The reason for doing this is that it may make solution of the problem easier or, sometimes, enable us to prove fundamental results on.
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. Inspired by the work of 9, 10, in the present paper we aim to establish some sufficient conditions for the boundedness of every solution of fractional order fuzzy integral equations as well as certain fuzzy differential equations. Solving a system of fuzzy integral equations by an analytic. As an application of our result, we study the existence and uniqueness of the solution to a nonlinear fredholm integral equation. Study materials integral equations mathematics mit. Solving fuzzy integral equations of the second kind by fuzzy.
Dec 30, 2015 in this paper, fuzzy fractional diffusion equations ffdes and fuzzy fractional wave equations ffwes, subjected to initial and boundary conditions are considered. Allahviranloo used a numerical method to solve fpde, that was based on the seikala derivative. A collocation method solving integral equation models for image restoration liu, yuzhen, shen, lixin, xu, yuesheng, and yang, hongqi, journal of integral equations and applications, 2016. A systematic spectraltau method for the solution of fuzzy. Fuzzy integral equations and strong fuzzy henstock integrals article pdf available in abstract and applied analysis 20143. A fuzzy set can be defined as the set of ordered pairs such that a x, a x x x, a x 0,1, where a x is called the membership function or grade of membership of x 12. Novel methods for solving linear and nonlinear integral equations. In this paper existence theorems for certain volterra integral equations and fredholm integral equation for the fuzzy set valued mappings are obtained. Numerical solutions to fuzzy differential and integral equations, fuzzy sets and systems, 106 1999, pp. We define an operator l as a map function from the vector space m to the vector space n.
S cd dc,t ef fe, where c and e contains the positive entries of a and b respectively, and d and f the absolute values of the negative entries of a and b, i. The first one starts by laying the groundwork of fuzzyintuitionistic fuzzy sets, fuzzy hedges, and fuzzy relations. Sanchez 3 put forward a solution of fuzzy equation by using extended operations. Kotsireasy june 2008 1 introduction integral equations arise naturally in applications, in many areas of mathematics, science and technology and have been studied extensively both at the theoretical and practical level. In this paper the substantiation of the averaging method for fuzzy integral equation using the second approach is considered. Find materials for this course in the pages linked along the left. In this book, we study the subject of smarandache fuzzy algebra. Also, the fuzzy integral equations have been studied by several authors, 14, 15. A fuzzy number is a convex normalized fuzzy set of the crisp set such that for only one x x, a x 1 and a x. Pdf solving linear fredholm fuzzy integral equations system by. Differential equations department of mathematics, hkust.
Novel methods for solving linear and nonlinear integral. Home browse by title periodicals advances in fuzzy systems vol. Originally, the revolutionary theory of smarandache notions was born as a paradoxist movement that challenged the status quo of existing mathematics. These proposed methods are used to solve a ham merstein integral equation by reducing to a system of nonlinear algebraic equations. Application of coupled fixed point technique in solving. Introduction the theory of fuzzy differential equations has focused much attention in. Pdf numerical solution of fuzzy integral equations researchgate.
In fact, as we will see, many problems can be formulated equivalently as either a differential or an integral equation. Pdf fuzzy integral equations and strong fuzzy henstock. The concept of fuzzy derivative was first introduced by chang and zadeh. Novel methods for solving linear and nonlinear integral equations saha ray, santanu, sahu, prakash kumar this book deals with the numerical solution of integral equations based on approximation of functions and the authors apply wavelet approximation to the unknown function of integral equations. Solving fuzzy integral equations of the second kind by.
Application of fuzzy laplace transforms for solving fuzzy. Particularly important examples of integral transforms include the fourier transform and the laplace transform, which we now. Recently the setvalued and fuzzy integral equations and inclusions began to be considered 614. As these equations have significant applications in physics and engineering, a methodical spectraltau scheme is utilized to obtain efficient solutions of ffde and ffwe. Mt5802 integral equations introduction integral equations occur in a variety of applications, often being obtained from a differential equation. Numerical solution of linear integral equations system. Theory and numerical solution of volterra functional. We define a measure of the fuzziness of a fuzzyvalued function and show that. The concept of fuzzy derivatives and fuzzy integration were studied in 11, 27 and then some generalization have been investigated in 10, 11, 24, 23. It can serve as an essential reference work for students, scholars, practitioners, researchers and academicians in engineering and science who need to model uncertain physical problems. One of the first applications of fuzzy integration was given by wu and ma who investigated the fuzzy fredholm integral equation of the second kind.
The intuitionistic fuzzy set is an ideal text for graduate and research students, as well as professionals, in image processing, decisionmaking, pattern recognition, and control system design. Recently, bede introduced a strongly generalized di. The goal of the fuzzylite libraries is to easily design and efficiently operate fuzzy logic controllers. Fuzzy differential equations and applications for engineers. Numerical examples are presented to illustrate the proposed model. In this paper, fuzzy fractional diffusion equations ffdes and fuzzy fractional wave equations ffwes, subjected to initial and boundary conditions are considered. The results reveal that the proposed method is very effective and simple. The method to obtain solution of system of linear equations with all the involved parameters being fuzzy is proposed here. It also contains elegant analytical and numerical methods, and an important topic of the variational principles. Fredholm integral equations are related to boundaryvalue problems for di.
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. Solution of fuzzy volterra integral equations in a. A collocation method solving integral equation models for image restoration liu, yuzhen, shen, lixin, xu, yuesheng, and yang, hongqi, journal of integral equations and applications, 2016 quadrature rules and iterative method for numerical solution of twodimensional fuzzy integral equations sadatrasoul, s. Section 5 and 6 describe an experiment pertaining to syllableproximity evaluation using the fuzzyintegrationbased aggregation. Kaleva, puri and ralescu introduced the notion of fuzzy derivative as an extension of the hukuhara derivative and the fuzzy integral, which was the same as that proposed by dubois and prade. That is why different ideas and methods to solve fuzzy differential equations have been developed.
The intuitionistic fuzzy set begins with an introduction, theory, and several examples to guide readers along. 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. A survey on solution methods for integral equations. Fuzzy linear systems have recently been studied by a good number of authors but only a few of them are mentioned here.
Solving a system of fuzzy integral equations by an analytic method. Since in some application mathematical problems finding the analytical solution is too complicated, in recent years a lot of attention has been devoted by researchers to find the numerical solution of this equations. So, in this special issue, we intend to consider the numerical methods to solve fuzzy integral equations and the related topics with real applications. Solving fuzzy fractional differential equations using zadehs. 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. We apply the trapezoidal rule to compute the riemann integrals. There has been a significant development in the study of fuzzy differential and integral equations see, for example, 5. Solving a system of fuzzy integral equations by an. The integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine.
Bounded solutions for fuzzy integral equations of fractional. Quadrature rules and iterative method for numerical solution of twodimensional fuzzy integral equations sadatrasoul, s. Numerical solution of interval and fuzzy system of linear. We establish a common coupled fixed point theorem for weakly compatible mappings on modified intuitionistic fuzzy metric spaces. Fuzzy differential equations and applications for engineers and scientists crc press book differential equations play a vital role in the modeling of physical and engineering problems, such as those in solid and fluid mechanics, viscoelasticity, biology, physics, and many other areas. 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. Theory and numerical solution of volterra functional integral. Section 7 provides a brief summary of the conclusions. Based on the parametric form of a fuzzy number, a fredholm.
One of them solves differential equations using zadehs extension principle buckleyfeuring 30, while another approach interprets fuzzy differential equations through differential inclusions. In this paper, we have studied a fuzzy fractional differential equation and presented its solution using zadehs extension principle. 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. Section 5 and 6 describe an experiment pertaining to syllableproximity evaluation using the fuzzy integrationbased aggregation. Together, they are the fuzzylite libraries for fuzzy logic control. This kind of integral equations convert to a linear system. Most methods for doing this rely on the local polynomial approximation of the solution and all the stability problems that were a concern for interpolation will be a concern for the. Section 4, we illustrate algorithm by solving some numerical examples. We then apply integration theory of fuzzyvalued functions to integral equations. First, the area is approximated by a sum of rectangle areas. M n introduce the following definitions concerning the operators in the vector. Pdf in this paper we intend to offer a numerical scheme to solve linear fredholm fuzzy integral equations system of the second kind.
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