An implicit method for solving fuzzy partial differential. Also fides in fuzzy setting are a natural way to model uncertainty of dynamical systems. First order non homogeneous ordinary differential equation. The method of upper and lower solutions is used to prove the unique solvability of a fuzzy urysohn integral equation. Introduction the solutions of integral equations have a major role in the field of science and engineering. Linear differential equations with fuzzy boundary values. The advantage of fuzzy laplace transform is that it solves the problem directly without determining a general solution. This is due to the significant role of nonlinear equations, where it is used to model many real life problems. A new method for optimal control of volterra integral equations s. A popular fuzzy number is the triangular fuzzy number v v m,v l,v u where v m denotes the modal value and the real values v l. By solving this equation it is possible that either e. In fact, as we will see, many problems can be formulated equivalently as either a differential or an integral equation.
To solve fuzzy fractional differential equation, fuzzy initial and boundary value problems, we use fuzzy laplace transform. Introduction for centuries, scientists have struggled to model real world phenomenon effectively and precisely. In these types of problems, an integral equation is appeared in the boundary conditions. Fuzzy integral equations and strong fuzzy henstock integrals. Introduction wu introduced in the improper fuzzy riemann integral and presented some of its elementary properties. Solving fuzzy integral equations of the second kind by. Solution of fuzzy volterra integral equations in a. For example, when the set is a single valued mapping, it is easy to see that the hukuhara derivative and the integral utilized in formulating the sdes reduce to the ordinary vector derivative and the integral and therefore, the results. The fuzzy differential and integral equations are important part of the fuzzy analysis theory and they have the important value of theory and application in control theory. Find out information about fuzzy relational equation. Solving fuzzy fractional differential equation with fuzzy. The linear mixed volterrafredholm integral equation is given by.
In section four a modi ed twostep simpson method and the corresponding convergence theorem of our method is presented. It is clear that can be easily calculated by determining the membership at the endpoints of the. Laplace transform is used for solving differential equations. For a function of one variable f fx, we use the following notation for the derivatives.
The cauchy problems for fuzzy differential equations have been studied by several authors 1 6 on the metric space of normal fuzzy convex set with the distance given. Mt5802 integral equations introduction integral equations occur in a variety of applications, often being obtained from a differential equation. The accuracy and efficiency of the proposed method is illustrated by solving a fuzzy initial value problem with trapezoidal fuzzy number. 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. M n introduce the following definitions concerning the operators in the vector. A method for solving fuzzy fredholm integral equations of the. Integral equations are one of the most useful mathematical tools in both pure and applied mathematics. Numerical solution of interval and fuzzy system of linear. Generalized abels integral equation, collocation method, taylor polynomials.
Monte carlo complexity of global solution of integral equations. First order non homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number sankar prasad mondal1, tapan kumar roy2 1department of mathematics, national institute of technology, agartala, jirania799046, tripura, india 2department of mathematics, indian institute of engineering and science and. Optimal control, volterra integral equation, discrete approximation. A role for symmetry in the bayesian solution of differential equations wang, junyang, cockayne, jon, and oates, chris.
Fuzzy number, trapezoidal fuzzy number, fuzzy differential equations, runge kutta method, higher order derivatives etc. Fuzzy sumudu transform for solving system of linear fuzzy. In this paper, we study the fuzzy laplace transforms introduced by the authors in allahviranloo and ahmadi in soft comput. Preliminaries let conv r n be a set of all nonempty convex compact subsets of the. However, the methods proposed for handling fies are still very limited and often involve complex calculations. 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. Then we solve a system of fuzzy integral equations by means of an analytic technique, namely the homotopy analysis method ham. 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. A physical even can be modelled by the differential equation, an integral equation.
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. The fides is obtained when a physical system is modeled under di. Solving linear fredholm fuzzy integral equations system by. Monte carlo methods are a classical tool of solving high dimensional integral equations.
Reservoir characterization and modeling studies in fuzziness and soft computing nikravesh, masoud, zadeh, lofti a. Bounded solutions for fuzzy differential and integral. We extend and use this method to solve secondorder fuzzy linear differential equations under generalized hukuhara differentiability. Method for solving fuzzy integro differential equation by. Application of fuzzyintegrationbased multipleinformation aggregation in automatic speech recognition shuangyu chang berkeley initiative in soft computing, eecs dept. This algorithm is implemented on some numerical examples by. Finally, an algorithm is presented to solve the fuzzy integral equation by using the trapezoidal rule. Fuzzy partial differential equations and relational. Among several modelling tools, scientists prefer to construct their models based on system of linear differential equations involving initial value problems 15.
Index terms fuzzy integral equations, system of volterra integral equation, bernstein polynomial i. However, in general, the process of finding the fuzzy set b fa is more complicated and cannot be gathered easily. Introduction the theory of fuzzy differential equations has focused much attention in the last decades since it provides good models for dynamical systems under uncertainty. Fuzzy relational equation article about fuzzy relational. In this paper the substantiation of the averaging method for fuzzy integral equation using the second approach is considered.
We can see the applications of nonlinear equations in many areas such as mathematics, medicines, engineering and social sciences. Obtain an initial basic feasible solution to the given problem in the form x b b b 1 and put it in the third column of the simplex table. Convert all the in equations of the constraints into equations by introducing slack and or surplus variables in the constraints. Homotopy method for solving finite level fuzzy nonlinear. We first express the necessary materials and definitions, and then consider our difference scheme and next the integrals in the boundary equations are approximated by the composite trapezoid rule. 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. Integral equations theory and numerical treatment wolfgang. In particular, it encourages interdisciplinary papers that cut across subdisciplines within mathematical analysis and to neighboring fields. They have enormous applications in many real problems. First, the authors transformed a fuzzy differential equation by two parametric ordinary differential equations and then solved by fuzzy eulers method. Recently, khastan and nieto 7 have found solutions for a large enough class of boundary value problems with the generalized derivative.
Many initial and boundary value problems associated with ordinary differential equation ode and partial differential equation pde can be transformed into problems of solving some. This is one of the types of nonlinear integral with respect to non additive measure. One of them solves differential equations using zadehs extension principle buckleyfeuring 30, while another approach interprets fuzzy. Reservoir characterization and modeling studies in. Practical applications of choquet and sugeno integrals 5. Some authors discussed the solution of fuzzy integrodifferential equation by fuzzy differential transform method in their research paper. Pdf solving fuzzy volterra integral equations via fuzzy.
Solving secondorder fuzzy differential equations by the. Solving a system of fuzzy integral equations by an. Fuzzy integral equations fies topic is an important branch in fuzzy mathematics. In this paper we present new techniques for solving fuzzy nonlinear volterra integral equations by using honotopy analysis method. We define an operator l as a map function from the vector space m to the vector space n. In the final part, we present an example for checking the. Application of fuzzy laplace transforms for solving fuzzy. Solving generalized abels integral equations of the first and second. An equation of the form a r b, where a and b are fuzzy sets, r is a fuzzy relation, and a r stands for the composition of a with r explanation of fuzzy relational equation. Method for solving fuzzy integrodifferential equation by. Optimal solution of a fully fuzzy linear fractional. That is why different ideas and methods to solve fuzzy differential equations have been developed.
Fuzzy partial differential equations and relational equations. There has been a significant development in the study of fuzzy differential and integral. Collocation method based on genocchi operational matrix for solving generalized fractional pantograph equations isah, abdulnasir, phang, chang, and phang, piau, international journal of differential equations, 2017. Arealfunction, 0,issaidtobeinthe space, r, if there exists a real number,such that 1,where 1. In this paper existence theorems for certain volterra integral equations and fredholm integral equation for the fuzzy set valued mappings are obtained. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. For example, if f is nonmonotone, then the problem can arise when two or more distinct points in x are mapped to the same point in y. Introduction integral equations appears in most applied areas and are as important as differential equations. In this chapter we will allow gx to be a fuzzy function andor. In this paper, we derived a new fuzzy version of eulers method by taking into account the dependency problem among fuzzy sets.
Issuu is a digital publishing platform that makes it simple to publish magazines, catalogs, newspapers, books, and more online. There has been a significant development in the study of fuzzy differential and integral equations see, for example, 5678, and the references therein. Application of fuzzy measure and fuzzy integral in. Choquet integral is more suitable for measure with probability related interpretation. Solution methods of fuzzy relational equations have been well studied in the literature cf.
Research article solving fuzzy fractional differential. The theory of integral equations has been an active research field for many years. Solving fuzzy fractional differential equations using. Communications in mathematical analysis is an international electronic research journal of rapid dissemination, devoted to the publication of high level research articles in all areas and subareas of mathematical analysis and their applications. 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. The concept of fuzzy derivative was first introduced by chang and zadeh. Our treatment of fuzzy nonlinear volterra ntegral equation centerel mainly on illustrations of the known methods of finding exact, or numerical solution. However as it is seen from the examples in mentioned article, these solutions are. Volterrafredholm integral equations in banach space.
Definition and background a fuzzy number is a fuzzy subset of the real line r i. Basic applications include neutron transport spanier and gelbard. 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. Recently the setvalued and fuzzy integral equations and inclusions began to be considered 614. In chapters 111 and 14, in the original integral equations, the independent variable is denoted by x, the integration variable by t, and the unknown function by y yx.
304 34 909 195 814 617 783 236 453 886 51 391 1156 1298 1138 1333 307 117 529 1534 589 505 1032 1137 668 1060 1472 948 620 1355 652 1427 719 146 997 132 575 886 67 822 165 803 458 1227 579 1342