We introduce a hypograph metric in the space of fuzzy sets and prove a theorem on. Linear systems of two secondorder partial differential equations. Solving fuzzy fractional differential equations using. As fpde adapt the fuzzy set theory by zadeh 4, it can be said that fpde are more powerful compare to partial differential equations. Preliminaries the prime information of an arbitrary fuzzy numbers is given in 6 and we give some necessary definitions which will be used in this paper. However as it is seen from the examples in mentioned article, these solutions are. Wu, existence and uniqueness of solutions to cauchy problem of fuzzy differential equations, fuzzy sets and systems 110 2000, 55 67. Pdf fuzzy differential equation with nonlocal conditions. In the continuous case we have two different approaches. Asweknow,the real systems are often faced with two kinds of uncertain. A theorem of convergence for the solution is stated and proved.
Laplace transform is used for solving differential equations. One of them solves differential equations using zadehs extension principle buckleyfeuring 30, while another approach interprets fuzzy differential equations through differential inclusions. This concept is based in the enlargement of the class of. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation. Fuzzy modeling for uncertainty nonlinear systems with. Numerical solution of firstorder linear differential. The continuous dependence on initial condition and stability properties are also established. The algorithm is illustrated by solving some linear and nonlinear fuzzy cauchy problems. The uncertain nonlinear systems can be modeled with fuzzy equations by incorporating the fuzzy set theory.
The topics of numerical methods for solving fuzzy differential equations have been rapidly growing in recent years. Differential equations with fuzzy parameters via differential. This paper is concerned with systems of ordinary differential equations with fuzzy parameters. It describes relations between variables and their derivatives. First order non homogeneous ordinary differential equation.
This means that as the time passes, the more fuzzy diffuse, uncertain the process becomes. This unique work provides a new direction for the reader in the use of basic concepts of fuzzy differential equations, solutions and its applications. Pdf solution of variable coefficient fuzzy differential. Fuzzy differential equations and applications for engineers. The first one historically was based on the hukuhara derivative introduced in puriralescu 123. Solutions of sequential conformable fractional differential equations around an ordinary point and conformable fractional hermite differential equation emrah unala b,ahmet gokdogan, ercan celikc a department of elementary mathematics education, artvin coruh university, 08100 artvin, turkey emrah. Systems of partial differential equations, linear eqworld. On a new fuzzy topological nonlinear differential equations sennimalai kalimuthu patta no. Solving secondorder fuzzy differential equations by the fuzzy. Fuzzy differential equations, fuzzy sets and systems, 110 1999, 4354. Sufficient conditions for stability and convergence of the proposed algorithms are given, and their applicability is illustrated with some examples. One of the most efficient ways to model the propagation of epistemic uncertainties in dynamical environmentssystems encountered in applied sciences, engineering and even social sciences is to employ fuzzy differential equations fdes. An implicit method for solving fuzzy partial differential. The ides are differential equations used to handle interval uncertainty that appears in.
The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations. In this paper, a scheme of partial averaging of fuzzy differential equations with maxima is considered. This approach does not reproduce the rich and varied behaviour of ordinary differential equations. List of nonlinear partial differential equations wikipedia. To solve fuzzy fractional differential equation, fuzzy initial and boundary value problems, we use fuzzy laplace transform. Solving fuzzy fractional differential equation with fuzzy. Solving secondorder fuzzy differential equations by the. In this paper numerical algorithms for solving fuzzy ordinary differential equations are considered. A numerical method for a partial integrodifferential.
The concept of a fuzzy derivative was first introduced by chang and zadeh 8 and others. Pdf solutions of fuzzy differential equations with lr. The numerical solutions are compared with igh and iigh differential exact solutions concepts system. First order homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number is described by mondal and roy 32. We also present the convergence analysis of the method. In section 4, we generalize this method in order to solve the nth order linear fuzzy differential equations with fuzzy initial values. See also nonlinear partial differential equation, list of partial differential equation topics and list of nonlinear ordinary differential equations. Analysis and computation of fuzzy differential equations. The theory of differential and difference equations forms two extreme representations of real world problems. Another approach to solution of fuzzy differential equations.
In the litreture, there are several approaches to study fuzzy differential equations. On new solutions of fuzzy differential equations victor ayala. Fuzzy volterra integrodifferential equations using. We study fuzzy differential equations fde using the concept of generalized h differentiability. We use parametric forms of fuzzy numbers to convert nonlinear fuzzy integro differential equations to a nonlinear system of integro differential equations in crisp case. Solving fuzzy fractional differential equations by fuzzy laplace transforms. System of type2 fuzzy differential equations and its. Applying the zadeh extension principle to the equations, we introduce the notions of fuzzy solutions and of componentwise fuzzy solutions. Numerical algorithms for solving firstorder fuzzy differential equations and hybrid fuzzy differential equations have been investigated. Pdf a note on numerical solutions for linear system of.
That relevance makes for rapid development of new ideas and theories. For instance, population dynamics in ecology and biology, mechanics of particles in physics, chemical reaction in chemistry, economics, etc. In this paper, the solutions of fdes are approximated by utilizing the fuzzy sumudu transform fst method. We consider the nonlinear fredholm and volterra integro differential equations of the second kind. Fuzzy differential equations and applications for engineers and scientists pdf adobe drm can be read on any device that can open pdf adobe drm files. Theory of fuzzy differential equations and inclusions. Miscellaneous topics fuzzy difference equations impulsive fuzzy differential equations fuzzy differential equations with delay hybrid fuzzy differential equations fixed points of fuzzy mappings boundary value problem fuzzy equations of volterra type a new concept of stability 6. In their formulas, first the crisp solution is found, fuzzified. In this paper, we study the fuzzy laplace transforms introduced by the authors in allahviranloo and ahmadi in soft comput.
First order linear homogeneous ordinary differential equation. Pdf solving fuzzy fractional differential equations by. The term fuzzy differential equation was introduced in1987 by kandel. We also propose a numerical method to approximate the solution of ffdes. 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. It has ample applications due to the fact that many practical problems. R fis said to be fuzzy continuous function if f exists for any. It provides a unique opportunity for soft computing researchers and oil industry practitioners to understand the significance of the changes in the fields by presenting recent accomplishments and new directions. A numerical method for fuzzy differential equations and.
Fuzzy differential equations are suggested as a way of modeling. Therefore, for such mathematical modeling, using fuzzy differential equations are necessary. The following approaches of fuzzy differential equations are depicted in this chapter. Solving fuzzy fractional riccati differential equations by. Rungekutta algorithms for solving fuzzy ordinary differential equations are considered. Numerical ways for solving fuzzy differential equations.
Fuzzy differential equation with nonlocal conditions and fuzzy semigroup article pdf available in advances in difference equations 20161 march 2016 with 147 reads how we measure reads. Differential equations department of mathematics, hkust. Fuzzy differential equations are used in modeling problems in science and. For example, a simple population model when represented as a differential equation shows the good behavior of solutions whereas the corresponding discrete analogue shows the chaotic behavior. In this paper, a new approach to solve the fuzzy linear system of differential equations based on pseudoinverse is presented. Solving system of fuzzy linear differential equations with. Modeling with stochastic fuzzy differential equations. The technique we use is the homotopy perturbation method hpm.
The concept of fuzzy derivative was first introduced by chang and zadeh in 10. This volume is a timely introduction to the subject that describes the current state of the theory of fuzzy differential equations and inclusions and. A novel approach for solving fuzzy differential equations. An operational matrix based on legendre polynomials for solving fuzzy fractionalorder differential equations ali ahmadian,1,2 mohamed suleiman,1 and soheil salahshour3 1institute for mathematical research, universiti putra malaysia, 43400 serdang, selangor, malaysia 2department of mathematics, science faculty, universiti putra malaysia, 43400 serdang, selangor, malaysia 3young researchers.
Fuzzy differential equations can be studied from a point of view discrete 2,27,28,42 or continuous 7,15,21,22,45. Recently fde has also used in many models such as hiv model 6, decay model 5, predatorprey model 10, etc. Pdf in this paper, a solution procedure for the solution of the system of fuzzy differential equations x. Partial averaging of fuzzy differential equations with maxima. Fuzzy sets and systems 24 1987 3017 301 northholland fuzzy differential equations osmo kaleva tampere university of technology, department of mathematics, p. Seikkala, towards the theory of fuzzy differential equations, fuzzy sets and systems 125 2002, 231 237. A pertinent approach to solve nonlinear fuzzy integro. The fuzzy laplace transform of derivative was used to solve second order variable. Averaging for fuzzy differential equations pdf paperity. It is seen that in recent years the topic of fuzzy differential equations fdes has been rapidly grown. We extend and use this method to solve secondorder fuzzy linear differential equations under generalized hukuhara differentiability.
Stability of solutions of fuzzy differential equations. Request pdf system of type2 fuzzy differential equations and its applications this paper develops the mathematical framework and the solution of a system of type2 fuzzy ordinary differential. In this paper we present the existence and uniqueness of solutions to the stochastic fuzzy differential equations driven by brownian motion. In this paper we propose a fuzzy laplace transform to solve variable coefficient fuzzy differential equations under strongly generalized differentiability concept. Reservoir characterization and modeling studies in fuzziness and soft computing nikravesh, masoud, zadeh, lofti a. 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. We study a fuzzy fractional differential equation ffde and present its solution using zadehs extension principle. We can see the applications of nonlinear equations in many areas such as mathematics, medicines, engineering and social sciences. Fuzzy transport equation is one of the simplest fuzzy partial differential equation, which may appear in many applications. Fuzzy number, fuzzy linear system, fuzzy differential equations, func tion of matrices.
Recently, khastan and nieto 7 have found solutions for a large enough class of boundary value problems with the generalized derivative. That is why different ideas and methods to solve fuzzy differential equations have been developed. Fractional riccati differential equation, international journal of differential equations volume 2010, article id 764738. If you want to learn differential equations, have a look at differential equations for engineers if your interests are matrices and elementary linear algebra, try matrix algebra for engineers if you want to learn vector calculus also known as multivariable calculus, or calculus three, you can sign up for vector calculus for engineers. Approximate solution of timefractional fuzzy partial. Fuzzy partial differential equations and relational equations. Baleanu, a jacobi operational matrix for solving a fuzzy linear fractional differential equation. Important ele ment of fuzzy differential equation is fuzzy derivative. The theory of fuzzy differential equations derived from this derivative is the most widespread and has been widely studied. In this note, it is shown that some obtained results in solaymani fard and ghaleh 2011 are not true.
Numerical solution of first order linear fuzzy differential equations using leapfrog method. The proposed approach reveals fast convergence rate and accuracy of the present method when compared with exact solutions of crisp. Solution of fuzzy partial differential equations using. The uncertain nonlinear systems can be modeled with fuzzy differential equations fdes and the solutions of these equations are applied to analyze many engineering problems.
J they have been many suggestions for definition of fuzzy derivative to study fuzzy differential equation. The book presents important steps in this direction by introducing fuzzy partial differential equations and relational equations. An example is provided to show the effectiveness of the proposed theory. Feuring 7,8 introduced two analytical methods for solving nthorder linear differential equations with fuzzy initial conditions. Fuzzy differential functions are applicable to realworld problems in engineering, computer science, and social science. Sunday, analysis and computation of fuzzy differential equations via interval differential equations with a generalized hukuharatype differentiability, american journal of computational and applied mathematics, vol. First, the authors transformed a fuzzy differential equation by two parametric ordinary differential equations and then solved by fuzzy eulers method. As an example of application we use some stochastic fuzzy differential equation in a model of population dynamics. Solution of fuzzy differential equations using fuzzy.
Pdf solving fuzzy differential equations by differential. Fuzzy number, fuzzy linear system, fuzzy differential equations, func tion of. Introduction fuzzy theory of fractional differential and integro differential equations is a new and important branch of fuzzy mathematics. Differential inclusions and fuzzy differential inclusions are two topics that are very interesting but they do not constitute the subject of the. In this paper, the fuzzy equations are applied as the models for the uncertain nonlinear systems. In this article, the control of the dynamic system by application of differential equations is shown. A new approach to linear interval differential equations as a first step toward solving fuzzy differential tiago m. Box 527, sf 33101 tampere, finland received january 1985 revised january 1986 this paper deals with fuzzy setvalued mappings of a real variable whose values are normal, convex, upper semicontinuous and compactly supported fuzzy. First order linear fuzzy differential equations are one of the simplest fuzzy differential equations, which appear in many applications. Fuzzy initial value problem for nth order linear differential equation, fuzzy sets and systems, 121 2001, pp. Differential equations with fuzzy parameters via differential inclusions. The fdes are special type of interval differential equations ides. The journal of differential equations is concerned with the theory and the application of differential equations. A tool for solving fuzzy differential equations nematollah kadkhoda, sedigheh sadeghi roushan, and hossein jafari international journal of applied and computational mathematics, 2018, volume 4, number 1.
In general, the parameters, variables and initial conditions within a model are considered as being defined exactly. Nthorder fuzzy differential equations under generalized. In the year 1987, the term fuzzy differential equation was. Solving fuzzy fractional differential equations by fuzzy. One is classical method and the other is extension principle method. Solutions of fuzzy differential equations with lr fuzzy numbers. On the one hand, this means that the precision of a model can easily. Pdf an extension of the differential transformation method dtm, which is an analyticalnumerical method for solving the fuzzy differential. Numerical solution of first order linear fuzzy differential. Impulsive differential equations ides are a new branch ofdifferentialequations. Most downloaded applied numerical mathematics articles elsevier. In this paper, we derived a new fuzzy version of eulers method by taking into account the dependency problem among fuzzy sets. We use the neural networks to approximate the coefficients of the fuzzy equations.
The proposed study extends the case of fuzzy differential equations of integer order. In this study, we develop perturbationiteration algorithm pia for numerical solutions of some types of fuzzy fractional partial differential equations ffpdes with generalized hukuhara derivative. Linear differential equations with fuzzy boundary values. There are several interpretations of a fuzzy differential equation. Minimal solution of fuzzy linear system of differential. However, it is very difficult to obtain solutions of fdes. Solving systems of fuzzy differential equation 1 introduction mhikari. Solution of the first order linear fuzzy differential equations by some. An operational matrix based on legendre polynomials for. Fuzzy differential equations with interactive derivative. This type of differential equation suffers from the defect of having solutions with increasing diameters over time. Fuzzy differential equations have been applied extensively in recent years to model uncertainty in mathematical models. First order linear homogeneous fuzzy ordinary differential. This paper deals with the solutions of fuzzy fractional differential equations ffdes under riemannliouville hdifferentiability by fuzzy laplace transforms.
We give a definition of solutions of ordinary differential equations in n con taining parameters which are described by changing in time fuzzy sets. Exact solutions systems of partial differential equations linear systems of two secondorder partial differential equations pdf version of this page. Idescanfindnumerousapplications in different branches of optimal control, electronics, economics,physics,chemistry,andbiologicalsciences. 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. Pdf solving systems of fuzzy differential equation researchgate. Solutions of sequential conformable fractional differential. System of differential equation with initial value as triangular intuitionistic fuzzy number and its application is solved by mondal and roy 30. Analysis and computation of fuzzy differential equations via. Boolean and fuzzy logic are based mainly on discrete transitions, whereas ordinary differential equations odes form a purely continuous model. In the chapter, the author considers an approach used in the studies of stochastic fuzzy differential equations. The first and most popular one is hukuhara derivative made by puri.
Citescore values are based on citation counts in a given year e. Here the solution of fuzzy differential equation becomes fuzzier as time goes on. This is due to the significant role of nonlinear equations, where it is used to model many real life problems. Her research interests include fuzzy differential equations, fuzzy fractional differential equations and numerical analysis. Averaging method, fuzzy differential equation with maxima.
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