The application of matrix functions have arisen in differential equation, markov models, control theory, non linear matrix equations, nuclear magnetic resonance, nonsymmetric eigenvalue problem. Apr 22, 2017 international journal of partial differential equations and applications. Equations of nonconstant coefficients with missing yterm if the yterm that is, the dependent variable term is missing in a second order linear equation, then the equation can be readily converted into a first order linear equation and solved using the integrating factor method. The first and most popular one is hukuhara derivative made by puri. Solving system of fuzzy linear differential equations with. 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. Solving fuzzy fractional differential equation with fuzzy laplace transform involving sine function dr. In this paper, we study analytical and numerical solutions of fuzzy differential equations based on the extension principle. Torvik, a theoretical basis for the application of fractional calculus to viscoelasticity,j. Differential equations department of mathematics, hkust. Solving fuzzy fractional differential equations by fuzzy.
The continuous dependence on initial condition and stability properties are also established. Solving fuzzy fractional riccati differential equations by. Solving fuzzy fractional differential equation with fuzzy. The ides are differential equations used to handle interval uncertainty that appears in. Linear systems of two secondorder partial differential equations. That is why different ideas and methods to solve fuzzy differential equations have been developed. Research article nonlinear fuzzy differential equation with.
Pdf we introduce the concept of differential equation in a metric space and apply it to the study of an initial value problem for a fuzzy differential. Using picard method of successive approximations, we shall prove the existence and uniqueness of solutions to rfdes with impulses under suitable conditions. The fdes are special type of interval differential equations ides. At present time, the study of fuzzy integro differential equations is an issue of remarkable consideration because it is one of the modern mathematical fields that arise from the modeling of uncertain physical, engineering and medical problems and are useful in studying the observability of dynamical control systems. The use and solution of differential equations is an important field of mathematics.
Box 527, sf 33101 tampere, finland received january 1985 revised january 1986 this paper deals with fuzzysetvalued mappings of a real variable whose values are normal, convex, upper semicontinuous and compactly. An implicit method for solving fuzzy partial differential. Another approach to solution of fuzzy differential equations. Using interpolation of fuzzy number for solving of fuzzy. A new technique to solve the initial value problems for.
Differential equations i department of mathematics. Consider the following fuzzy fractional differential equation of order 0 fuzzy differential equations via interval differential equations with a generalized hukuharatype differentiability, american journal of computational and applied mathematics, vol. Artificial neural network approach for solving fuzzy. Research article nonlinear fuzzy differential equation. Much of the material of chapters 26 and 8 has been adapted from the widely used textbook elementary differential equations and boundary value problems. 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. In this paper, a new approach for solving hybrid fuzzy differential equation initial value problems under generalized differentiability is proposed. Fuzzy sets and systems 24 1987 3017 301 northholland fuzzy differential equations osmo kaleva tampere university of technology, department of mathematics, p. Consider a firstorder fuzzy initial value differential equation is given by.
Linear differential equations with fuzzy boundary values. J they have been many suggestions for definition of fuzzy derivative to studyfuzzy differential equation. System of differential equation with initial value as triangular intuitionistic fuzzy number and its application is solved by mondal and roy 30. Pdf solving systems of fuzzy differential equation dr. 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. Systems of partial differential equations, linear eqworld. Numerical solution of first order linear fuzzy differential. In this paper, a solution procedure for the solution of the system of fuzzy differential equations. Definition and background a fuzzy number is a fuzzy subset of the real line r i.
First order linear homogeneous ordinary differential. Artificial neural network for solving fuzzy differential. The concept of a fuzzy derivative was first introduced by chang and zadeh 8 and others. In this paper, we derived a new fuzzy version of eulers method by taking into account the dependency problem among fuzzy sets. Fuzzy transport equation is one of the simplest fuzzy partial differential equation, which may appear in many applications. In this way, by taking an alphacut of initial value, the given differential equation is converted to a differential inclusion and the obtained solution is accepted as the alphacut of the fuzzy solution. As an example of application we use some stochastic fuzzy differential equation in a model of population dynamics. Fuzzy partial differential equations and relational equations. A differential equation that governs the population growth taking into account these aspects is dx dt fxrx1. A fuzzy differential approach to strong allee effect based. A trial solution of this system is written as a sum of two parts. In this paper we present the existence and uniqueness of solutions to the stochastic fuzzy differential equations driven by brownian motion.
First order linear homogeneous ordinary differential equation. Then kumar in 19 obtained exact solution of fully fuzzy linear system by solving a linear programming. Fuzzy differential equations were first formulated by kaleva 9 and seikkala 10 in time dependent form. For this purpose, new procedures for solving the system are proposed. The theory of differential and difference equations forms two extreme representations of real world problems. A pertinent approach to solve nonlinear fuzzy integro. On fuzzy laplace transforms for fuzzy differential. First order linear homogeneous ordinary differential equation in fuzzy environment sankar prasad mondal 1, sanhita banerjee 2 and tapan kumar roy 3 1, 2, 3 department of mathematics, bengal engineering and science university, shibpur, howrah711103, west bengal, india corresponding author, email. Solving hybrid fuzzy fractional differential equations by. 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. The output of the network is computed using a black box differential equation solver. This family of solutions is called the general solution of the differential equation. These continuousdepth models have constant memory cost, adapt their.
The differentiability concept used in this paper is the generalized differentiability since a fuzzy differential equation under this differentiability can have two solutions. A firstorder initial value problem is a differential equation whose solution must satisfy an initial condition. At present time, the study of fuzzy integrodifferential equations is an issue of remarkable consideration because it is one of the modern mathematical fields that arise from the modeling of uncertain physical, engineering and medical problems and are useful in studying the observability of dynamical control systems. Some of the properties of solution of rfdes with impulses are studied. Puri and ralescu 3 introduced the concept of the differential of a fuzzy function.
The solutions would be mill stones both in topology and nonlinear dynamics. The second part involves a feedforward neural network containing adjustable parameters the weights. Dehgan in 8, 9, 10 introduced full fuzzy system in which b and a are fuzzy vector and fuzzy matrix, respectively. Homogeneous differential equations of the first order. Differential equations with fuzzy parameters via differential. In general, the parameters, variables and initial conditions within a model are considered as being defi. One of them solves differential equations using zadehs extension principle buckleyfeuring 30, while another approach interprets fuzzy differential equations through differential inclusions.
A new method for solving fuzzy linear differential equations. Impulsive differential equations ides are a new branch ofdifferentialequations. Idescanfindnumerousapplications in different branches of optimal control, electronics, economics,physics,chemistry,andbiologicalsciences. Exact solutions systems of partial differential equations linear systems of two secondorder partial differential equations pdf version of this page. 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. The numerical solutions are compared with igh and iigh differential exact solutions concepts system. Numerical methods for fractional differential equations. In the litreture, there are several approaches to study fuzzy differential equations. Fuzzy partial differential equations and relational. Our purpose in this article is solving fuzzy partial differential equation fpde.
We presented an implicit method for solving this equation, and we considered necessary conditions for stability of this method. The authors sincerely believe that the nonlinear partial differential equations that we have formulated above may be become a major clue and a tool to study this problematic problem. Also we compared the approximate solution and exact solution. Pdf fuzzy differential equation with nonlocal conditions. Depending upon the domain of the functions involved we have ordinary di. Here the solution of fuzzy differential equation becomes fuzzier as time goes on. A numerical example is carried out for solving system adapted from fuzzy.
We consider the random fuzzy differential equations rfdes with impulses. Neural ordinary differential equations nips proceedings neurips. Also, many researchers have worked on the theoretical and numerical solutions of fuzzy differential equations such as 4,5 and 6. Nov, 2010 in this paper, a novel operator method is proposed for solving fuzzy linear differential equations under the assumption of strongly generalized differentiability. Differential equations are called partial differential equations pde or or dinary differential equations ode according to whether or not they. Solving hybrid fuzzy differential equations by chebyshev. Misukoshi et al 11 have proved that, under certain.
We extend and use this method to solve secondorder fuzzy linear differential equations under generalized hukuhara differentiability. The following corollary shows a new technique to find the exact solutions of intervalvalued delay fractional differential equation by using the solutions of intervalvalued delay integer order differential equation. Some authors discussed the solution of fuzzy integro differential equation by fuzzy differential transform method in their research paper. For linear fuzzy differential equations, we state some results on the behaviour of the solutions and study their relationship with the generalised hukuhara derivative. Partial averaging of fuzzy differential equations with maxima. 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. Regrettably mathematical and statistical content in pdf files is unlikely to be accessible. Using interpolation of fuzzy number for solving of fuzzy differential equations by pstage rungekutta method mojtaba ranjbar.
First order homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number is described by mondal and roy 32. It is easily seen that the differential equation is homogeneous. The aim of this work is to present a novel approach based on the artificial neural network for finding the numerical solution of first order fuzzy differential equations under generalized hderivation. First, the authors transformed a fuzzy differential equation by two parametric ordinary differential equations and then solved by fuzzy eulers method. In last section we given an example for consider numerical results. Fuzzy differential equations and applications for engineers. Josephs college autonomoustrichy india assistant2 professor, department of mathematics, a. Finally, an example is presented to illustrate the results.
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 scheme of partial averaging of fuzzy differential equations with maxima is considered. Reservoir characterization and modeling studies in fuzziness and soft computing nikravesh, masoud, zadeh, lofti a. Numerical solution of firstorder linear differential. In this paper, we study the fuzzy laplace transforms introduced by the authors in allahviranloo and ahmadi in soft comput. Figure 1 shows the behaviour of function f and its differences with the logistic equation.
Fractional riccati differential equation, international journal of differential equations volume 2010, article id 764738. Asweknow,the real systems are often faced with two kinds of. Numerical solution of first order linear fuzzy differential equations using leapfrog method. A novel approach for solving fuzzy differential equations. 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. A role for symmetry in the bayesian solution of differential equations wang, junyang, cockayne, jon, and oates, chris. Averaging method, fuzzy differential equation with maxima.
Research article nonlinear fuzzy differential equation with time delay and optimal control problem wichaiwitayakiattilerd department of mathematics, faculty of science, king mongkut s institute of technology ladkrabang, bangkok, ai land correspondence should be addressed to wichai witayakiattilerd. Solving secondorder fuzzy differential equations by the. The accuracy and efficiency of the proposed method are demonstrated by applying it to some different numerical experiments. The system with fuzzy constant coefficients is interpreted under strongly generalized differentiability. The method is based on chebyshev wavelets approximations. To this end, the equivalent integral form of the original problem is obtained then by using its lower and upper functions the solutions in the parametric forms are determined. Numerical solution of fuzzy differential equations by taylor method. Secondorder differential equations the open university. Analysis and computation of fuzzy differential equations. In this paper, we employ fuzzy sumudu transform for solving system of linear fuzzy differential equations with fuzzy constant coefficients.
902 674 1439 1467 508 589 129 1307 420 1420 1075 1014 840 248 470 326 568 918 1412 517 132 958 480 534 1079 1332 16 1206 45 683 803 610 1397 641 201 22 73 1237 879 710 1109 593 1472