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A Numerical Scheme for Fuzzy Cauchy Problems
Abstract
In this paper, we use power series method to solve fuzzy Cauchy differential equations of first order. Theoretical consideration is discussed and some examples are presented to show the ability of the method for fuzzy Cauchy differential equations. We use Matlab for numerical calculations.
... The stability properties and analytical results of HFDEs can be found in [15,16,20]. The numerical methods of fuzzy differential equation are studied by numerous authors such as [4,1,5]. Furthermore, there are some numerical techniques to solve hybrid fuzzy differential equations [17,18,19,14]. ...
In this paper, we study a numerical method for hybrid fuzzy differential equations (HFDEs) by an application of the variational iteration method (VIM). We state a convergence result and give numerical examples to illustrate the theory. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. The method is easy to apply and produces very accurate numerical results.
... In section 3 we define the problem, this is a fuzzy Cauchy problem [11,9] whose numerical solution is the main interest of this work and we apply the standard Euler method for systems [1,7,8,13] followed by a complete error analysis and show that the numerical solution converges to the unique solution. ...
... Subsequently, by using the lateral H-derivatives, there are two different interpretations of a fuzzy differential equation generating new solutions for a fuzzy differential equation. Numerical solution of FIVPS under generalised differentiability concept has been studied in [11][12][13][14][15][16][17][18][19][20][21].Fuzzy set theory is a useful tool to describe the situation in which data are imprecise or vague or uncertain. A membership function of a classical fuzzy set assigns to each element of the universe of discourse a number from the unit interval to indicate the degree of belongingness to the set under consideration. ...
In this paper, an Intuitionistic Fuzzy
Differential Equation (IFDE) with initial condition is
solved numerically through fourth order Runge-Kutta
method under generalised differentiability concept. The
efficiency of the proposed method over the Euler and
Modified Euler methods is shown by illustrating an
example.
... Subsequently, by using the lateral H-derivatives, there are two different interpretations of a fuzzy differential equation generating new solutions for a fuzzy differential equation. Numerical solution of FIVPS under generalised differentiability concept has been studied in [11][12][13][14][15][16][17][18][19][20][21]. In this paper, non-autonomous fuzzy Cauchy problem is solved numerically by a new fourth order Runge-Kutta like formula in which higher order derivative terms have been taken as new parameters in order to increase the accuracy, under generalised differentiability concept. ...
This paper presents numerical solution for Fuzzy Differential Equation under generalized
differentiability by various second orders Runge-Kutta methods such as Arithmetic Mean, Centroid mean, Harmonic
Mean, Contra Harmonic Mean and Geometric Mean with new parameters that increase the order of accuracy of the
solution. The accuracy and efficiency of the proposed methods are illustrated by solving a first order FDE.
... Subsequently, by using the lateral H-derivatives, there are two different interpretations of a fuzzy differential equation generating new solutions for a fuzzy differential equation. Numerical solution of FIVPS under generalised differentiability concept has been studied in [11][12][13][14][15][16][17][18][19][20][21]. In this paper, non-autonomous fuzzy Cauchy problem is solved numerically by a new fourth order Runge-Kutta like formula in which higher order derivative terms have been taken as new parameters in order to increase the accuracy, under generalised differentiability concept. ...
In this paper a new Runge-Kutta –like
formula of order 4 with higher order derivatives is
derived for non-autonomous system of ordinary
differential equations. This formula is used for finding
the numerical solution of fuzzy differential equations
under generalized differentiability concept. The
proposed formula is illustrated with an example.
... Subsequently, by using the lateral H-derivatives, there are two different interpretations of a fuzzy differential equation generating new solutions for a fuzzy differential equation. Numerical solution of FIVPS under generalised differentiability concept has been studied in [11][12][13][14][15][16][17][18][19][20][21].Fuzzy set theory is a useful tool to describe the situation in which data are imprecise or vague or uncertain. A membership function of a classical fuzzy set assigns to each element of the universe of discourse a number from the unit interval to indicate the degree of belongingness to the set under consideration. ...
In this paper, an Intuitionistic Fuzzy
Differential Equation (IFDE) with initial condition is
solved numerically through Modified Euler method
under generalised differentiability concept. The
efficiency of the proposed method over the Euler
method is shown by illustrating an example.
... At times, it may be difficult or not possible to find exact solution of FDE, also the real-life applications may have only the observations for the dynamical processes involving imprecisionthen for solving such problems use of numerical methods becomes inevitable. Numerical method for solving FDE with fuzzy initial condition given by many authors [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and analytical technique used in [33]. In [25], [26], [27], [28], [29] authors solved system of fuzzy differential equations with/or fuzzy parameters and fuzzy initial condition. ...
This paper investigates system of differential equations with fuzzy parameters and fuzzy initial condition for numerical solution. Here,the function, can be nonlinear or it can be linear with some or all matrices entries as fuzzy number. In this paper, we propose the numerical technique which is based on approximation of Hukuhara difference, for both kind of dynamical systems (linear and nonlinear). In dynamical system, uncertainty of possibilistic type can be realized efficiently using fuzzy parameters and such systems are mathematical models for various application in varied domains. For such systems, we get the scheme for existence of solution and its convergence. Lastly, illustrative examples are solved by using proposed scheme and compared with crisp solution.
... . (29) ε = 10 −4 . ...
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... In recent years, many works have been performed by several authors in numerical solutions of fuzzy differential equations (Fard, 2009a,b; Fard et al., 2009, 2010; Fard and Kamyad, 2010; Friedman et al., 1999; Hullermeier, 1999). Furthermore, there are some numerical techniques to solve hybrid fuzzy differential equations, for example, Pederson and Sambandham (2007, 2008) have investigated the numerical solution of HFDEs by using the Euler and Runge–Kutta methods,respectively, and Prakash and Kalaiselvi (2009) have studied the predictor–corrector method for hybrid fuzzy differential equations. ...
In this paper, the Nyström method is developed to approximate the solutions for hybrid fuzzy differential equation initial value problems (IVPs) using the Seikkala derivative. A proof of convergence of this method is also discussed in detail. The accuracy and efficiency of the proposed method are demonstrated by applying it to two different numerical experiments.
... In such situations, FDEs are common tools if the underlying structure is not probabilistic. In recent years, several investigators have studied FDEs [1][2][3][4][7][8][9][10][11][12]14,15,[17][18][19][20][21][22][23][25][26][27][28][29][30][37][38][39][40][41]44,45,47,48,50,53,57,58]. FDEs are used to analyze the behavior of phenomena that are subject to imprecise or uncertain factors, ranging from particle physics [24], chaotic systems [54,55] and engineering [48] to medicine [6] and computational biology [5,13,16]. ...
In this paper, a novel iterative method is proposed to obtain approximate-analytical solutions for the linear systems of first-order fuzzy differential equations (FDEs) with fuzzy constant coefficients (FCCs) while avoiding the complexities of eigen-value computations. A theorem for the convergence and the validity of the approach is also presented in detail. Numerical experiments and comparisons with exact solutions reveal that the proposed method is capable of generating accurate results.
In this research, we adopted the Picard method for solving the quadratic Riccati and the Painlevé I equations under fuzzy environment and generalized H-differentiability. We provide the existence and uniqueness theorem for solving the differential equations. The convergence of the proposed fuzzy Picard method is also investigated. Finally, various computational experiments are carried out to assess the proposed method.
In this paper, the fuzzy variational iteration method is proposed to solve the nonlinear fuzzy differential equation (NFDE). The convergence and the maximum absolute truncation error of the proposed method are proved in details. Some examples are investigated to verify convergence results and to illustrate the efficiently of the method.
In this paper, Picard method is proposed to solve the Cauchy reaction-diffusion equation with fuzzy initial condition under generalized H-differentiability. The existence and uniqueness of the solution and convergence of the proposed method are proved in details. Some examples are investigated to verify convergence results and to illustrate the efficiently of the method. Also, we obtain the switching points in examples.
In this paper, numerical algorithms for solving “fuzzy ordinary differential
equations” are considered. A scheme based on the Taylor method of order p is
discussed in detail and this is followed by a complete error analysis. The algorithm is
illustrated by solving some linear and nonlinear fuzzy Cauchy problems.
In this paper the Rådström embedding theorem (Proc. Amer. Math. Soc. 3 (1952), 165) is generalized and is used to define the concept of the differential of a fuzzy function.
This paper first presents a new solution to the fuzzy first-order initial value problem. Elementary properties of this new solution are given. We then compare various derivatives of fuzzy functions that have been presented in the literature followed by comparing the different solutions one may obtain to the fuzzy initial value problem using these derivatives. Examples are given, including linear and non-linear fuzzy first-order differential equations, where we find both our new solution and other solutions to the fuzzy initial value problem.
A fuzzy mapping from X to Y is a fuzzy set on X Ã Y. The concept is extended to fuzzy mappings of fuzzy sets on X to Y, fuzzy function and its inverse, fuzzy parametric functions, fuzzy observation, and control. Set theoretical relations are obtained for fuzzy mappings, fuzzy functions, and fuzzy parametric functions. It is shown that under certain conditions a precise control goal can be attained with fuzzy observation and control as long as the observations become sufficiently precise when the goal is approached.
Using the embedding method, numerical procedures for solving fuzzy differential equations (FDEs) and fuzzy integral equations (FIEs) with arbitrary kernels have been investigated. Sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples. This work and its conclusions may narrow the gap between the theoretical research on FDEs and FIEs and the practical applications already existing in the design of various fuzzy dynamical systems.
The initial value problem x′(t) = f(t,x(t)), x(0)= x0, with fuzzy initial value and with deterministic or fuzzy function f is considered. Two different approaches, viz. the extension principle and the use of extremal solutions of deterministic initial value problems, are applied. Generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated.
Fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets. Topological properties of the resulting metric space are studied. The notion of θ-crisp fuzzy numbers is introduced, and, in particular, it is shown that this class of fuzzy numbers is homeomorphic to the Hilbert cube.
The classical Peano theorem states that in finite dimensional spaces the Cauchy problem x ' (t)=f(t,x(t)), x(t 0 )=x 0 , has a solution provided f is continuous. In addition, Godunov has shown that each Banach space in which the Peano theorem holds true is finite dimensional. For differential inclusions, the existence of a solution to the Cauchy problem is also guaranteed under various assumptions on the right-hand side. In this paper, we study the Cauchy problem for fuzzy differential equations. To be more specific, let U be a subspace of normal, convex, upper semicontinuous, compactly supported fuzzy sets defined in ℝ n and assume that f:[t 0 ,t 0 +a]×U→U is continuous. We show that the Cauchy problem has a solution if and only if U is locally compact.
Numerical algorithms for solving ‘fuzzy ordinary differential equations’ (FODE) are considered. A scheme based on the classical Euler method is discussed in detail, and this is followed by a complete error analysis. The algorithm is illustrated by solving several linear and nonlinear fuzzy Cauchy problems.
In this paper, a numerical method for solving ‘fuzzy differential inclusions’ is considered.Fuzzy reachable set can be approximated by proposed method with complete error analysis, which is discussed in detail. The method is illustrated by solving some linear and nonlinear fuzzy initial value problems.
We study the effect of the forcing term to the solution of a fuzzy differential equation.
The validity of the classical Peano theorem is noted for differential equations on the metric space (E(n), d(p)) of normal fuzzy convex sets in R(n), where d(p) is the p-th mean of the Hausdorff distances between corresponding level sets. This contrasts with differential equations on the space (E(n), D) where D is the max metric, which is not locally compact so more than just continuity is required to ensure the existence of solutions. Some additional properties, differing from that given by Kaleva, are mentioned.
In the present note it is shown that the examples presented in a recent paper by T. Allahviranloo et al. [Inf. Sci. 177, No. 7, 1633–1647 (2007; Zbl 1183.65090)], are incorrect. Namely, the “exact solutions” proposed by the authors are not solutions of the given fuzzy differential equations (FDEs). The correct exact solutions are also presented here, together with some results for characterizing solutions of FDEs under Hukuhara differentiability by an equivalent system of ODEs. In this way a new direction for the numerical solutions of FDEs is proposed.
In this paper three numerical methods to solve ''The fuzzy ordinary differential equations'' are discussed. These methods are Adams-Bashforth, Adams-Moulton and predictor-corrector. Predictor-corrector is obtained by combining Adams-Bashforth and Adams-Moulton methods. Convergence and stability of the proposed methods are also proved in detail. In addition, these methods are illustrated by solving two fuzzy Cauchy problems.
The extension of algebraic and analytical concepts to the theory of fuzzy sets appears to play a central role in the investigation of nondeterministic techniques. Since an exact description of any real physical situation is virtually impossbile, it is necessary to develop schemes which deal analytically with decision processes in an imprecise environment. In this paper we are concerned with formulating such a framework and with introducing a bibliography of 570 items, all classified with fuzzy set theory and its applications, which will help the readers to come to grips with the literature explosion on the subjects of fuzzy sets, fuzzy algebra, fuzzy statistics, and closely related applications.
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