Solving fuzzy differential equations by differential transformation method

An extension of the differential transformation method (DTM), which is an analytical-numerical method for solving the fuzzy differential equation (FDE), is given. The concept of generalised H-differentiability is used. This concept is based on an enlargement of the class of differentiable fuzzy mappings; to define this, the lateral Hukuhara derivatives are considered. The proposed algorithm is illustrated by numerical examples, and some error comparisons are made with other methods for solving a FDE.     

具有推广Hukuhara导数的模糊微分方程的数值解

在推广Hukuhara导数概念下研究了一阶模糊微分方程的模糊初值问题,利用预估-校正算法给出了模糊初值问题的数值解,文中的例子说明了方法的可行性及实用性。     

Note on “Numerical solutions of fuzzy differential equations by predictor-corrector method”

In the present note it is shown that the examples presented in a recent paper by Allahviranloo et al., 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.     

Nth-order fuzzy linear differential equations

In this paper a numerical method for solving nth-order linear differential equations with fuzzy initial conditions is considered. The idea is based on the collocation method. The existence theorem of the fuzzy solution is considered. This method is illustrated by solving several examples.     

Toward the existence and uniqueness of solutions of second-order fuzzy differential equations

In this paper the existence and uniqueness of solutions for second-order fuzzy differential equations with initial conditions under generalized H-differentiability is proved. To this end, the concept of second-order generalized differential equation is defined, which is based on an enlargement of the class of differentiable fuzzy mappings.     

Artificial neural network approach for solving fuzzy differential equations

The current research attempts to offer a novel method for solving fuzzy differential equations with initial conditions based on the use of feed-forward neural networks. First, the fuzzy differential equation is replaced by a system of ordinary differential equations. A trial solution of this system is written as a sum of two parts. The first part satisfies the initial condition and contains no adjustable parameters. The second part involves a feed-forward neural network containing adjustable parameters (the weights). Hence by construction, the initial condition is satisfied and the network is trained to satisfy the differential equations. This method, in comparison with existing numerical methods, shows that the use of neural networks provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples.     

A method for solving nth order fuzzy linear differential equations

In this paper, an analytic method (eigenvalue–eigenvector method) for solving nth order fuzzy differential equations with fuzzy initial conditions is considered. In this method, three cases are introduced, in each case, it is shown that the solution of differential equation is a fuzzy number. In addition, the method is illustrated by solving several numerical examples.     

Numerical solution of fuzzy differential equations by predictor-corrector method

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.     

Comparison results for fuzzy differential equations

In this paper, we present several comparison results for the solutions of fuzzy differential equations, by using the Hukuhara derivative. These results constitute the extension to the fuzzy context of some comparison results for the solutions of linear differential equations. Besides, some side-results about the Hukuhara difference and the partial orderings are given. We illustrate the applicability of the new results by showing several examples.     

Solving fuzzy equations using evolutionary algorithms and neural nets

 In this paper we use evolutionary algorithms and neural nets to solve fuzzy equations. In Part I we: (1) first introduce our three solution methods for solving the fuzzy linear equation AˉXˉ + Bˉ= Cˉ; for Xˉ and (2) then survey the results for the fuzzy quadratic equations, fuzzy differential equations, fuzzy difference equations, fuzzy partial differential equations, systems of fuzzy linear equations, and fuzzy integral equations; and (3) apply an evolutionary algorithm to construct one of the solution types for the fuzzy eigenvalue problem. In Part II we: (1) first discuss how to design and train a neural net to solve AˉXˉ + Bˉ= Cˉ for Xˉ and (2) then survey the results for systems of fuzzy linear equations and the fuzzy quadratic.     

Initial value problem for fuzzy differential equations under dissipative conditions

A new concept of inner product on the fuzzy space (Eⁿ,D) is introduced, studied and used to prove several theorems stating the existence, uniqueness and boundedness of solutions of fuzzy differential equations. A stability result is also proved in the same context.     

Eigenvalue problems of second order impulsive differential equations

This paper is concerned with an eigenvalue problem for second order differential equations with impulse. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved.     

On d-solvability for linear differential equations

The aim of this paper is to investigate the possibility of solving a linear differential equation of degree n$n$ by means of differential equations of degree less than or equal to a fixed d$d$, 1≤d<n$1\le d<n$. This paper recovers and extends work of G. Fano, M. F. Singer and E. Compoint. Representations of algebraic Lie algebras are the main tool.     

Numerical solution of a fuzzy system of linear equations with polynomial parametric form

A systems of linear equations are used in many fields of science and industry, such as control theory and image processing, and solving a fuzzy linear system of equations is now a necessity. In this work we try to solve a fuzzy system of linear equations having fuzzy coefficients and crisp variables using a polynomial parametric form of fuzzy numbers.     

具有广义Hukuhara导数的模糊微分系统的近似解

利用广义Hukuhara导数研究了一阶模糊线性微分系统的模糊初值问题,将一阶模糊线性微分系统转化成2n个等价的分明线性微分系统,给出了模糊初值问题近似解析解的微分变换解法;给出了具体算例。     

Taylor polynomial solutions of systems of linear differential equations with variable coefficients

A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients. Using the Taylor collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence by finding the Taylor coefficients, the Taylor polynomial approach is obtained. Also, the method can be used for the linear systems in the normal form. To illustrate the pertinent features of the method, examples are presented and results are compared.     

Linear systems of first order ordinary differential equations: fuzzy initial conditions

 We present two types of fuzzy solutions to linear systems of first order differential equations having fuzzy initial conditions. The first solution, called the extension principle solution, fuzzifies the crisp solution and then checks to see if its α-cuts satisfy the differential equations. The second solution, called the classical solution, solves the fuzzified differential equations and then checks to see if the solution always defines a fuzzy number. Three applications are presented: (1) predator–prey models; (2) the spread of infectious diseases; and (3) modeling an arms race.     

On solutions to fuzzy stochastic differential equations with local martingales

This paper deals with a class of fuzzy stochastic differential equations (FSDEs) driven by a continuous local martingale under the Lipschitzian condition. Such equations can be useful in modeling hybrid systems, where the phenomena are simultaneously subjected to two kinds of uncertainties: randomness and fuzziness. The solutions of the FSDEs are the fuzzy stochastic processes, and their uniqueness is considered to be in a strong sense. Thus, the existence and uniqueness of solutions to the FSDEs under the Lipschitzian condition is first proven. Moreover, some asymptotic properties of the solutions to the FSDEs are investigated. Finally, an illustrating example on the interest term model is provided.     

Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

We describe locally one-dimensional (LOD) time integration schemes for parabolic differential equations in two space dimensions, based on the generalized trapezoidal formulas (GTF(α)). We describe the schemes for the diffusion equation with Dirichlet and Neumann boundary conditions, for nonlinear reaction-diffusion equations, and for the convection-diffusion equation in two space dimensions. The obtained schemes are second order in time and unconditionally stable for all α ∈ [0, 1]. Numerical experiments are given to illustrate the obtained schemes and to compare their performance with the better known LOD Crank-Nicolson scheme. While the LOD Crank-Nicolson scheme can give unwanted oscillations in the computed solution, our present LOD-GTF(α) schemes provide both stable and accurate approximations for the true solution.