Solution Of Fuzzy Differential Equations Research Articles

Numerical Solutions of Fuzzy Differential Equations by Harmonic Mean and Cubic Mean of Modified Euler' s Method

We aimed to solve first-order differential equations using two novel techniques: the harmonic mean and the cubic mean of Euler' s modified approach for fuzzy primary value in this research proposal. We present a new formulation of Euler' s classic approach based on Zadeh' s extension concept to address this dependency issue in a fuzzy situation. In the literature, numerical approaches for solving differential equations with fuzzy main values often disregard this issue. With a few examples, we show how our approach outperforms more traditional fuzzy approaches based on Euler' s method.

VARIATIONAL ITERATION METHOD FOR SOLVING FUZZY BOUNDARY VALUE PROBLEMS

Abstract: 
 This paper will find the approximate solution to the linear and nonlinear fuzzy boundary value problems using the variational iteration method. The numerical scheme is based on analyzing the fuzzy problem into two crisp sub-problems. The first is for the upper solution and the second is for the lower solution of the fuzzy solution. Also, the convergence of the obtained variational iteration formula has been proved to converge to the exact solution of the problem under consideration in each illustrative example, since there is no general formula that may be obtained for the correction functional related to the problem under consideration which is due to the variation of the general Lagrange multiplier from one problem to another.
 Keywords:Fuzzy boundary value problems, variational iteration method, solution of fuzzy differential equations using variational iteration method.

Fuzzy Yang transform for second order fuzzy differential equations of integer and fractional order

In this work, a Laplace-like transform in a fuzzy environment called Yang transform is introduced to solve fuzzy differential equations (FDEs) with the order θ ∈ (1, 2] involving the Caputo fractional derivative in the sense of gH-differentiability. Some basic properties of Yang transform for integer and fractional derivatives are also provided. Furthermore, by utilizing the combination between the Adomian decomposition method (ADM) and the Yang transform method, a general algorithm called the hybrid Yang transform method (HYTM) to solve the solutions of FDEs in the nonlinear form is proposed. For the validity and accuracy of this novel method, some examples and their simulations are given.

Self-Adjusting Formulae for the Solution of Fuzzy Differential Equations with Triangular Fuzzy Numbers

Self-Adjusting Formulae for the Solution of Fuzzy Differential Equations with Triangular Fuzzy Numbers

APPLICATION OF AN EFFECTIVE METHOD ON THE SYSTEM OF NONLINEAR FUZZY INTEGRO-DIFFERENTIAL EQUATIONS

In real world physical applications purpose, it is complicated to acquire an exact solution of fuzzy differential equations due to complexities in fuzzy arithmetic and therefore creating the need for the use of reliable and efficient techniques in the solution of fuzzy differential equations. The purpose of this research paper is to utilize the reliable analytic approach of homotopy perturbation Sumudu transform method for better understanding of systems of non-linear fuzzy integro-differential equations, while using the concept of fuzzy parameter in certain dynamical problems to remove the hurdles faced in numerical approach. These mathematical models are of great interest in engineering and physics. Some numerical examples are also given to demonstrate the efficiency and superiority of the method, followed by graphical representation of the comparison of exact and approximated solution by using Maple 2017

Milstein method for solving fuzzy differential equation

To solve fuzzy differential equations driven by Liu process, three Milstein schemes are proposed in this work, which are explicit Milstein scheme, semi-implicit Milstein scheme and improved Milstein scheme. Improved Milstein scheme is constructed by correcting the error with semi-implicit method, the error is the difference between the exact solution of fuzzy differential equations and the solution derived from Milstein scheme. These numerical methods are proved to have strong convergence with order two. Accompanying the results above, the concept of mean-stability of numerical schemes for fuzzy differential equations is put forward and analyzed. For a linear test equation, it is showed the mean-stability region of improved Milstein scheme is bigger than explicit Milstein scheme. Finally, the accuracy and effectiveness of these schemes are confirmed through numerical examples.

Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives

In this paper, we propose the fuzzy Shehu transform method (FSTM) using Zadeh’s decomposition theorem and fuzzy Riemann integral of real-valued functions on finite intervals. As an alternative to standard fuzzy Laplace transform and the fuzzy Sumudu integral transform, we established some potential useful (new or known) properties of the FSTM and validate their applications. Furthermore, the FSTM is coupled with the well-known homotopy analysis method to obtain the approximate and exact solutions of fuzzy differential equations of integer and non-integer order derivatives. The convergence analysis and the error analysis of the suggested technique are provided and supported by graphical solutions. Comparison of the numerical simulations of exact and approximate solutions of two fuzzy fractional partial differential equations are tabulated to further justify the reliability and efficiency of the proposed method.

(2002-5686) Milstein method for solving fuzzy differential equation

To solve fuzzy differential equations driven by Liu process, three Milstein schemes are proposed in this work, which are explicit Milstein scheme, semi-implicit Milstein scheme and improved Milstein scheme. Improved Milstein scheme is constructed by correcting the error with semi-implicit method, the error is the difference between the exact solution of fuzzy differential equations and the solution derived from Milstein scheme. These numerical methods are proved to have strong convergence with order two. Accompanying the results above, the concept of mean-stability of numerical schemes for fuzzy differential equations is put forward and analysed. For a linear test equation, it is showed the mean-stability region of improved Milstein scheme is bigger than explicit Milstein scheme. Finally, the accuracy and effectiveness of these schemes are confirmed through numerical examples.

Numerical Solutions of Fuzzy Differential Equation under Generalized Differentiability concept using a Third Order Runge - Kutta Method with Contra Harmonic Mean

Numerical Solutions of Fuzzy Differential Equation under Generalized Differentiability concept using a Third Order Runge - Kutta Method with Contra Harmonic Mean

Modified Fractional Runge–Kutta Method for Solving Fuzzy Differential Equations of Fractional Order

The purpose of this research work is to solve fuzzy differential equations of fractional order under the Caputo-type. The modified fractional Runge–Kutta (R–K) method is proposed based on a generalized Runge–Kutta formula and modified trapezoidal rule. Moreover, the approach is followed to obtain a solution for fuzzy differential equations of fractional order in the sense of Caputo-type fuzzy fractional derivatives. Finally, illustrative example is provided to demonstrate the applicability, accuracy and efficiency of the proposed method.

Calculus for linearly correlated fuzzy function using Fréchet derivative and Riemann integral

In this manuscript we study integration and derivative theories for interactive fuzzy processes. These theories are based on the Fréchet derivative and the Riemann integral. In addition, we present a connection between these two theories, i.e., some problems may be formulated in both ways. We establish the fundamental theorem of calculus, theorem of existence and the local uniqueness of the solution of fuzzy differential equations and some techniques to solve fuzzy initial value problems. To illustrate the usefulness of the developed theory, we investigate the radioactive decay model.

Numerical Solution of Fuzzy Differential Equations by Seventh Order Runge-Kutta Method

Numerical Solution of Fuzzy Differential Equations by Seventh Order Runge-Kutta Method

Numerical solution of fuzzy differential equations using orthogonal polynomials

Numerical solution of fuzzy differential equations using orthogonal polynomials

Numerical solution of fuzzy differential equations using orthogonal polynomials

The present paper proposed a new method to solve nth order fuzzy differential equations using collocation type of method. In the solution procedure, Gram Schmidt orthogonalisation process is used with Legendre and Chebyshev polynomials. Known example problems are solved and compared with the exact results to illustrate the efficiency and reliability of the proposed method.

Numerical solution of fuzzy differential equations by Milne's predictor-corrector method and the dependency problem

Numerical solution of fuzzy differential equations by Milne's predictor-corrector method and the dependency problem

Fuzzy Euler approximation and its local convergence

Fuzzy differential equation driven by Liu process is an important tool to deal with dynamic system in fuzzy environment. In many cases, however, it is difficult to find analytic solution of fuzzy differential equation driven by Liu process. Based on credibility theory, the Taylor series of fuzzy differential driven by Liu process is given and Euler approximation that use Taylor expansion as the recursion formula is obtained in this paper. Then the local convergence of Euler approximation is deduced.

Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives

In this paper, the solution of fuzzy differential equations is approximated numerically using diagonally implicit multistep block method of order four. The multistep block method is well known as an efficient and accurate method for solving ordinary differential equations, hence in this paper the method will be used to solve the fuzzy initial value problems where the initial value is a symmetric triangular fuzzy interval. The triangular fuzzy number is not necessarily symmetric, however by imposing symmetry the definition of a triangular fuzzy number can be simplified. The symmetric triangular fuzzy interval is a triangular fuzzy interval that has same left and right width of membership function from the center. Due to this, the parametric form of symmetric triangular fuzzy number is simple and the performing arithmetic operations become easier. In order to interpret the fuzzy problems, Seikkala’s derivative approach is implemented. Characterization theorem is then used to translate the problems into a system of ordinary differential equations. The convergence of the introduced method is also proved. Numerical examples are given to investigate the performance of the proposed method. It is clearly shown in the results that the proposed method is comparable and reliable in solving fuzzy differential equations.

Solution of Fuzzy Differential Equations Using Fuzzy Sumudu Transforms

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. However, it is very difficult to obtain solutions of FDEs. In this paper, the solutions of FDEs are approximated by utilizing the fuzzy Sumudu transform (FST) method. Significant theorems are suggested in order to explain the properties of FST. The proposed method is validated with three real examples.

Numerical Solution of Fuzzy Differential Equations with Z-numbers using Fuzzy Sumudu Transforms

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. However, it is very difficult to obtain solutions of FDEs. In this paper, the solutions of FDEs are approximated by utilizing the fuzzy Sumudu transform (FST) method. Here, the uncertainties are in the sense of Z-numbers. Important theorems are laid down to illustrate the properties of FST. The theoretical analysis and simulation results show that this new technique is effective to estimate the solutions of FDEs.

Numerical solution of fuzzy differential equations by Taylor's method

Numerical solution of fuzzy differential equations by Taylor's method