FUZZY DELAY DIFFERENTIAL EQUATIONS WITH HYBRID SECOND AND THIRD ORDERS RUNGE-KUTTA METHOD

This paper considers fuzzy delay differential equations with known statedelays. A dynamic problem is formulated by time-delay differential equations and an efficient scheme using a hybrid second and third orders Runge-Kutta method is developed and applied. Runge-Kutta is well-established methods and can be easily modified to overcome the discontinuities, which occur in delay differential equations. Our objective is to develop a scheme for solving fuzzy delay differential equations. A numerical example was run, and the solutions were validated with the exact solution. The numerical results from C program will show that the hybrid Runge-Kutta scheme able to calculate the fuzzy solutions successfully

Hybrid runge-kutta method for solving linear fuzzy delay differential equations with unknown state-delays

In this research, a new method to solve the Fuzzy Delay Differential Equations (FDDEs) with unknown state-delays constrained optimization problem is introduced. This method is based on the coupling of second and third orders Runge- Kutta (RK) method called hybrid RK method. The main goal of this thesis is to identify the unknown state-delays using experimental data. RK methods are chosen because they are well-established and can be easily modified to overcome the discontinuities which occur in Delay Differential Equations (DDEs) especially outside uniform nodes with delay step-size. Numerical results of FDDEs from the hybrid RK methods are compared with exact solutions derived from stepwise approach using Maple software. The relative errors are calculated for the purpose of accuracy checking on these numerical schemes. In this study, a dynamic optimization problem in which the state-delays are decision variables is also imposed; with its formulated cost function. The gradient of the cost function is computed by solving auxiliary FDDEs. By exploiting the results, the state-delay identification problem can be solved efficiently and accurately using a gradient-based optimization method. In addition, a C program has been developed based on hybrid RK methods for solving these problems. Consequently, the results show that the new hybrid scheme is an efficient numerical technique in solving all the problems above with acceptable errors

Numerical Solution of First-Order Linear Differential Equations in Fuzzy Environment by Runge-Kutta-Fehlberg Method and Its Application

The numerical algorithm for solving “first-order linear differential equation in fuzzy environment” is discussed. A scheme, namely, “Runge-Kutta-Fehlberg method,” is described in detail for solving the said differential equation. The numerical solutions are compared with (i)-gH and (ii)-gH differential (exact solutions concepts) system. The method is also followed by complete error analysis. The method is illustrated by solving an example and an application

Application of Adomian decomposition method to solve hybrid fuzzy differential equations

AbstractIn this paper, we study the numerical solution of hybrid fuzzy differential equations by using Adomian decomposition method (ADM). This is powerful method which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution. Several numerical examples are given and by comparing the numerical results obtained from ADM and predictor corrector method (PCM), we have studied their accuracy

A kernel least mean square algorithm for fuzzy differential equations and its application in earth's energy balance model and climate

Abstract This paper concentrates on solving fuzzy dynamical differential equations (FDDEs) by use of unsupervised kernel least mean square (UKLMS). UKLMS is a nonlinear adaptive filter which works by applying kernel trick to LMS adaptive filter. UKLMS estimates multivariate function which is embedded to estimate the solution of FDDE. Adaptation mechanism of UKLMS helps for finding solution of FDDE in a recursive scenario. Without any desired response, UKLMS finds nonlinear functions. For this purpose, an approximate solution of FDDE is constructed based on adaptable parameters of UKLMS. An optimization algorithm, optimizes the values of adaptable parameters of UKLMS. The proposed algorithm is applied for solving Earth energy balance model (EBM) which is considered as a fuzzy differential equation for the first time. The method in comparison with the other existing approaches (such as numerical methods) has some advantages such as more accurate solution and also that the obtained solution has a functional form, thus the solution can be obtained at each time in training interval. Low error and applicability of developed algorithm are examined by applying it for solving several problems. After comparing the numerical results, with relative previous works, the superiority of the proposed method will be illustrated

Numerical solution of n’th order fuzzy initial value problems by six stages

The purpose of this paper is to present a numerical approach to solve fuzzy initial value problems (FIVPs) involving n-th order ordinary differential equations.The idea is based on the formulation of the six stages Runge-Kutta method of order five (RKM56) from crisp environment to fuzzy environment followed by the stability definitions and the convergence proof.It is shown that the n-th order FIVP can be solved by RKM56 by transforming the original problem into a system of first-order FIVPs. The results indicate that the method is very effective and simple to apply.An efficient procedure is proposed of RKM56 on the basis of the principles and definitions of fuzzy sets theory and the capability of the method is illustrated by solving second-order linear FIVP involving a circuit model problem

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

Numerical And Approximate- Analytical Solution Of Fuzzy Initial Value Problems

Persamaan pembezaan kabur ( FDEs ) digunakan untuk memodel masalah tertentu dalam bidang sains dan kejuruteraan dan telah dikaji oleh ramai penyelidik . Masalah tertentu memerlukan penyelesaian FDEs yang memenuhi keadaan awal kabur menimbulkan masalah awal kabur ( FIVPs ). Contoh masalah seperti ini boleh didapati dalam fizik, kejuruteraan, model penduduk, dinamik reaktor nuklear, masalah perubatan, rangkaian neural dan teori kawalan. Walau bagaimanapun, kebanyakan masalah nilai awal kabur tidak boleh diselesaikan dengan tepat. Tambahan pula, penyelesaian analisis tepat yang diperoleh juga mungkin begitu sukar untuk dinilai dan oleh itu kaedah berangka dan analisis hampiran perlu untuk memperoleh penyelesaian. Fuzzy differential equations (FDEs) are used for the modeling of some problems in science and engineering and have been studied by many researchers. Certain problems require the solution of FDEs which satisfy fuzzy initial conditions giving rise to fuzzy initial problems (FIVPs). Examples of such problems can be found in physics, engineering, population models, nuclear reactor dynamics, medical problems, neural networks and control theory. However, most fuzzy initial value problems cannot be solved exactly. Furthermore, exact analytical solutions obtained may also be so difficult to evaluate and therefore numerical and approximate- analytical methods may be necessary to evaluate the solution

(SI10-123) Comparison Between the Homotopy Perturbation Method and Variational Iteration Method for Fuzzy Differential Equations

In this article, the authors discusses the numerical simulations of higher-order differential equations under a fuzzy environment by using Homotopy Perturbation Method and Variational Iteration Method. The fuzzy parameter and variables are represented by triangular fuzzy convex normalized sets. Comparison of the results are obtained by the homotopy perturbation method with those obtained by the variational iteration method. Examples are provided to demonstrate the theory