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

New Multi-Step Runge-Kutta Method For Solving Fuzzy Differential Equations

This paper presents solution for the first order fuzzy differential equation by Runge Kutta method of order two with new parameters which are used in the main formula in order to increase the order of accuracy of the solution. This method is discussed in detail followed by a complete error analysis. The accuracy and efficiency of the proposed method is illustrated by solving a fuzzy initial value problem. Keywords: Fuzzy differential equations, multi-step Runge-Kutta method, higher order derivative approximations, harmonic mean

An impulsive approach for numerical investigation of hybrid fuzzy differential equations and intuitionistic treatment for fuzzy ordinary and partial differential equations

Many evolution processes are characterized by the fact that at certain moments of time, they experience a change of state abruptly. It is assume naturally, that those perturbations act instantaneously, in the form of impulses. The impulsive differential equations, by means differential equations involving impulse effects, are seen as a natural description of observed evolution phenomenon of several real world problems. For example, systems with impulse effect have applications in physics, biotechnolagy, industrial robotics, pharmacokinetics, population dynamics, ecology, optimal control production theory and many others. Therefore, it is beneficial to study the theory of impulsive differential equations as a well deserved discipline, due to the increase applications of impulsive differential equations in various fields in the future. However, in many mathematical modelling of the real world problems, fuzziness and impulsiveness occurs simultaneously. This problem would be better modelled by impulsive fuzzy differential equations. Therefore, this research applies the theory of impulsive fuzzy differential equations by combining the theories of impulsive differential equations and fuzzy differential equations. The numerical algorithms are developed and the solutions are verified by comparing the results with the analytical solutions. The novel method for the first order linear impulsive hzzy differential equations under generalized differentiability is also proposed analytically and numerically, The convergence theor~m for the impulsive fuzzy differential equations (FDE) under generalized differentiability is defined. In this study, Ant Colony Programming (ACP) was used to find the optimal solution of FDE. Results obtained show that the method is effective in solving fuzzy differential equation. The solution in this method is equivaIent to the exact solution of the problem. Modified Romberg's method and Modified Two-step Simpson's 318 method are used to solve FDE with hzzy IVP has been successfully derived. The result has been shown that Modified Rornberg's method gave smaller error than the Standard Euler's method. Therefore Modified Romberg's method can estimate the solution of fizzy differential equation more effectively than the Euler's method in solving fuzzy differential equation. Meanwhile, by using the modified wo-step Simpson's 318 methods, it has been shown that the solution of FDE provide more accurate approximation to the exact solution and it also gives better results than the Runge-Kutta method. In other words, Modified Twostep Simpson's 318 method is an effective method to solve fuzzy differential equation compared to the Runge-Kutta method

Numerical algorithm for solving second order nonlinear fuzzy initial value problems

The purpose of this analysis would be to provide a computational technique for the numerical solution of second-order nonlinear fuzzy initial value (FIVPs). The idea is based on the reformulation of the fifth order Runge Kutta with six stages (RK56) from crisp domain to the fuzzy domain by using the definitions and properties of fuzzy set theory to be suitable to solve second order nonlinear FIVP numerically. It is shown that the second order nonlinear FIVP can be solved by RK56 by reducing the original nonlinear equation intoa system of couple first order nonlinear FIVP. The findings indicate that the technique is very efficient and simple to implement and satisfy the Fuzzy solution properties. The method’s potential is demonstrated by solving nonlinear second-order FIVP

Numerical Solution of NTH - Order Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Basesd On Contrahamonic Mean

In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Contra-harmonic Mean (RKCoM4) is used to find the numerical solution of this problem and the convergence and stability of the method is proved. This method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits well to find the numerical solution of Nth - order FIVPs

Non-constructive interval simulation of dynamic systems

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