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

Déferlement de vague : approche multi-pas

Nous simulons numériquement le déferlement de vagues par un modèle d'écoulement multi fluide à faible Mach grâce à une formulation explicite efficacement parallélisable. Le modèle repose sur un schéma par volumes finis de type Godunov du second ordre en temps et en espace et éventuellement un raidissement de l'interface. Nous introduisons une approche multi-pas qui autorise de conséquents gains en temps de calcul. Cette approche est validée par des confrontations expérience /simulation sur le déferlement de vague solitaire 2D sur plan incliné et la rupture de barrage 3D avec obstacle

Modified Artificial Neural Networks For Solving Fuzzy Differential Equations

In this paper, we introduce a novel approach based on modified  neural networks  to solve fuzzy differential equations. Using modified  neural network makes that training points should be selected over an open interval  without training the network in the range of first and end points. Therefore, the calculating volume involving computational error is reduced. In fact, the training points depending on the distance selected for training neural network are converted to similar points in the open interval  by using a new approach, then the network is trained in these similar areas. In comparison with existing similar neural networks proposed model provides solutions with high accuracy. The proposed method is illustrated by three numerical examples. Keywords: Fuzzy  differential  equation, Modified  neural  network, Feed-forward  neural  network, BFGS Teqnique, Hyperbolic tangent   function

Numerical Solution of Fuzzy Arbitrary Order Predator-Prey Equations

This paper seeks to investigate the numerical solution of fuzzy arbitrary order predator-prey equations using the Homotopy Perturbation Method (HPM). Fuzziness in the initial conditions is taken to mean convex normalised fuzzy sets viz. triangular fuzzy number. Comparisons are made between crisp solution given by others and fuzzy solution in special cases. The results obtained are depicted in plots and tables to demonstrate the efficacy and powerfulness of the methodology

Numerical Solution of Fuzzy Differential Equations Based on Taylor Series by Using Fuzzy Neural Networks

In this paper a new method based on learning algorithm of Fuzzy neural network and Taylor series has been developed for obtaining numerical solution of fuzzy differential equations.A fuzzy trial solution of the fuzzy initial value problem is written as a sum of two parts.The first part satisfies the fuzzy initial condition,it contains Taylor series and involves no fuzzy adjustable parameters.The second part involves a feed-forward fuzzy neural network containing fuzzy adjustable parameters (the fuzzy weights).Hence by construction,the fuzzy initial condition is satisfied and the fuzzy network is trained to satisfy the fuzzy differential equation . In comparison with existing similar neural networks,the proposed method provides solutions with high accuracy.Finally , we illustrate our approach by two numerical examples

Numerical Solution of Fuzzy Equations with Z-numbers using Neural Networks

In this paper, the uncertainty property is represented by the Z-number as the coefficients of the fuzzy equation. This modification for the fuzzy equation is suitable for nonlinear system modeling with uncertain parameters. We also extend the fuzzy equation into dual type, which is natural for linear-in-parameter nonlinear systems. The solutions of these fuzzy equations are the controllers when the desired references are regarded as the outputs. The existence conditions of the solutions (controllability) are proposed. Two types of neural networks are implemented to approximate solutions of the fuzzy equations with Z-number coefficients

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

Approximate Solution of th-Order Fuzzy Linear Differential Equations

The approximate solution of nth-order fuzzy linear differential equations in which coefficient functions maintainthe sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results

Approximate Solution of n

The approximate solution of nth-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results