Abstract
In this article, we propose a new approach to solve the hybrid fuzzy differential equations based on the feed-forward neural networks. We first replace it by a system of ordinary differential equations. A trial solution of this system involves two parts. The first part satisfies the initial condition and contains no adjustable parameters; however, the second part involves a feed-forward neural network containing adjustable parameters (the weights). This method shows that using neural networks provides solutions with good generalization and the high accuracy.
Highlights
IntroductionFuzzy differential equations (FDEs) are significant for studying and solving large proportions of problems in many topics of applied mathematics, in relation to physics, geography, medicine, biology, control chaotic systems, bioinformatics and computational biology, synchronize hyperchaotic systems, economics and finance, and so on.[1,2,3] In lots of applications, some of the parameters are represented by fuzzy numbers rather than the crisp numbers, and it is essential to develop mathematical models and numerical procedures which would have appropriately treated to general FDEs
Some of the parameters are represented by fuzzy numbers rather than the crisp numbers, and it is essential to develop mathematical models and numerical procedures which would have appropriately treated to general Fuzzy differential equations (FDEs)
These control systems which are capable of controlling complex systems have discrete event dynamics as well as continuous time dynamics that can be modeled by hybrid systems
Summary
Fuzzy differential equations (FDEs) are significant for studying and solving large proportions of problems in many topics of applied mathematics, in relation to physics, geography, medicine, biology, control chaotic systems, bioinformatics and computational biology, synchronize hyperchaotic systems, economics and finance, and so on.[1,2,3] In lots of applications, some of the parameters are represented by fuzzy numbers rather than the crisp numbers, and it is essential to develop mathematical models and numerical procedures which would have appropriately treated to general FDEs. In this article, using the characterization theorem, we generalize a fourth-order Runge–Kutta method that originally presented to solve the HFDEs. That is, we substitute the original initial value problem with two parametric hybrid ordinary differential systems. The function approximation capability of feedforward neural networks is used by expressing the trial solutions for system (7) as the sum of two terms (see equation (9)). The second term involves a feed-forward neural network to be trained, so satisfies the differential equations Since it is known as a multilayer perceptron with one hidden layer which can approximate any function to arbitrary accuracy, the multilayer perceptron is used as the type of the network architecture. The hybrid fuzzy initial problem (equation (14)) is equivalent to the following system of FIVPs. In equation (14), y(x) + m(x)lk(yxk ) is a continuous function of x, y, and lk(yxk ). It is deduced that the results are very close to the exact solutions which confirm the validity and feasibility of this method
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