Solving First-Order Differential Equations of Z-Numbers’ Initial Value Using Radial Basic Function

Leila Qalehe,Mozhdeh Afshar Kermani,Tofigh Allahviranloo

doi:10.1155/2020/5924847

Abstract

In this paper, a method was proposed based on RBF for numerical solution of first-order differential equations with initial values that are valued by Z-numbers. The proposed method consists of two parts. The first part has stated the amount of limitation of the fragmentation solution, while the second part has described the assurance of the first part. The limitation section also has two parts. The first part has included the initial condition of the problem, while the second part has included the RBF network. The confidence interval was also considered as a function based on the probability function, which has calculated the confidence level of the first part (limitation). The RBF network or the radial-base grid network has three distinct layers: the input layer that is the set of elementary nodes (sensory units); the second layer is the hidden layers with high dimensions, in which the output layer that has responded to the network response and the activation patterns used in the input layer. The advantage of using RBF is that the use of this technique does not require sufficient information. It only relies on the domain and the boundary. In an example, we have showed that our proposed approach could approximate the problem with acceptable confidence.

Highlights

radial basis function (RBF) network RBF neuronsBxT(t) 1 − t􏼒e− λAxT(t) − e− λAxT(t)􏼓. e value of BxT(t) is a fuzzy number, which must be computed
Radial basis functions are methods, which are based on the location method for interpolation of discrete data while have a high convergence rate [9]. is method is one of the most widely used methods for approximating the functions in the theory of modern approximation [10]
It is not necessary to produce a regular network in the domain of the problem, which, due to the high computational cost of network production, is the main advantage of these methods to finite difference methods and finite elements, and so on. e geometric feature used in the radial basis function (RBF) approximation is the distance between points. e distance in each space dimension is calculated, resulting in higher

Summary

RBF network RBF neurons

BxT(t) 1 − t􏼒e− λAxT(t) − e− λAxT(t)􏼓. e value of BxT(t) is a fuzzy number, which must be computed. BxT(t) 1 − t􏼒e− λAxT(t) − e− λAxT(t)􏼓. E value of BxT(t) is a fuzzy number, which must be computed. We want to derive the derivative of the function [xT(t)]Z i.e. We derive from the functions of (23), so we have. To optimize the weights wj and wj, we use the minimization of the sum-squared error function:.

Ae min n

Concluding Remarks