First Order Linear Homogeneous Ordinary Differential Equation in Fuzzy Environment Based On Laplace Transform

In this paper the First Order Linear Ordinary Differential Equations (FOLODE) are described in fuzzy environment. Here coefficients and /or initial condition of FOLODE are taken as Generalized Triangular Fuzzy Numbers (GTFNs).The solution procedure of the FOLODE is developed by Laplace transform. It is illustrated by numerical examples. Finally imprecise bank account problem and concentration of drug in blood problem are described

Fuzzy modelling on the depletion of forest biomass and forest-dependent wildlife population

A project report submitted in Franklin Open Volume 4, September 2023,This paper presents a system of non-linear differential equations describing the depletion of forest biomass and forest-dependent wildlife population caused by human population and its associated activities. The model incorporates the imprecise nature of the parameters, which are treated as triangular fuzzy numbers to reflect the inherent uncertainty. We utilised cut to transform these imprecise parameters into intervals. Subsequently, employing the principles of interval mathematics, we effectively converted the related differential equation into a pair of distinct differential equations. By leveraging the signed distance of the fuzzy numbers, we further simplified the equations, resulting in a single differential equation, which led to the formulation of a defuzzified model. The existence of equilibrium points with their stability behaviour is presented. Furthermore, the existence of trans-critical bifurcation is analysed. Through numerical simulations, we observe significant differences between the solutions of system in crisp and fuzzy environments. These findings highlight the importance of using fuzzy models to accurately represent the dynamics of complex natural systems. Consequently, we conclude that fuzzy models provide a trustworthy representation of the dynamics of complex natural systems

Uncertain non near system control with Fuzzy Differential Equations and Z-numbers

In this paper, the solutions of fuzzy differential equations (FDEs) are estimated by using two types of Bernstein neural networks. Here, the uncertainties are in the form of Z numbers. Firstly, we transform the FDE to four ordinary differential equations (ODEs) at par with Hukuhara differentiability. After that we develop neural models having the structure of ODEs. By using modified backpropagation technique for Z number variables, the training of neural networks are carried out. The results of the simulation illustrate that these innovative models, Bernstein neural networks, are efficient to approximate the solutions of FDEs which are on the basis of Z-numbers

Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better