Fuzzy Initial Conditions Research Articles

On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

<abstract><p>This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional ($ \mathcal{GPF} $) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-$ \mathcal{GPF} $ operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations $ (\mathcal{FVFIE}s) $ via generalized fuzzy Hilfer-$ \mathcal{GPF} $ Hukuhara differentiability ($ \mathcal{HD} $) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy $ \mathcal{FVFIE}s $ which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.</p></abstract>

Comparison for the Approximate Solution of the Second-Order Fuzzy Nonlinear Differential Equation with Fuzzy Initial Conditions

This research focuses on the approximate solutions of second-order fuzzy differential equations with fuzzy initial condition with two different methods depending on the properties of the fuzzy set theory. The methods in this research based on the Optimum homotopy asymptotic method (OHAM) and homotopy analysis method (HAM) are used implemented and analyzed to obtain the approximate solution of second-order nonlinear fuzzy differential equation. The concept of topology homotopy is used in both methods to produce a convergent series solution for the propped problem. Nevertheless, in contrast to other destructive approaches, these methods do not rely upon tiny or large parameters. This way we can easily monitor the convergence of approximation series. Furthermore, these techniques do not require any discretization and linearization relative with numerical methods and thus decrease calculations more that can solve high order problems without reducing it into a first-order system of equations. The obtained results of the proposed problem are presented, followed by a comparative study of the two implemented methods. The use of the methods investigated and the validity and applicability of the methods in the fuzzy domain are illustrated by a numerical example. Finally, the convergence and accuracy of the proposed methods of the provided example are presented through the error estimates between the exact solutions displayed in the form of tables and figures.

Solution and stability analysis of non-homogeneous difference equation followed by real life application in fuzzy environment

The study fuzzy difference equation becomes very important as huge numbers of real-life problems in the field of engineering; ecology social science, etc. can be mathematically represented in the form of difference equation where impreciseness is inherently involved. In this paper, we have focused on the solution techniques of non-homogeneous fuzzy linear difference equation with different cases involving fuzzy initial conditions, fuzzy forcing function and fuzzy coefficient. The idea of fuzzy equilibrium point is introduced and its stability analysis has been performed. The whole theoretical work is followed by real-life applications which show the impact of fuzzy concepts in mathematical modelling for better understanding the behaviour of the system in an elegant manner.

Analysis for fractional-order predator-prey model with uncertainty.

Here, the authors analyse the fractional‐order predator–prey model with uncertainty, due to the vast applications in various ecological systems. The most of the ecological model do not have exact analytic solution, so they proposed a numerical technique for an approximate solution. In the proposed method, they have implemented the higher order term into the fractional Euler method to enhance the precise solution. Further, the present attempt is aimed to discuss the solutions of the FPPM with uncertainty (fuzzy) initial conditions. The initial conditions of the predator–prey model were taken as fuzzy initial conditions due to the fact that the ecological model highly depends on uncertain parameters such as growth/decay rate, climatic conditions, and chemical reactions. Finally, the numerical example manifest that the proposed method is authentic, applicable, easy to use from a computational viewpoint and the acquired outcomes are balanced with the existing method (HPM), which shows the efficiency of the proposed method.

On discrete time Beverton-Holt population model with fuzzy environment

In this work, dynamical behaviors of discrete time Beverton-Holt population model with fuzzy parameters are studied. It provides a flexible model to fit population data. For three different fuzzy parameters and fuzzy initial conditions, according to a generalization of division (g-division) of fuzzy number, it can represent dynamical behaviors including boundedness, global asymptotical stability and persistence of positive solution. Finally, two examples are given to demonstrate the effectiveness of the results obtained.

TRANSPORT EQUATION WITH FUZZY DATA

In this paper, we use the generalized differentiability concept tostudy the fuzzy transport equation. We consider transport equationin the homogeneous and non-homogeneous cases with fuzzy initialcondition. We also present the solution when speed parameter is a fuzzynumber. Our method is based on the construction of the solutionsby employing Zadeh's extension principle.

Solving shallow water equations with crisp and uncertain initial conditions

Purpose This paper aims to deal with the application of variational iteration method and homotopy perturbation method (HPM) for solving one dimensional shallow water equations with crisp and fuzzy uncertain initial conditions. Design/methodology/approach Firstly, the study solved shallow water equations using variational iteration method and HPM with constant basin depth and crisp initial conditions. Further, the study considered uncertain initial conditions in terms of fuzzy numbers, which leads the governing equations to fuzzy shallow water equations. Then using cut and parametric concepts the study converts fuzzy shallow water equations to crisp form. Then, HPM has been used to solve the fuzzy shallow water equations. Findings Results obtained by both methods HPM and variational iteration method are compared graphically in crisp case. Solution of fuzzy shallow water equations by HPM are presented in the form triangular fuzzy number plots. Originality/value Shallow water equations with crisp and fuzzy initial conditions have been solved.

Fuzzy solution of homogeneous heat equation having solution in Fourier series form

While solving practical problems, we often come across situations where the system involves fuzziness. The mathematical models resulting in partial differential equations, involve fuzzy parameters and variables. In available literature, methods are presented mainly for solving non-homogeneous fuzzy partial differential equations (see Allahviranloo in Comput Methods Appl Math 2(3):233–242, 2002; Allahviranloo and Taheri in Int J Contemp Math Sci 4(3):105–114, 2009; Allahviranloo and Afshar in Iran J Fuzzy Syst 7(3):33–50, 2010; Allahviranloo et al. in Appl Soft Comput 11:2186–2192, 2011). We present a method to find the solution of homogeneous fuzzy heat equations with fuzzy Dirichlet boundary conditions. We consider the fuzziness in zero in the homogeneous equation as well as in the boundary conditions. The initial conditions are also in fuzzy form. Further, we study the solution of fuzzy heat equation when the fuzzy initial conditions are represent as a Fourier series.

Fuzzy picard’s method for derivatives of second order

In this paper we proposed an iterative scheme for the solution of second order fuzzy differential equations with fuzzy initial conditions using fuzzy picard method. The convergence is also discussed under generalized H – differentiability. Finally this method evidenced is illustrated by solving some examples to show its efficiency

Exact Solution of Semi-linear Fuzzy System

In this paper we consider a semi-linear dynamical system with fuzzy initial condition. We discuss the results regarding the existence of the solution and obtain the best possible solution for such systems. We give a real life supportive illustration of population model, justify the need for fuzzy setup for the problem, and discuss the solution for it.

Solution of second order linear fuzzy ordinary differential equation by Lagrange multiplier method with application in mechanics

In this paper the solution of second order linear fuzzy ordinary differential equation is described. The solution procedure is described by Lagrange multiplier method and extension principle method. Further two mechanics problem with fuzzy initial condition are briefly illustrated. The solutions are defuzzified by a well min of α-cut defuzzification method.

Solution of Fuzzy Fractional Integro-Differential Equations Using A Domian Decomposition Method

In this paper, we present Adomian decomposition method to solve linear fuzzy fractional integro-differential equation with fuzzy initial conditions. Results are compared with the results obtained using Fuzzy Laplace transform method.

Analysis of Fractional Schrödinger Equation Occurring in Quantum Mechanics

In this paper, we present Adomian decomposition method to solve linear fuzzy fractional integro-differential equation with fuzzy initial conditions. Results are compared with the results obtained using Fuzzy Laplace transform method.

Solution of first order system of differential equation in fuzzy environment and its application

In this paper, we solve a system of differential equation of first order in fuzzy environment using generalised Hukuhara derivative approach. Four different cases are discussed for the said differential equation: 1) coefficient is positive crisp number and initial condition is fuzzy number; 2) coefficient is negative crisp number and initial condition is fuzzy number; 3) coefficient is fuzzy number and initial condition is fuzzy number; 4) coefficient is interval number and initial condition is fuzzy number. Finally, we apply the results in Lanchaster combat model. The solution that comes in the application are defuzzified by removal area method.

Solving fuzzy differential equations by using Picard method

In this paper, The Picard method is proposed to solve the system of first-order fuzzy differential equations $(FDEs)$ with fuzzy initial conditions under generalized $H$-differentiability. Theexistence and uniqueness of the solution and convergence of theproposed method are proved in details. Finally, the method is illustrated by solving some examples.

A dye removal model with a fuzzy initial condition

A fuzzy model for dye removal is suggested to study a transport model of the direct textile industry wastewater, and the variational iteration method is adopted to obtain its analytical solutions. The concentration depends upon not only the parameters in the governing equation, but also the pair of the initial condition.

Fuzzy neural network-based system identification of multi-storey shear buildings

This paper uses fuzzy neural network modelling for the identification of structural parameters of multi-storey shear buildings. First the identification has been done using response of the structure subject to ambient vibration with fuzzy initial condition. Then forced vibration with horizontal displacement in fuzzified form has been used to investigate the identification procedure. The model has been developed to handle the data in fuzzified form for multi-storey shear structure, and the procedure is tested for the identification of the stiffness parameters of simple example problems using the prior values of the design parameters.

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 .

Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations

The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 < α ≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.

A note on “A class of linear differential dynamical systems with fuzzy initial condition”

In this note, we show that the proposed approach in J. Xu et al. (2007) [1] fails to obtain the stable solutions to the stable linear dynamical systems with fuzzy initial conditions. As it will be explained, this shortcoming is caused by neglecting to move the stability property of the fuzzy system to the quasi-level-wise system. Moreover, to handle this, another approach is proposed similar to the previous one for the stable and unstable systems.