Fuzzy Fractional Differential Equations Research Articles

Fuzzy fractional differential equations with the generalized Atangana-Baleanu fractional derivative

In this paper, we introduce a generalization of Atangana-Baleanu type fractional calculus with respect to the generalized Mittag-Leffler kernel which has been named as the generalized Atangana-Baleanu (GAB) type fractional calculus. Existence and uniqueness results for the initial value problems of fuzzy differential equations involving a GAB fractional derivative in the Caputo sense are established by employing the method of successive approximation and by means of fixed point theorems. To visualize the theoretical results, some examples and numerical simulations are given.

On solutions of fuzzy fractional order complex population dynamical model

AbstractUncertainty always involved in our life activities because we cannot measure a physical quantity accurately. This situation has handled by fuzzy systems and fuzzy differential equations. Recently, fuzzy fractional differential equations got tremendous attention of the researchers of the current century because these operators model the real phenomenon more accurately than integer‐order and fractional‐order operators. Therefore, we investigate the complex population dynamical model under the fuzzy Caputo fractional derivative. Since the Laplace transform has a high convergence rate among all transform methods, so we use fuzzy Laplace transform along with Adomian decomposition to obtain general numerical results for the proposed model. We provide two examples to support the proposed procedure. We simulate the numerical results in terms of graphs for the various fractional‐order and at uncertaintyr ∈ [0, 1].

Neutral fuzzy fractional functional differential equations

In this paper, a class of fuzzy fractional functional differential equations of neutral type is investigated. The existence, uniqueness and iterative formula in the case of linear equations are obtained. Meanwhile, the continuous dependence of solutions on initial data, parameters and functions involved in right hand side of equation is also discussed. Several examples are given to illustrate the applications of our results.

Interpretation of exact solution for fuzzy fractional non-homogeneous differential equation under the Riemann–Liouville sense and its application on the inventory management control problem

This article is presented for manifestation of the non-homogeneous linear fractional differential equation under fuzzy uncertainty. Taking the initial values and the coefficients of the fractional differential equations to be fuzzy numbers, the solutions are categorized into different problems and sub-problems. Based on the three different combinations of the fuzzy initial values and fuzzy coefficients, the theoretical foundation in this article is primarily divided into three major problems. Then, each problem again contains two different sub-problems according to the sign of the coefficients. Finally, the solutions for each of the sub-problems are given in the sense of two different cases of $${}^{\text{RL}}[\left(i\right)-\alpha ]$$ and $${}^{\text{RL}}[\left(ii\right)-\alpha ]$$ differentiability. The strong and the weak solution criteria have been discussed here. Also, the existence and uniqueness criterion for solution of the initial-valued fuzzy fractional differential equation has been established in this article. Finally, an EPQ model of deteriorated items is discussed for its production phase only as a proper validation of proposed theory. The greatness of the consideration of fuzzy fractional differential equation over crisp integer differential equation, fuzzy integer differential equation and crisp fractional differential equation to describe the EPQ model is established in this current article. In this context, a new defuzzification technique is developed to compare the fuzzy and crisp phenomena related to the EPQ model.

Fuzzy fractional-order model of the novel coronavirus

In this paper, a novel coronavirus infection system with a fuzzy fractional differential equation defined in Caputo’s sense is developed. By using the fuzzy Laplace method coupled with Adomian decomposition transform, numerical results are obtained for better understanding of the dynamical structures of the physical behavior of COVID-19. Such behavior on the general properties of RNA in COVID-19 is also investigated for the governing model. The results demonstrate the efficiency of the proposed approach to address the uncertainty condition in the pandemic situation.

Bandwidth optimization of information application system under fine integral method of fuzzy fractional order ordinary differential equations

In order to solve the problems that the network bandwidth of previous information application systems can’t guarantee the quality of big data transmission, resulting in low transmission efficiency and slow data processing, etc., an improved information application system is proposed by optimizing the fine integration method of fuzzy fractional ordinary differential equations and combining with software-defined networking (SDN). Firstly, the fractional-order fuzzy differential equation fine integration method of Pade approximation is derived. Then, based on the error analysis theory, the optimization formula for the adaptive selection of weighted parameter N and expanded item number q is obtained. Combined with SDN, an improved information application system is designed, and the improved algorithm and system are detected by example simulation and performance test. The results show that the numerical accuracy and the computational efficiency of the improved algorithm are higher than that of the improved algorithm. The improved port data merge rate and task completion efficiency under different bandwidth are also significantly higher than the original system. It shows that the information application system proposed in this research can better solve the problems of insufficient bandwidth and low communication speed of traditional systems, and provide a new perspective for bandwidth optimization of big data.

Evaluation of one dimensional fuzzy fractional partial differential equations

This manuscript is related to investigate analytical solutions to some linear fractional partial fuzzy differential equations under certain conditions. For the concerned investigation, we utilize Laplace transform along with some decomposition method to compute series type solution under fuzzy concept. Some examples are solved to demonstrate the proposed method.

Role of fuzzy fractional differential equation in the construction of low carbon economy statistical evaluation system

This study aims to explore the application of fractional order chaotic system (FOCS), based on the T-S fuzzy model, in the design of secure communication (SC) and this SC system’s role in the construction of efficiency evaluation system for dispatching operation of energy saving and power generation under a low carbon economy (LCE). First, the definition of fractional order differential equations and the stability theory of FOCS are discussed. T-S fuzzy control is introduced to analyse the stability of FOCS. Then, based on the adaptive synchronisation theory, the combined FOCS is designed for SC, and the performance of the system constructed here is analysed in a MATLAB environment. Next, an evaluation index system of the dispatching operation effect of energy saving and power generation under LCE is constructed. The confidential communication system constructed here is used to transmit the index data. The particle swarm optimisation (PSO) algorithm is then used to optimise the back propagation neural network (BPNN) model. The BPNN algorithm based on PSO (PSO-BPNN) is obtained, and the evaluation model is constructed. After training this model, it is applied to the evaluation of the dispatching operation effect for energy saving and power generation in 10 provinces in China. The simulation results show that the FOCS based on the T-S fuzzy model can acquire the criterion of system stability. Furthermore, the constructed SC system can effectively undertake the secure transmission of square and sine waves. The performance of the PSO-BPNN model seems to be better than other algorithms. After analysing the data on energy saving and power generation from 10 provinces in China using this model, the comprehensive evaluation value of province 5 is found to be the highest and good (0.802), while that of province 4 is the lowest and general (0.600). Summarising, the FOCS based on T-S fuzzy model constructed here is applied to the design of SC. This can improve the confidentiality of data transmission and reduce the transmission error. Moreover, the BPNN evaluation model based on PSO can effectively achieve the evaluation of the dispatching operation effect of energy saving and power generation under LCE.

The optimization effect of fuzzy factional-order ordinary differential equation in block chain financial cross-border E-commerce payment mode

To solve the problems of low execution efficiency and insufficient security of traditional electronic payment mode, in this study, an improved mode of cross-border e-commerce payment based on fuzzy fractional-order ordinary differential equation and secure electronic transaction protocol is proposed. Firstly, the competition model of cross-border e-commerce payment platform is constructed using fuzzy fractional order ordinary differential equation. Based on the theory of Hopf bifurcation, the stability of the model is analyzed. Then, the cross-border e-commerce improved payment mode is built based on the secure electronic transaction protocol. Finally, the efficiency and security performance of the improved payment mode are compared with that of the traditional payment mode with face through simulation experiments. The results show that the improved mode takes less time in encryption and decryption than the traditional mode; the time spent on the security payment of the improved mode remains around 1.2 s, while that of the traditional mode is around 2.7 s; the improved mode security payment takes significantly less time than the traditional mode, which indicates that the payment mode proposed in this research based on fuzzy fractional order ordinary differential equation is not only more secure than the traditional payment mode, but also can effectively improve the payment speed of users and greatly enhance their shopping experience.

Automatic cyber security risk assessment based on fuzzy fractional ordinary differential equations

Real-time network security risk detection is of great significance to the research of dynamic network security, and it is one of the current research hotspots of network security. Based on the idea of artificial immune, a dynamic network intrusion detection and prediction model based on fuzzy fractional ordinary differential equations is proposed. The uniqueness of the solution is studied in the square integrable equation space using the principle of compression operator, and then the second type of solution is proposed. The differential-type piecewise Taylor series expansion method of the linear Fredholm integral equation is used to obtain the approximate solution expressions and perform convergence and error estimation. The process of network attack detection, antibody concentration calculation method and risk prediction process based on time series autoregressive moving average model are given. The experimental results show that the model can quantitatively analyze the current security situation of the network in real time and make predictions about the risks faced by the network. The prediction effect of catastrophe network risk is better than the GM (1, 1) model, and it is close to the actual risk situation, with high prediction accuracy.

New fractional signal smoothing equations with short memory and variable order

In this paper, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application.

AN ALGORITHM FOR THE SOLUTION OF FUZZY FRACTIONAL DIFFERENTIAL EQUATION

In this paper the variational iteration method is used to compute the solution for the fuzzy fractional order partial differential equations with fuzzy coefficients, fuzzy initial values and fuzzy forcing functions.We propose an algorithm based on $\alpha$-cut of a fuzzy set. Finally we present some examples by usingour proposed algorithm.

Existence of unique solution to nonlinear mixed Volterra Fredholm-Hammerstein integral equations in complex-valued fuzzy metric spaces

The purpose of this article is to investigate the existence of unique solution for the following mixed nonlinear Volterra Fredholm-Hammerstein integral equation considered in complex plane; (0.1) ξ ( τ ) = g ( t ) + ρ ∫ 0 τ K 1 ( τ , ℘ ) ϝ 1 ( ℘ , ξ ( ℘ ) ) d ℘ + ϱ ∫ 0 1 K 2 ( τ , ℘ ) ϝ 2 ( ℘ , ξ ( ℘ ) ) d ℘ , such that ξ = ξ 1 + ξ 2 , ξ 1 , ξ 2 ∈ ( C ( [ 0 , 1 ] ) , R ) g = g 1 + g 2 , g l : [ 0 , 1 ] → R , l = 1 , 2 , ϝ l ( ℘ , ξ ( ℘ ) ) = ϝ l 1 * ( ℘ , ξ 1 * ) + i ϝ l 2 * ( ℘ , ξ 2 * ) , ϝ lj * : [ 0 , 1 ] × R → R for l , j = 1 , 2 , and ξ 1 * , ξ 2 * ∈ ( C ( [ 0 , 1 ] ) , R ) K l ( t , ℘ ) = K l 1 * ( t , ℘ ) + iK l 2 * ( t , ℘ ) , for l , j = 1 , 2 and K lj * : [ 0 , 1 ] 2 → R , where ρ and ϱ are constants, g (t), the kernels Kl (τ, ℘) and the nonlinear functions ϝ1 (℘, ξ (℘)), ϝ 2 (℘, ξ (℘)) are continuous functions on the interval 0 ≤ τ ≤ 1. In this direction we apply fixed point results for self mappings with the concept of (ψ, ϕ) contractive condition in the setting of complex-valued fuzzy metric spaces. This study will be useful in the development of the theory of fuzzy fractional differential equations in a more general setting.

On the initial value problem for fuzzy differential equations of non-integer order $$\alpha \in (1,2)$$

The purpose of this paper is to present some basic theories of an initial value problem of fuzzy fractional differential equations involving the Caputo-fuzzy-type concept of fractional derivative in the case of the order $$\alpha \in (1,2).$$ The existence and uniqueness results of the solution for the given problem are presented. Finally, some examples are given to illustrate our main results.

Invariant solutions of hyperbolic fuzzy fractional differential equations

We consider the hyperbolic type fuzzy fractional differential equation and derive the second-order fuzzy fractional differential equation using scaling transformation. We present a theoretical and a numerical method to find the invariant solutions of such equations. Also, we prove the existence and uniqueness results using Banach fixed point theorem. Numerical solutions are approximated using finite difference method. Finally, numerical examples are given to illustrate the obtained results.

Spline collocation methods for systems of fuzzy fractional differential equations

Abstract In this paper, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application.

Solution of boundary value problem for fuzzy fractional differential equations by using differential transform method (DTM)

In this study, we implement a well-known transformation technique differential transform method (DTM), to the area of fractional differential equations, in this Paper, the solution to fuzzy fractional Initial Value Problem (FFIVP) under copout-type fuzzy fractional derivatives are define based on Hukuhara difference and strongly gene fuzzy differentiability. Also numerical examples are carried out for various types of problems including the Bagley-Torvik, Ricatti and composite fractional oscillation equations for the application of the method.

On the fuzzy fractional differential equation with interval Atangana–Baleanu fractional derivative approach

Abstract The fuzzy systems with interval approach use an infinite valued parameter in the range of [0,1] as a confidence degree of belief. This parameter makes more complicity but plays the main role in creating the fuzzy solution of the fuzzy systems. In solving process of the model, the Atangana–Baleanu derivative in the fractional case of differential equations has a memory to use all the previous information. Therefore this is as a key point and advantage of using this derivative to reduce the complicity of numerical results in comparison with other known derivatives. In this paper, first, the ABC fractional derivative on fuzzy set-valued functions in parametric interval form is defined. Then it is applied for proving the existence and uniqueness of the solution of fuzzy fractional differential equation with ABC fractional derivative. In general, it is shown that the last interval model is as a coupled system of nonlinear equations. To solve the final system an efficient numerical method called ABC-PI is used. For more illustration, several examples are solved numerically and analyzed by the figures.

Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation

In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order 0<α<1 is considered. The uniqueness of solution is established by using the contraction mapping principle and the existence of solution is obtained by Schauder fixed point theorem.

Existence, uniqueness and Eq-Ulam type stability of fuzzy fractional differential equations with parameters1

Existence, uniqueness and Eq-Ulam type stability of fuzzy fractional differential equations with parameters1