scholarly journals A Neural Computational Intelligence Method Based on Legendre Polynomials for Fuzzy Fractional Order Differential Equation

2016 ◽  
Vol 12 (2) ◽  
pp. 67-82
Author(s):  
N. A. Khan ◽  
A. Shaikh ◽  
M. A. Zahoor Raja ◽  
S. Khan
Keyword(s):  
Simulated Annealing ◽  
Fractional Order ◽  
Computational Intelligence ◽  
Legendre Polynomials ◽  
Order Differential Equation ◽  
Activation Function ◽  
Fractional Order Differential Equation ◽  
Initial Value ◽  
Fractional Order Differential Equations ◽  
Taylor Series Approximation

AbstractIn this article, Legendre simulated annealing, neural network (LSANN) is designed for fuzzy fractional order differential equations, which is employed on fractional fuzzy initial value problem (FFIVP) with triangular condition. Here, Legendre polynomials are used to modify the structure of neural networks with a Taylor series approximation of the tangent hyperbolic as activation function while the network adaptive coefficients are trained in the procedure of simulated annealing to optimize the residual error. The computational results are depicted in terms of numerical values to compare them with previous results.

A Numerical Algorithm to Initial Value Problem of Caputo Fractional-Order Differential Equation

2015 ◽  
Author(s):  
Lu Bai ◽  
Dingyü Xue
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Algorithm ◽  
Initial Value Problem ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Initial Value

A numerical algorithm is presented to solve the initial value problem of linear and nonlinear Caputo fractional-order differential equations. Firstly, nonzero initial value problem should be transformed into zero initial value problem. Error analysis has been done to polynomial algorithm, the reason has been found why the calculation error of the algorithm is large. A new algorithm called exponential function algorithm is proposed based on the analysis. The obtained fractional-order differential equation is transformed into difference equation. If the differential equation is linear, the obtained difference equation is explicit, the numerical solution can be solved directly; otherwise, the obtained difference equation is implicit, the predictor of the numerical solution can be obtained with extrapolation algorithm, substitute the predictor into the equation, the corrector can be solved. Error analysis has been done to the new algorithm, the algorithm is of first order.

scholarly journals A Refinement of Quasilinearization Method for Caputo's Sense Fractional-Order Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Coşkun Yakar ◽  
Ali Yakar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Unique Solution ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Quasilinearization Method ◽  
Fractional Order Differential Equation ◽  
Fractional Order Differential Equations ◽  
Quasilinearization Technique

The method of the quasilinearization technique in Caputo's sense fractional-order differential equation is applied to obtain lower and upper sequences in terms of the solutions of linear Caputo's sense fractional-order differential equations. It is also shown that these sequences converge to the unique solution of the nonlinear Caputo's sense fractional-order differential equation uniformly and semiquadratically with less restrictive assumptions.

scholarly journals Existence and Uniqueness of Abstract Stochastic Fractional-Order Differential Equation

2019 ◽  
Vol 16 ◽  
pp. 8280-8287
Author(s):  
Mahmoud Mohammed Mostafa El-Borai ◽  
A. Tarek S.A.
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Stochastic Analysis ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Analysis Techniques ◽  
Fractional Order Differential Equations

In this paper, the existence and uniqueness about the solution for a class of abstract stochastic fractional-order differential equations                                           where  in and  are given functions, are investigated, where the fractional derivative is described in Caputo sense. The fractional calculus, stochastic analysis techniques and the standard $Picard's$ iteration method are used to obtain the required.

A Series of New Chaotic Attractors via Switched Linear Integer Order and Fractional Order Differential Equations

2015 ◽  
Vol 25 (01) ◽  
pp. 1550008 ◽  
Author(s):  
Fei Xu ◽  
Ross Cressman ◽  
Xiao-Bao Shu ◽  
Xinzhi Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
Fractional Order Differential Equation ◽  
Integer Order ◽  
Fractional Order Differential Equations ◽  
Dynamical Behaviors

This article investigates the design of a series of new chaotic attractors. A switching control with different switching surfaces is designed to link two systems of linear integer order differential equations. Under such control, the linked systems have rich dynamical behaviors such as chaos. We also investigate the dynamical behaviors of the corresponding linear fractional order differential equation systems with switching controls. It is shown that such fractional order systems have chaotic behaviors as well.

scholarly journals Left- and Right-Shifted Fractional Legendre Functions with an Application for Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Haidong Qu ◽  
Xiaopeng Yang ◽  
Zihang She
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Legendre Polynomials ◽  
Legendre Functions ◽  
Initial Value ◽  
Orthogonal Functions ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
Left And Right ◽  
The Right

Two new orthogonal functions named the left- and the right-shifted fractional-order Legendre polynomials (SFLPs) are proposed. Several useful formulas for the SFLPs are directly generalized from the classic Legendre polynomials. The left and right fractional differential expressions in Caputo sense of the SFLPs are derived. As an application, it is effective for solving the fractional-order differential equations with the initial value problem by using the SFLP tau method.

A Numerical Algorithm to Initial Value Problem of Linear Caputo Fractional-Order Differential Equation

2015 ◽  
Author(s):  
Lu Bai ◽  
Dingyü Xue
Keyword(s):  
Differential Equation ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Algorithm ◽  
Taylor Series ◽  
Initial Value Problem ◽  
Order Differential Equation ◽  
Closed Form Solution ◽  
Fractional Order Differential Equation ◽  
Initial Value

A numerical algorithm is presented to solve the initial value problem of linear Caputo fractional-order differential equations. Error analysis has been done to Taylor series algorithm, the reason has been found why the error of the algorithm is large, the condition of using Taylor series algorithm is presented. A new algorithm called exponential function algorithm is proposed based on the analysis. Nonzero initial value problem could be transformed into zero initial value problem. The obtained fractional-order differential equation is transformed into difference equation, the numerical solution can be found with closed form solution formula. The error of the numerical solution can be modified with prediction-correction algorithm.

scholarly journals Fractional-Derivative Approximation of Relaxation in Complex Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Kin M. Li ◽  
Mihir Sen ◽  
Arturo Pacheco-Vega
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
System Identification ◽  
Complex Systems ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Initial Conditions ◽  
Time Dependent ◽  
Fractional Order Differential Equation

In this paper, we present a system identification (SI) procedure that enables building linear time-dependent fractional-order differential equation (FDE) models able to accurately describe time-dependent behavior of complex systems. The parameters in the models are the order of the equation, the coefficients in it, and, when necessary, the initial conditions. The Caputo definition of the fractional derivative, and the Mittag-Leffler function, is used to obtain the corresponding solutions. Since the set of parameters for the model and its initial conditions are nonunique, and there are small but significant differences in the predictions from the possible models thus obtained, the SI operation is carried out via global regression of an error-cost function by a simulated annealing optimization algorithm. The SI approach is assessed by considering previously published experimental data from a shell-and-tube heat exchanger and a recently constructed multiroom building test bed. The results show that the proposed model is reliable within the interpolation domain but cannot be used with confidence for predictions outside this region. However, the proposed system identification methodology is robust and can be used to derive accurate and compact models from experimental data. In addition, given a functional form of a fractional-order differential equation model, as new data become available, the SI technique can be used to expand the region of reliability of the resulting model.

scholarly journals THE EXISTENCE OF THE EXTREMAL SOLUTION FOR THE BOUNDARY VALUE PROBLEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH CAUSAL OPERATOR

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040025
Author(s):  
JINGFEI JIANG ◽  
JUAN L. G. GUIRAO ◽  
TAREQ SAEED
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Existence Results ◽  
Extremal Solution ◽  
Variable Order ◽  
Fractional Order Differential Equation ◽  
Causal Operator ◽  
Linear Differential

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.

scholarly journals Boundary Value Problems of Fractional Order Differential Equation with Integral Boundary Conditions and Not Instantaneous Impulses

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Peiluan Li ◽  
Changjin Xu
Keyword(s):  
Boundary Conditions ◽  
Fractional Order ◽  
Fixed Point Theorems ◽  
Order Differential Equation ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Fractional Order Differential Equation ◽  
Uniqueness Of Solutions ◽  
Integral Boundary

We investigate the existence of mild solutions for fractional order differential equations with integral boundary conditions and not instantaneous impulses. By some fixed-point theorems, we establish sufficient conditions for the existence and uniqueness of solutions. Finally, two interesting examples are given to illustrate our theory results.

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