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.

scholarly journals Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 914
Author(s):  
Oana Brandibur ◽  
Roberto Garrappa ◽  
Eva Kaslik
Keyword(s):  
Differential Equations ◽  
Detailed Analysis ◽  
Linear Systems ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Limit Case ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Stability

Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results.

scholarly journals Iterative Bernstein splines technique applied to fractional order differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zoltan Satmari
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Initial Value Problems ◽  
Spline Approximation ◽  
Initial Value ◽  
Numerical Examples ◽  
Fractional Order Differential Equations ◽  
Banach’S Fixed Point Theorem

<p style='text-indent:20px;'>In this work we will discuss about an approximation method for initial value problems associated to fractional order differential equations. For this method we will use Bernstein spline approximation in combination with the Banach's Fixed Point Theorem. In order to illustrate our results, some numerical examples will be presented at the end of this article.</p>

scholarly journals Solutions for Impulsive Fractional Differential Equations via Variational Methods

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Peiluan Li ◽  
Hui Wang ◽  
Zheqing Li
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Critical Points ◽  
Existence Results ◽  
Fractional Order Differential Equations ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin and Willem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.

scholarly journals Non-instantaneous impulses in Caputo fractional differential equations

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Real World ◽  
Existence Results ◽  
Advantages And Disadvantages ◽  
Fractional Order Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

AbstractRecent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of the solutions. Both approaches are compared and their advantages and disadvantages are illustrated with examples. Also some existence results are derived.

New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Ali Bhrawy ◽  
Yahia Alhamed ◽  
Dumitru Baleanu ◽  
Abdulrahim Al-Zahrani
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Laguerre Polynomials ◽  
Orthogonal Functions ◽  
Spectral Techniques ◽  
Fractional Differential ◽  
Definition Of ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

AbstractFractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

scholarly journals Application of Said Ball Curve for Solving Fractional Differential-Algebraic Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1926
Author(s):  
Fateme Ghomanjani ◽  
Samad Noeiaghdam
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Convergence Theorem ◽  
Differential Algebraic Equations ◽  
Bézier Curve ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
Curve Method ◽  
Fractional Differential

The aim of this paper is to apply the Said Ball curve (SBC) to find the approximate solution of fractional differential-algebraic equations (FDAEs). This method can be applied to solve various types of fractional order differential equations. Convergence theorem of the method is proved. Some examples are presented to show the efficiency and accuracy of the method. Based on the obtained results, the SBC is more accurate than the Bezier curve method.

Identification of Systems of Arbitrary Real Order: A New Method Based on Systems of Fractional Order Differential Equations and Orthogonal Distance Fitting

2009 ◽  
Author(s):  
Tomas Skovranek ◽  
Vladimir Despotovic
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Main Idea ◽  
Measured Data ◽  
New Method ◽  
Multidimensional Space ◽  
3 Dimensional ◽  
Fractional Order Differential Equations ◽  
Data Points ◽  
Fractional Differential

A new method for identification of systems of arbitrary real order based on numerical solution of systems of nonlinear fractional order differential equations (FODEs) and orthogonal distance fitting is presented. The main idea is to fit experimental or measured data using a solution of a system of fractional differential equations. The parameters of these equations, including the orders of derivatives, are subject to optimization process, where the criterion of optimization is the minimal sum of orthogonal distances of the data points from the fitting line. Once the minimal sum is found, the identified parameters are considered as optimal. The so called orthogonal distance fitting, known also under the names of total least squares or orthogonal regression is naturally used in the fitting criterion, since it is the most suitable tool for fitting lines and surfaces in multidimensional space. The examples illustrating the methods are presented in 2-dimensional and 3-dimensional problems.

Analysis of iterative learning control with high-order internal models for fractional differential equations

2016 ◽  
Vol 24 (6) ◽  
pp. 1145-1161 ◽  
Author(s):  
Shengda Liu ◽  
JinRong Wang
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Internal Models ◽  
High Order ◽  
Time Interval ◽  
Fractional Order Differential Equations ◽  
Design Learning ◽  
Fractional Differential ◽  
P Type

In this paper, we design learning law with high-order internal models for fractional order differential equations to track the varying reference accurately by adopting a few iterations in a finite time interval. We establish sufficient conditions of convergence for the P-type and PD-type updating law for different fractional order differential equations. Finally, we give some numerical examples to demonstrate the validity of the designed method.

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