scholarly journals Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Iman Malmir
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Operational Matrices ◽  
Chebyshev Wavelets ◽  
Fractional Delay

A spectral framework for the solution of fractional optimal control and variational problems involving Mittag–Leffler nonsingular kernel

2020 ◽  
pp. 107754632097481
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Caputo Fractional Derivative ◽  
Variational Problems ◽  
Computational Technique ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control

A new fractional-order Dickson functions are introduced for solving numerically fractional optimal control and variational problems involving Mittag–Leffler nonsingular kernel. The type of fractional derivative in the proposed problems is the Atangana–Baleanu–Caputo fractional derivative. In the process of the method, we use fractional-order Dickson functions and their properties to provide an accurate computational technique for calculating operational matrices, at first. Then, with the help of operational matrices and the Lagrange multiplier method, these problems are reduced to a system of algebraic equations. At last, to demonstrate the effectiveness of the new method, we enforce the proposed algorithm for several examples.

Generalized fractional order Chebyshev wavelets for solving nonlinear fractional delay-type equations

2019 ◽  
Vol 17 (03) ◽  
pp. 1950014 ◽  
Author(s):  
Amir Saeed ◽  
Umer Saeed
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Operational Matrices ◽  
Chebyshev Wavelets ◽  
Method Of Steps ◽  
Type Differential Equation ◽  
Fractional Delay ◽  
Delay Differential ◽  
Nonlinear Delay ◽  
Quasilinearization Technique

In this paper, we develop the generalized fractional order Chebyshev wavelets (GFCWs) from generalized fractional order of Chebyshev polynomials. The operational matrices for the presented wavelets are constructed and derived. We also proposed a technique by utilizing the GFCWs, the method of steps and quasilinearization technique for solving nonlinear fractional delay-type differential equations. According to the development, the method of step is used to transform the fractional nonlinear delay-type differential equation to a fractional nonlinear non-delay differential equation, and then apply the quasilinearization technique to discretize the obtained nonlinear equation. The GFCW method is utilized in each iteration of quasilinearization method for the improvement of solution. We perform the error analysis for the proposed technique. Procedure of implementation for the present method is also provided. Numerical simulation of some examples will be presented to demonstrate the benefits of computing with the present technique over existing methods in literature.

scholarly journals Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Operational Matrix ◽  
Classical Solutions ◽  
Operational Matrices ◽  
Fractional Differential

We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

scholarly journals Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1866
Author(s):  
Mohamed Jleli ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
A Priori Estimates ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Differential Inequalities ◽  
Structure Of Solutions ◽  
Nonlinear Capacity ◽  
Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

scholarly journals Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 130
Author(s):  
Suphawat Asawasamrit ◽  
Yasintorn Thadang ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Boundary Conditions ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Uniqueness Results ◽  
Nonlinear Alternative

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence of a unique solution is obtained via Banach’s contraction mapping principle, while an existence result is established by using Leray–Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

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