Qualitative analysis of fractional relaxation equation and coupled system with Ψ-Caputo fractional derivative in Banach spaces

Abstract This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative

AIMS Mathematics ◽

10.3934/math.2021301 ◽

2021 ◽

Vol 6 (5) ◽

pp. 5088-5105

Author(s):

Tamer Nabil ◽

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Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Caputo Fractional Derivative ◽

Coupled System ◽

Nonlinear Coupled System ◽

Fractional Differential

Analysis of Coupled System of Implicit Fractional Differential Equations Involving Katugampola–Caputo Fractional Derivative

Complexity ◽

10.1155/2020/9285686 ◽

2020 ◽

Vol 2020 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Manzoor Ahmad ◽

Jiqiang Jiang ◽

Akbar Zada ◽

Syed Omar Shah ◽

Jiafa Xu

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Sufficient Conditions ◽

Coupled System ◽

Uniqueness Of Solutions ◽

Fractional Differential System ◽

Fractional Differential ◽

Implicit Fractional Differential Equations

In this paper, we study the existence and uniqueness of solutions to implicit the coupled fractional differential system with the Katugampola–Caputo fractional derivative. Different fixed-point theorems are used to acquire the required results. Moreover, we derive some sufficient conditions to guarantee that the solutions to our considered system are Hyers–Ulam stable. We also provided an example that explains our results.

A numerical study of fractional relaxation–oscillation equations involving $$\psi $$ ψ -Caputo fractional derivative

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-018-0590-0 ◽

2018 ◽

Vol 113 (3) ◽

pp. 1873-1891 ◽

Cited By ~ 5

Author(s):

Ricardo Almeida ◽

Mohamed Jleli ◽

Bessem Samet

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Numerical Study ◽

Relaxation Oscillation ◽

Fractional Relaxation

On Complete Monotonicity of Solution to the Fractional Relaxation Equation with the nth Level Fractional Derivative

Mathematics ◽

10.3390/math8091561 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1561 ◽

Cited By ~ 1

Author(s):

Yuri Luchko

Keyword(s):

Fractional Derivative ◽

Initial Value Problems ◽

Monotone Functions ◽

Initial Value ◽

Relaxation Equation ◽

Explicit Formulas ◽

Linear Combinations ◽

Completely Monotone Functions ◽

Completely Monotone ◽

Fractional Relaxation

In this paper, we first deduce the explicit formulas for the projector of the nth level fractional derivative and for its Laplace transform. Afterwards, the fractional relaxation equation with the nth level fractional derivative is discussed. It turns out that, under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag–Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the second level derivative.

Cauchy problem with $\psi $--Caputo fractional derivative in Banach spaces

Advances in the Theory of Nonlinear Analysis and its Application ◽

10.31197/atnaa.706292 ◽

2020 ◽

Author(s):

Choukri DERBAZİ ◽

Zidane BAİTİCHE ◽

Mouﬀak BENCHOHRA

Keyword(s):

Cauchy Problem ◽

Fractional Derivative ◽

Banach Spaces ◽

Caputo Fractional Derivative

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

Mathematics ◽

10.3390/math9090979 ◽

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.

Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽

10.3390/e23020211 ◽

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.

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

Mathematics ◽

10.3390/math9161866 ◽

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.

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 ◽

10.3390/axioms10030130 ◽

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.