Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

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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.

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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.

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Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative

Mathematics ◽

10.3390/math7070630 ◽

2019 ◽

Vol 7 (7) ◽

pp. 630

Author(s):

Dandan Yang ◽

Chuanzhi Bai

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Differential Inclusions ◽

Existence Of Solutions ◽

Sufficient Conditions ◽

Fractional Differential Inclusions ◽

Nonlinear Alternative ◽

Contractive Map ◽

Fractional Differential ◽

Definition Of

In this paper, we investigate the existence of solutions for a class of anti-periodic fractional differential inclusions with ψ -Riesz-Caputo fractional derivative. A new definition of ψ -Riesz-Caputo fractional derivative of order α is proposed. By means of Contractive map theorem and nonlinear alternative for Kakutani maps, sufficient conditions for the existence of solutions to the fractional differential inclusions are given. We present two examples to illustrate our main results.

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Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Mathematical Problems in Engineering ◽

10.1155/2021/8608447 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

Khalid Hattaf

Keyword(s):

Stability Analysis ◽

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Fractional Derivatives ◽

Direct Method ◽

Lyapunov Direct Method ◽

Science And Engineering ◽

Fractional Differential ◽

The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

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Correction: FRACTIONAL ORDER BARBALAT’S LEMMA AND ITS APPLICATIONS IN THE STABILITY OF FRACTIONAL ORDER NONLINEAR SYSTEMS

Mathematical Modelling and Analysis ◽

10.3846/13926292.2017.1329755 ◽

2017 ◽

Vol 22 (4) ◽

pp. 503-513 ◽

Cited By ~ 5

Author(s):

Fei Wang ◽

Yongqing Yang

Keyword(s):

Nonlinear Systems ◽

Fractional Derivative ◽

Fractional Order ◽

Caputo Fractional Derivative ◽

Barbalat’S Lemma ◽

Liouville Derivative ◽

Liouville Fractional Derivative ◽

The Stability ◽

The Relationship ◽

Theoretical Results

This paper investigates fractional order Barbalat’s lemma and its applications for the stability of fractional order nonlinear systems with Caputo fractional derivative at first. Then, based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative, fractional order Barbalat’s lemma with Riemann-Liouville derivative is derived. Furthermore, according to these results, a set of new formulations of Lyapunov-like lemmas for fractional order nonlinear systems are established. Finally, an example is presented to verify the theoretical results in this paper.

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The Dirihlet problem for the fractional Poisson’s equation with Caputo derivatives: A finite difference approximation and a numerical solution

Thermal Science ◽

10.2298/tsci110421076b ◽

2012 ◽

Vol 16 (2) ◽

pp. 385-394 ◽

Cited By ~ 3

Author(s):

V.D. Beibalaev ◽

R.P. Meilanov

Keyword(s):

Monte Carlo ◽

Finite Difference ◽

Fractional Derivative ◽

Monte Carlo Methods ◽

Caputo Fractional Derivative ◽

Fractional Derivatives ◽

Finite Difference Approximation ◽

Difference Approximation ◽

Difference Problem ◽

The Stability

A finite difference approximation for the Caputo fractional derivative of the 4-?, 1 < ? ? 2 order has been developed. A difference schemes for solving the Dirihlet?s problem of the Poisson?s equation with fractional derivatives has been applied and solved. Both the stability of difference problem in its right-side part and the convergence have been proved. A numerical example was developed by applying both the Liebman and the Monte-Carlo methods.

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Stability of solutions for generalized fractional differential problems by applying significant inequality estimates

Advances in Difference Equations ◽

10.1186/s13662-021-03533-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Mohammed D. Kassim ◽

Thabet Abdeljawad ◽

Saeed M. Ali ◽

Mohammed S. Abdo

Keyword(s):

Fractional Derivative ◽

Power Function ◽

Sufficient Conditions ◽

Research Paper ◽

Stability Of Solutions ◽

Initial Value ◽

Regularization Techniques ◽

Fractional Differential ◽

The Stability ◽

Generalized Fractional Derivative

AbstractIn this research paper, we intend to study the stability of solutions of some nonlinear initial value fractional differential problems. These equations are studied within the generalized fractional derivative of various orders. In order to study the solutions’ decay to zero as a power function, we establish sufficient conditions on the nonlinear terms. To this end, some versions of inequalities are combined and generalized via the so-called Bihari inequality. Moreover, we employ some properties of the generalized fractional derivative and appropriate regularization techniques. Finally, the paper involves examples to affirm the validity of the results.

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Initialization Issues of the Caputo Fractional Derivative

Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2005-84348 ◽

2005 ◽

Cited By ~ 2

Author(s):

B. N. Narahari Achar ◽

Carl F. Lorenzo ◽

Tom T. Hartley

Keyword(s):

Differential Equation ◽

Fractional Calculus ◽

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Fractional Derivatives ◽

Past History ◽

The Past ◽

History Of ◽

Fractional Differential ◽

Generally True

The importance of proper initialization in taking into account the history of a system whose time evolution is governed by a differential equation of fractional order, has been established by Lorenzo and Hartley, who also gave the method of properly incorporating the effect of the past (history) by means of an initialization function for the Riemann-Liouville and the Grunwald formulations of fractional calculus. The present work addresses this issue for the Caputo fractional derivative and cautions that the commonly held belief that the Caputo formulation of fractional derivatives properly accounts for the initialization effects is not generally true when applied to the solution of fractional differential equations.

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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.

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A Note on the Stability Analysis of Fuzzy Nonlinear Fractional Differential Equations Involving the Caputo Fractional Derivative

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2021/7488524 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Ali El Mfadel ◽

Said Melliani ◽

M’hamed Elomari

Keyword(s):

Stability Analysis ◽

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Caputo Fractional Derivative ◽

Direct Method ◽

Stability Result ◽

Nonlinear Fractional Differential Equations ◽

Fractional Differential ◽

The Stability

In this paper, we present and establish a new result on the stability analysis of solutions for fuzzy nonlinear fractional differential equations by extending Lyapunov’s direct method from the fuzzy ordinary case to the fuzzy fractional case. As an application, several examples are presented to illustrate the proposed stability result.

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