Analytical Solution of Linear Fractional Systems with Variable Coefficients Involving Riemann–Liouville and Caputo Derivatives

This paper deals with the initial value problem for linear systems of fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives. Some basic properties of fractional derivatives and antiderivatives, including their non-symmetry w.r.t. each other, are discussed. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by examples.

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Solution of linear fractional order systems with variable coefficients

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0037 ◽

2020 ◽

Vol 23 (3) ◽

pp. 753-763

Author(s):

Ivan Matychyn ◽

Viktoriia Onyshchenko

Keyword(s):

Linear Systems ◽

Fractional Order ◽

Initial Value Problem ◽

Transition Matrix ◽

State Transition ◽

Variable Coefficients ◽

Explicit Solutions ◽

Fractional Order Systems ◽

Initial Value ◽

Theoretical Results

AbstractThe paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville derivatives. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by an example.

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Some misinterpretations and lack of understanding in differential operators with no singular kernels

Open Physics ◽

10.1515/phys-2020-0158 ◽

2020 ◽

Vol 18 (1) ◽

pp. 594-612 ◽

Cited By ~ 3

Author(s):

Abdon Atangana ◽

Emile Franc Doungmo Goufo

Keyword(s):

Fractional Calculus ◽

Power Law ◽

Initial Value Problems ◽

Differential Operators ◽

Integral Operators ◽

Initial Value ◽

Complex Processes ◽

Singular Kernels ◽

Fractional Differential ◽

Theoretical Results

AbstractHumans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

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A Universal Predictor-corrector Algorithm for Numerical Simulation of Generalized Fractional Differential Equations

10.21203/rs.3.rs-375580/v1 ◽

2021 ◽

Author(s):

Zaid Odibat

Keyword(s):

Numerical Simulation ◽

Fractional Derivatives ◽

Fractional Integral ◽

Differential Operators ◽

Numerical Solutions ◽

Fractional Operators ◽

Initial Value ◽

Generalized Fractional Integral ◽

Fractional Differential ◽

Predictor Corrector

Abstract This study introduces some remarks on generalized fractional integral and differential operators, that generalize some familiar fractional integral and derivative operators, with respect to a given function. We briefly explain how to formulate representations of generalized fractional operators. Then, mainly, we propose a predictor-corrector algorithm for the numerical simulation of initial value problems involving generalized Caputo-type fractional derivatives with respect to another function. Numerical solutions of some generalized Caputo-type fractional derivative models have been introduced to demonstrate the applicability and efficiency of the presented algorithm. The proposed algorithm is expected to be widely used and utilized in the field of simulating fractional-order models.

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Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0004 ◽

2020 ◽

Vol 23 (1) ◽

pp. 103-125 ◽

Cited By ~ 1

Author(s):

Latif A-M. Hanna ◽

Maryam Al-Kandari ◽

Yuri Luchko

Keyword(s):

Differential Equation ◽

Fractional Differential Equation ◽

Fractional Derivatives ◽

Fractional Integrals ◽

Operational Method ◽

Initial Value ◽

Right Hand ◽

Fractional Differential ◽

Left And Right ◽

Fractional Integrals And Derivatives

AbstractIn this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the four-parameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

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The numerical algorithms for discrete Mittag-Leffler functions approximation

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0006 ◽

2019 ◽

Vol 22 (1) ◽

pp. 95-112 ◽

Cited By ~ 1

Author(s):

Ang Li ◽

Yiheng Wei ◽

Zongyang Li ◽

Yong Wang

Keyword(s):

Transition Matrix ◽

State Transition ◽

Numerical Algorithms ◽

The State ◽

State Transition Matrix ◽

Numerical Implementation ◽

Science And Engineering ◽

Implementation Method ◽

Theoretical Results

Abstract Motivated essentially by the success of the applications of the discrete Mittag-Leffler functions (DMLF) in many areas of science and engineering, the authors present, in a unified manner, a detailed numerical implementation method of the Mittag-Leffler function. With the proposed method, the overflow problem can be well solved. To further improve the practicability, the state transition matrix described by discrete Mittag-Leffler functions are investigated. Some illustrative examples are provided to verify the effectiveness of the proposed theoretical results.

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Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-014-0156-6 ◽

2014 ◽

Vol 17 (1) ◽

Cited By ~ 3

Author(s):

Myong-Ha Kim ◽

Guk-Chol Ri ◽

Hyong-Chol O

Keyword(s):

Differential Equation ◽

Fractional Differential Equations ◽

Initial Value Problem ◽

Fractional Derivatives ◽

Operational Calculus ◽

Operational Method ◽

Initial Value ◽

Constant Coefficients ◽

Generalized Fractional Derivatives ◽

Fractional Differential

AbstractThis paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski’s type. We prove that the initial value problem has the solution if and only if some initial values are zero.

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Existence of solutions for fractional differential inclusions with initial value condition and non-instantaneous impulses

Filomat ◽

10.2298/fil1917499l ◽

2019 ◽

Vol 33 (17) ◽

pp. 5499-5510 ◽

Cited By ~ 2

Author(s):

Danfeng Luo ◽

Zhiguo Luo

Keyword(s):

Differential Inclusions ◽

Existence Of Solutions ◽

Mean Value ◽

Mean Value Theorem ◽

Fractional Differential Inclusions ◽

Initial Value ◽

Generalized Gronwall Inequality ◽

Nonlinear Alternative ◽

Fractional Differential ◽

Theoretical Results

In this paper, we mainly consider the existence of solutions for a kind of ?-Hilfer fractional differential inclusions involving non-instantaneous impulses. Utilizing another nonlinear alternative of Leray-Schauder type, we present a new constructive result for the addressed system with the help of generalized Gronwall inequality and Lagrange mean-value theorem, and some achievements in the literature can be generalized and improved. As an application, a typical example is delineated to demonstrate the effectiveness of our theoretical results.

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New Method for Numerical Solution of Z-Fractional Differential Equations

New Mathematics and Natural Computation ◽

10.1142/s1793005721500034 ◽

2020 ◽

pp. 1-17

Author(s):

Parisa Keshavarz ◽

Tofigh Allahviranloo ◽

Farajollah M. Yaghoobi ◽

Ali Barahmand

Keyword(s):

Distribution Function ◽

Differential Equations ◽

Fractional Differential Equations ◽

Fractional Derivatives ◽

Probability Function ◽

Trapezoidal Rule ◽

Taylor Formula ◽

Initial Value ◽

Euler’S Method ◽

Fractional Differential

In this paper, at first, we introduce fractional differential equations with [Formula: see text]-valuation. Then, we propose a numerical method to approximate the solution. The proposed method is a hybrid method based on the corrected fractional Euler’s method and the probability distribution function. Moreover, the corrected fractional Euler’s method based on the generalized Taylor formula and the modified trapezoidal rule is proposed that this method can be used in the problems’ limitation section of the [Formula: see text]-fractional Initial value problem of order [Formula: see text] with the fuzzy Caputo fractional differential (fractional derivatives are defined on the basis of the Hukuhara differences and the generalized fuzzy derivatives). The probability function is based on exponential distribution function and used to represent the reliability of the problem limitation part. Finally, by two examples, we show that the proposed method can arbitrarily approximate the fractional differential equations with [Formula: see text]-valuation.

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COMPARISON THEOREMS FOR CAPUTO–HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽

10.1142/s0218348x19500361 ◽

2019 ◽

Vol 27 (03) ◽

pp. 1950036 ◽

Cited By ~ 2

Author(s):

LI MA

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Continuous Dependence ◽

Fractional Derivatives ◽

Comparison Theorems ◽

Hadamard Fractional Differential Equations ◽

Fractional Differential ◽

The One ◽

Lipschitz Conditions ◽

Theoretical Results

The main purpose of this paper is to investigate the comparison theorems for fractional differential equations involving Caputo–Hadamard fractional derivatives. First, we indicate the continuous dependence on parameters of solutions for Caputo–Hadamard fractional differential equations (C-HFDEs). Then, the first and second comparison theorems for C-HFDEs are proposed and proved, respectively. In addition, we establish the generalized comparisons for C-HFDEs under the one-side Lipschitz conditions. At last, the corresponding examples are also provided to verify the theoretical results.

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

Axioms ◽

10.3390/axioms10040322 ◽

2021 ◽

Vol 10 (4) ◽

pp. 322

Author(s):

Ricardo Almeida ◽

Ravi P. Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Lyapunov Functions ◽

Fractional Derivatives ◽

Sufficient Conditions ◽

Hopfield Neural Network ◽

Quadratic Lyapunov Functions ◽

Fractional Differential ◽

The Stability ◽

Theoretical Results

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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