scholarly journals A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Entropy ◽  
2016 ◽  
Vol 18 (10) ◽  
pp. 345 ◽  
Author(s):  
Waleed Abd-Elhameed ◽  
Youssri Youssri
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Fibonacci Polynomials ◽  
Caputo Fractional Derivatives ◽  
Fractional Differential ◽  
Derivatives Of

scholarly journals EXISTENCE AND STABILITY RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO FRACTIONAL DERIVATIVES

2019 ◽  
pp. 341
Author(s):  
Mohamed Houas ◽  
Mohamed Bezziou
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Contraction Mapping ◽  
Nonlocal Boundary ◽  
Stability Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Existence And Stability ◽  
Fractional Differential ◽  
Stability Results

In this paper, we discuss the existence, uniqueness and stability of solutions for a nonlocal boundary value problem of nonlinear fractional differential equations with two Caputo fractional derivatives. By applying the contraction mapping and O’Regan fixed point theorem, the existence results are obtained. We also derive the Ulam-Hyers stability of solutions. Finally, some examples are given to illustrate our results.

scholarly journals A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
D. Baleanu ◽  
A. H. Bhrawy ◽  
T. M. Taha
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Laguerre Polynomials ◽  
Collocation Methods ◽  
Half Line ◽  
Caputo Fractional Derivatives ◽  
Polynomial Degree ◽  
Generalized Laguerre Polynomials ◽  
Fractional Differential

This paper deals with modified generalized Laguerre spectral tau and collocation methods for solving linear and nonlinear multiterm fractional differential equations (FDEs) on the half line. A new formula expressing the Caputo fractional derivatives of modified generalized Laguerre polynomials of any degree and for any fractional order in terms of the modified generalized Laguerre polynomials themselves is derived. An efficient direct solver technique is proposed for solving the linear multiterm FDEs with constant coefficients on the half line using a modified generalized Laguerre tau method. The spatial approximation with its Caputo fractional derivatives is based on modified generalized Laguerre polynomialsLi(α,β)(x)withx∈Λ=(0,∞),α>−1, andβ>0, andiis the polynomial degree. We implement and develop the modified generalized Laguerre collocation method based on the modified generalized Laguerre-Gauss points which is used as collocation nodes for solving nonlinear multiterm FDEs on the half line.

Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives

2021 ◽  
Vol 24 (2) ◽  
pp. 483-508
Author(s):  
Mohammed D. Kassim ◽  
Nasser-eddine Tatar
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Logarithmic Function ◽  
Fractional Differential ◽  
Derivatives Of ◽  
L’Hôpital’S Rule

Abstract The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1675
Author(s):  
Nur Amirah Zabidi ◽  
Zanariah Abdul Majid ◽  
Adem Kilicman ◽  
Faranak Rabiei
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lagrange Interpolation ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Third Order ◽  
Fractional Differential ◽  
Derivatives Of

Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept of a third-order Adam–Bashforth numerical scheme by implementing Lagrange interpolation for fractional case, where the fractional derivatives are defined in the Caputo sense. Furthermore, the study includes a discussion on stability and convergence analysis of the method. Several numerical examples are also provided in order to validate the reliability and efficiency of the proposed method. The examples in this study cover solving linear and nonlinear fractional differential equations for the case of both single order as α∈(0,1) and higher order, α∈1,2, where α denotes the order of fractional derivatives of Dαy(t). The comparison in terms of accuracy between the proposed method and other existing methods demonstrate that the proposed method gives competitive performance as the existing methods.

scholarly journals A New Operational Matrix of Fractional Derivatives to Solve Systems of Fractional Differential Equations via Legendre Wavelets

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 238 ◽  
Author(s):  
Aydin Secer ◽  
Selvi Altun
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Matrix Method ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Operational Matrix Method ◽  
Fractional Differential

This paper introduces a new numerical approach to solving a system of fractional differential equations (FDEs) using the Legendre wavelet operational matrix method (LWOMM). We first formulated the operational matrix of fractional derivatives in some special conditions using some notable characteristics of Legendre wavelets and shifted Legendre polynomials. Then, the system of fractional differential equations was transformed into a system of algebraic equations by using these operational matrices. At the end of this paper, several examples are presented to illustrate the effectivity and correctness of the proposed approach. Comparing the methodology with several recognized methods demonstrates that the advantages of the Legendre wavelet operational matrix method are its accuracy and the understandability of the calculations.

Infinitely many solutions to boundary value problem for fractional differential equations

2018 ◽  
Vol 21 (6) ◽  
pp. 1585-1597 ◽  
Author(s):  
Diego Averna ◽  
Angela Sciammetta ◽  
Elisabetta Tornatore
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Infinitely Many Solutions ◽  
Caputo Fractional Derivatives ◽  
Fractional Order Differential Equations ◽  
Critical Point Theorems ◽  
Fractional Differential

Abstract Variational methods and critical point theorems are used to discuss existence of infinitely many solutions to boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. An example is given to illustrate our result.

scholarly journals On existence–uniqueness results for proportional fractional differential equations and incomplete gamma functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zaid Laadjal ◽  
Thabet Abdeljawad ◽  
Fahd Jarad
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Uniqueness Of Solutions ◽  
Gamma Functions ◽  
Uniqueness Results ◽  
Incomplete Gamma Functions ◽  
Fractional Differential ◽  
Derivatives Of

AbstractIn this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form $$ D^{\rho }= (1-\rho )+ \rho D, \quad \rho \in [0,1], $$ D ρ = ( 1 − ρ ) + ρ D , ρ ∈ [ 0 , 1 ] , where D is the ordinary differential operator.

scholarly journals Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Atimad Harir ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Higher Order ◽  
Conformable Derivatives ◽  
Fuzzy Solution ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, the generalized concept of conformable fractional derivatives of order q ∈ n , n + 1 for fuzzy functions is introduced. We presented the definition and proved properties and theorems of these derivatives. The fuzzy conformable fractional differential equations and the properties of the fuzzy solution are investigated, developed, and proved. Some examples are provided for both the new solutions.

scholarly journals A numerical study of a coupled system of fractional differential equations

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2585-2600
Author(s):  
Amal Alshabanat ◽  
Bessem Samet
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Study ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Coupled System ◽  
Numerical Approach ◽  
Coupled Systems ◽  
Caputo Fractional Derivatives ◽  
Fractional Differential

We consider a certain class of coupled systems of fractional differential equations involving ?-Caputo fractional derivatives. A numerical approach is provided for solving this class of systems. The method is based on operational matrix of fractional integration of an arbitrary ?-polynomial basis. A theoretical study related to the numerical scheme and the convergence of the method is presented. Next, several numerical examples are given using different types of polynomials aiming to confirm the efficiency of our approach.

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