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.

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.

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 Operational shifted hybrid Gegenbauer functions method d for solving multi-term time fractional differential equations

2019 ◽  
Vol 38 (4) ◽  
pp. 97-110
Author(s):  
Nasibeh Seyedi ◽  
Habibollah Saeedi
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Good Deal ◽  
Operational Matrices ◽  
Tau Method ◽  
Caputo Fractional Derivatives ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this paper, we propose an efficient operational formulation of spectral tau method d for solving multi-term time fractional differential equations with initial-b boundary conditions. The shifted hybrid Gegenbauer functions (SHGFs) operational matrices of Riemann-Liouville fractional integral and Caputo fractional derivatives are presented. By using these operational matrices, the shifted hybrid Gegenbauer functions tau method for both temp oral and spatial discretization are presented, which allow us to introduce an efficient spectral method for solving such problems. Finally, numerical results show good deal with the theoretical analysis.

scholarly journals Hyers–Ulam–Mittag-Leffler Stability for a System of Fractional Neutral Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Manzoor Ahmad ◽  
Jiqiang Jiang ◽  
Akbar Zada ◽  
Zeeshan Ali ◽  
Zhengqing Fu ◽  
...  
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Coupled System ◽  
Caputo Fractional Derivatives ◽  
Picard Operator ◽  
Neutral Differential Equations ◽  
Chebyshev Norm ◽  
Fractional Differential

This article concerns with the existence and uniqueness for a new model of implicit coupled system of neutral fractional differential equations involving Caputo fractional derivatives with respect to the Chebyshev norm. In addition, we prove the Hyers–Ulam–Mittag-Leffler stability for the considered system through the Picard operator. For application of the theory, we add an example at the end. The obtained results can be extended for the Bielecki norm.

scholarly journals Eigenvalue problems for fractional differential equations with mixed derivatives and generalized p-Laplacian

2018 ◽  
Vol 23 (6) ◽  
pp. 830-850 ◽  
Author(s):  
Yupin Wang ◽  
Shutang Liu ◽  
Zhenlai Han
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Laplacian Operator ◽  
Caputo Fractional Derivatives ◽  
Wide Range ◽  
Mixed Derivatives ◽  
Fractional Differential

This paper reports the investigation of eigenvalue problems for two classes of nonlinear fractional differential equations with generalized p-Laplacian operator involving both Riemann–Liouville fractional derivatives and Caputo fractional derivatives. By means of fixed point theorem on cones, some sufficient conditions are derived for the existence, multiplicity and nonexistence of positive solutions to the boundary value problems. Finally, an example is presented to further verify the correctness of the main theoretical results and illustrate the wide range of their potential applications.

scholarly journals An Accurate Spectral Galerkin Method for Solving Multiterm Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
A. H. Bhrawy ◽  
A. S. Alofi
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Differential Equations ◽  
Jacobi Polynomials ◽  
Initial Conditions ◽  
Solution Technique ◽  
Caputo Fractional Derivatives ◽  
Polynomial Degree ◽  
Generalized Jacobi Polynomials ◽  
Fractional Differential

This paper reports a new formula expressing the Caputo fractional derivatives for any order of shifted generalized Jacobi polynomials of any degree in terms of shifted generalized Jacobi polynomials themselves. A direct solution technique is presented for solving multiterm fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using spectral shifted generalized Jacobi Galerkin method. The homogeneous initial conditions are satisfied exactly by using a class of shifted generalized Jacobi polynomials as a polynomial basis of the truncated expansion for the approximate solution. The approximation of the spatial Caputo fractional order derivatives is expanded in terms of a class of shifted generalized Jacobi polynomialsJnα,−β(x)withx∈(0,1), andnis the polynomial degree. Several numerical examples with comparisons with the exact solutions are given to confirm the reliability of the proposed method for multiterm FDEs.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

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