scholarly journals A short review on analytical methods for fractional equations with he’s fractional derivative

2017 ◽  
Vol 21 (4) ◽  
pp. 1567-1574 ◽  
Author(s):  
Yan Wang ◽  
Yu-Feng Zhang ◽  
Jian-Gen Liu ◽  
Muhammad Iqbal
Keyword(s):  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Methods ◽  
Function Method ◽  
Solution Process ◽  
Short Review ◽  
Fractional Equations ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method ◽  
Fractional Differential

He?s fractional derivative is adopted in this paper, and analytical methods for fractional differential equations are briefly reviewed, two modifications of the exp-function method (the generalized Kudryashov method and generalized expo-nential rational function method) are emphasized, and fractional Benjamin-Bo-na-Mahony equation with He?s fractional derivative is used as an example to elucidate the solution process.

A note on exp-function method combined with complex transform method applied to fractional differential equations

2015 ◽  
Vol 4 (3) ◽  
pp. 201-208 ◽  
Author(s):  
Ozkan Guner ◽  
Ahmet Bekir ◽  
Halis Bilgil
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Nonlinear Ordinary Differential Equation ◽  
Transform Method ◽  
Fractional Equations ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

AbstractIn this article, the fractional derivatives in the sense of modified Riemann–Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.

scholarly journals Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations

2021 ◽  
Vol 5 (2) ◽  
pp. 37
Author(s):  
Snezhana Hristova ◽  
Stepan Tersian ◽  
Radoslava Terzieva
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Initial Point ◽  
Lipschitz Stability ◽  
Fractional Equations ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Derivatives Of

A system of nonlinear fractional differential equations with the Riemann–Liouville fractional derivative is considered. Lipschitz stability in time for the studied equations is defined and studied. This stability is connected with the singularity of the Riemann–Liouville fractional derivative at the initial point. Two types of derivatives of Lyapunov functions among the studied fractional equations are applied to obtain sufficient conditions for the defined stability property. Some examples illustrate the results.

scholarly journals Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner ◽  
Adem C. Cevikel
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Fractional Complex Transform ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Fractional Differential

The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.

scholarly journals Soliton solutions of nonlinear fractional differential equations with its applications in mathematical physics

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 422
Author(s):  
A. C. Çevikel ◽  
E. Aksoy
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Soliton Solutions ◽  
Mathematical Physics ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Complex Transform ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method ◽  
Present Technique ◽  
Fractional Differential

Generalized Kudryashov method has been used to private type of nonlinear fractional differential equations. Firstly, we proposed a fractional complex transform to convert fractional differential equations into ordinary differential equations. Three applications were given to demonstrate the effectiveness of the present technique. As a result, abundant types of exact analytical solutions are obtained.

scholarly journals Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Chunye Gong ◽  
Weimin Bao ◽  
Guojian Tang ◽  
Yuewen Jiang ◽  
Jie Liu
Keyword(s):  
Parallel Computing ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Computer Science ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Variable Order ◽  
Fractional Equations ◽  
Fractional Differential

We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations areO(N2M),O(NM2), andO(NM(M+N)) compared withO(MN) for the classical partial differential equations with finite difference methods, whereM,Nare the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator), short memory principle, fast Fourier transform (FFT) based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

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.

scholarly journals Conformable fractional derivative and its application to fractional Klein-Gordon equation

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3745-3749
Author(s):  
Kangle Wang ◽  
Shaowen Yao
Keyword(s):  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Gordon Equation ◽  
Variational Iteration Method ◽  
Conformable Fractional Derivative ◽  
Fractional Complex Transform ◽  
Klein Gordon ◽  
Fractional Differential ◽  
Fractional Equation ◽  
Klein Gordon Equation

This paper adopts conformable fractional derivative to describe the fractional Klein-Gordon equations. The conformable fractional derivative is a new simple well-behaved definition. The fractional complex transform and variational iteration method are used to solve the fractional equation. The result shows that the proposed technology is very powerful and efficient for fractional differential equations.

Sign in / Sign up

Export Citation Format

Share Document