scholarly journals Note on the generalized conformable derivative

2021 ◽  
pp. 443-457
Author(s):  
Alberto Fleitas ◽  
Juan E. Nápoles Valdés ◽  
José M. Rodríguez ◽  
José María Sigarreta-Almira
Keyword(s):  
Conformable Derivative

A variety of wave amplitudes for the conformable fractional (2 + 1)-dimensional Ito equation

2021 ◽  
pp. 2150254
Author(s):  
Emad A. Az-Zo’bi ◽  
Wael A. Alzoubi ◽  
Lanre Akinyemi ◽  
Mehmet Şenol ◽  
Basem S. Masaedeh
Keyword(s):  
Mapping Method ◽  
Jacobi Elliptic Function ◽  
Conformable Derivative ◽  
Hyperbolic Functions ◽  
Fractional Complex Transform ◽  
New Exact Solutions ◽  
Higher Dimensional ◽  
Fractional Orders ◽  
Ito Equation ◽  
Ordinary Derivative

The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically. The Jacobi elliptic function method and Riccati equation mapping method are successfully used for this purpose. New exact solutions in terms of linear, rational, periodic and hyperbolic functions for the wave amplitude are derived. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. Numerical simulations of some obtained solutions with special choices of free constants and various fractional orders are displayed.

A new non-conformable derivative based on Tsallis’s q- exponential function

2021 ◽  
Vol 2 (2) ◽  
pp. 106-118
Author(s):  
Cristina de Andrade Santos Reis ◽  
Rinaldo Vieira da Silva Junior
Keyword(s):  
Exponential Function ◽  
Conformable Derivative

Neste artigo, uma nova derivada do tipo local é proposta e algumas propriedades básicas são estudadas. Esta nova derivada satisfaz algumas propriedades do cálculo de ordem inteira, por exemplo linearidade, regra do produto, regra do quociente e a regra da cadeia. Devido à função exponencial generalizada de Tsallis, podemos estender alguns dos resultados clássicos, a saber: teorema de Rolle, teorema do valor médio. Apresentamos a correspondente Q-integral a partir da qual surgem novos resultados. Especificamente, generalizamos a propriedade de inversão do teorema fundamental do cálculo e provamos um teorema associado à integração clássica por partes. Finalmente, apresentamos uma aplicação envolvendo equações diferenciais lineares por meio da Q-derivada.

scholarly journals Newly Developed Analytical Scheme and Its Applications to the Some Nonlinear Partial Differential Equations with the Conformable Derivative

2021 ◽  
Vol 5 (4) ◽  
pp. 238
Author(s):  
Li Yan ◽  
Gulnur Yel ◽  
Ajay Kumar ◽  
Haci Mehmet Baskonus ◽  
Wei Gao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rational Function ◽  
Analytical Approach ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Conformable Derivative ◽  
Partial Differential ◽  
Analytical Scheme

This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail.

scholarly journals An Extension of the Double G ′ / G ,   1 / G -Expansion Method for Conformable Fractional Differential Equations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Altaf A. Al-Shawba ◽  
Farah A. Abdullah ◽  
Amirah Azmi ◽  
M. Ali Akbar
Keyword(s):  
Wave Motion ◽  
Nonlinear Models ◽  
Expansion Method ◽  
Periodic Wave ◽  
Relativistic Wave ◽  
Conformable Derivative ◽  
Fractional Systems ◽  
Periodic Wave Solutions ◽  
Engineering Problems ◽  
Fractional Differential

The phenomena, molecular path in a liquid or a gas, fluctuating price stoke, fission and fusion, quantum field theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic (CRWP) equation, nonlinear time fractional SharmaTassoOlver (STO) equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, the G ′ / G ,   1 / G -expansion method is found to be ensuing, effective, and capable to provide functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the G ′ / G ,   1 / G -expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also confirm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and efficient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs.

scholarly journals A Variety of Explicit Exact and Soliton Type Solutions of Conformable Fractional Equal-Width Equations

2018 ◽  
Author(s):  
Alper Korkmaz ◽  
Asim Zafar ◽  
Hadi Rezazadeh
Keyword(s):  
Fluid Dynamics ◽  
Surface Waves ◽  
Space Time ◽  
Function Approach ◽  
Hyperbolic Function ◽  
Trigonometric Functions ◽  
Conformable Derivative ◽  
Wave Type

Exact and soliton type solutions have great importance in propagation of surface waves, fluid dynamics, optics, and many other elds of nonlinear sciences. In this study, the explicit and exact soliton type solutions for two space-time fractional Equal- Width (FEW) equations with conformable derivative are procured via the hyperbolic function approach. The wave type solutions are represented in some hyperbolic and trigonometric functions.

First integral method for non-linear differential equations with conformable derivative

2018 ◽  
Vol 13 (1) ◽  
pp. 14 ◽  
Author(s):  
H. Yépez-Martínez ◽  
J.F. Gómez-Aguilar ◽  
Abdon Atangana
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Integral Method ◽  
General Scheme ◽  
First Integral ◽  
Fractional Partial Differential Equations ◽  
Conformable Derivative ◽  
First Integral Method ◽  
Analytic Treatment ◽  
Linear Differential

In this paper, we present an analysis based on the first integral method in order to construct exact solutions of the nonlinear fractional partial differential equations (FPDE) described by beta-derivative. A general scheme to find the approximated solutions of the nonlinear FPDE is showed. The results obtained showed that the first integral method is an efficient technique for analytic treatment of nonlinear beta-derivative FPDE.

scholarly journals Consensus of Multiagent Systems Described by Various Noninteger Derivatives

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
G. Nava-Antonio ◽  
G. Fernández-Anaya ◽  
E. G. Hernández-Martínez ◽  
J. J. Flores-Godoy ◽  
E. D. Ferreira-Vázquez
Keyword(s):  
Multiagent Systems ◽  
Directed Graphs ◽  
Lyapunov Stability Theory ◽  
Order Derivative ◽  
Variable Order ◽  
Conformable Derivative ◽  
External Disturbances ◽  
Recent Developments ◽  
Variable Order Derivative ◽  
Distributed Order

In this paper, we unify and extend recent developments in Lyapunov stability theory to present techniques to determine the asymptotic stability of six types of fractional dynamical systems. These differ by being modeled with one of the following fractional derivatives: the Caputo derivative, the Caputo distributed order derivative, the variable order derivative, the conformable derivative, the local fractional derivative, or the distributed order conformable derivative (the latter defined in this work). Additionally, we apply these results to study the consensus of a fractional multiagent system, considering all of the aforementioned fractional operators. Our analysis covers multiagent systems with linear and nonlinear dynamics, affected by bounded external disturbances and described by fixed directed graphs. Lastly, examples, which are solved analytically and numerically, are presented to validate our contributions.

Local generalization of transversality conditions for optimal control problem

2019 ◽  
Vol 14 (3) ◽  
pp. 310
Author(s):  
Beyza Billur İskender Eroglu ◽  
Dіlara Yapişkan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Control Problems ◽  
Hamiltonian Formalism ◽  
Variational Calculus ◽  
Control Problems ◽  
Conformable Derivative ◽  
Transversality Conditions ◽  
Control Functions

In this paper, we introduce the transversality conditions of optimal control problems formulated with the conformable derivative. Since the optimal control theory is based on variational calculus, the transversality conditions for variational calculus problems are first investigated and then supported by some illustrative examples. Utilizing from these formulations, the transversality conditions for optimal control problems are attained by using the Hamiltonian formalism and Lagrange multiplier technique. To illustrate the obtained results, the dynamical system on which optimal control problem constructed is taken as a diffusion process modeled in terms of the conformable derivative. The optimal control law is achieved by analytically solving the time dependent conformable differential equations occurring from the eigenfunction expansions of the state and the control functions. All figures are plotted using MATLAB.

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