In this communication, one shows that there exists in the literature a certain form of deformed derivative that can here be identified as the dual of conformable derivative. The deformed subtraction is used here, together with the duality concept, as the basic definitions and starting points in order to obtain the connected dual operators. The q-exponential, in the context of generalized statistical mechanics, is the eigenfunction of this dual conformable derivative. The basic properties of the dual deformed-derivatives and also some perspective of applications and simple models are presented. The importance of this deformed derivative for position-dependent models is highlighted. An outlook of potential applications and developments is presented.
The purpose of this work is to show that the Khalil and Katagampoula conformable derivatives are equivalent to the simple change of variables $x$ $\rightarrow $ $x^{α}/α,$ where $α$ is the order of the derivative operator, when applied to differential functions. Although this means no \textquotedblleft new mathematics\textquotedblright\ is obtained by working with these derivatives, it is a second purpose of this work to argue that there is still significant value in exploring the mathematics and physical applications of these derivatives. This work considers linear differential equations, sel…
In this article, a modification of the rapidly convergent approximation method is proposed to solve a coupled Korteweg-de Vries equations with conformable derivative that govern shallow-water waves. Based on the Leibniz and chain rule of conformable derivative, these equations reduced into ODEs with integer-order using traveling wave transformation. Adopting the modified scheme a new novel exact solution of the reduced coupled ordinary differential equations is obtained in terms of exponential functions. Finally, by putting them back into traveling wave transformation the solutions of the cons…
Delay Modeling with Conformable and Caputo Derivatives: Analytical and Computational Insights
Universidad Católica del Norte（天主教主教大学）；Universidad de Antofagasta（安托法加斯塔大学）；Durban University of Technology（德班理工大学）；North-West University（西北大学）；Universidad Iberoamericana Ciudad de México（墨西哥城伊比利亚美洲大学）；Universidad de Tarapaca（塔拉帕卡大学）
This work oversees with the coupled Schrödinger-Boussinesq equations with conformable derivative, which have lots of applications in laser and plasma. The said equations are reduced to a coupled stationary form using complex travelling wave transformation. Next Painlevé test applied to derived the integrable cases of the reduced equation, after that using RCAM derived the solution of reduced equations integrable and nonintegrable cases. Few theorems have been presented and proved to ensure their boundedness. All presented boundedness cases have been checked and explained by plotting the soluti…
Definitions of fractional derivative of order $α$ ($0 < α\leq 1$) using non-singular kernels have been recently proposed. In this note we show that these definitions cannot be useful in modelling problems with a initial value condition (like, for example, the fractional diffusion equation) because the solutions obtained for these equations do not satisfy the initial condition (except for the integer case $α= 1$). In order to satisfy an arbitrary initial condition the definitions of fractional derivative must necessarily involve a singular kernel.
After reviewing the definition of two differential operators which have been recently introduced by Caputo and Fabrizio and, separately, by Atangana and Baleanu, we present an argument for which these two integro-differential operators can be understood as simple realizations of a much broader class of fractional operators, i.e. the theory of Prabhakar fractional integrals. Furthermore, we also provide a series expansion of the Prabhakar integral in terms of Riemann-Liouville integrals of variable order. Then, by using this last result we finally argue that the operator introduced by Caputo an…
We discuss some of the mathematical properties of the fractional derivative defined by means of Fourier transforms. We first consider its action on the set of test functions $\Sc(\mathbb R)$, and then we extend it to its dual set, $\Sc'(\mathbb R)$, the set of tempered distributions, provided they satisfy some mild conditions. We discuss some examples, and we show how our definition can be used in a quantum mechanical context.
