Conformable Fractional Calculus Research Articles

The Fractional Navier-Stokes Equations with delay conditions

Throught this paper, we study the cauchy problem for the conformable fractional Navier-Stokes Equations (FNSE) with finite delay external forces, contains some hereditary features, on a bounded domain. We prove that there exist a unique local mild solutions for the initial datum, by using semigroup theory, conformable fractional calculus and Banach contraction theorem.With more conditions on delay external forces we establish the globality and continuation of the mild solutions.

On a Novel Fractional Calculus and its Applications to Well-known Problems

The objective of this study is to introduce three novel and comprehensive models utilizing the concept of conformable fractional calculus. These models encompass the fractional population growth model (FPGM), the fractional body cooling model (FBCM), and the fractional heat differential equation (FHDE). Firstly, using new results, a type of new conformable fractional derivative introduced in \cite{5}:\begin{center}$\left(\mathcal{D}^\alpha H\right)(y)=\lim _{z \longrightarrow 0} \frac{H\left(y+z e^{(\alpha-1) y}\right)-H(y)}{z},$\end{center}where $\alpha \in(0,1]$ and $H$ is a function. We obtain exact solutions of the considered models. The results feature excellent agreement of exact solutions. Finally, it is shown that the proposed method provides a more powerful mathematical tool for solving models in mathematical physics.

On Conformable Fractional Milne-Type Inequalities

Building upon previous research in conformable fractional calculus, this study introduces a novel identity. Using this identity as a foundation, we derive a set of conformable fractional Milne-type inequalities specifically designed for differentiable convex functions. The obtained results recover some existing inequalities in the literature by fixing some parameters. These novel contributions aim to enrich the analytical tools available for studying convex functions within the realm of conformable fractional calculus. The derived inequalities reflect an inherent symmetry characteristic of the Milne formula, further illustrating the balanced and harmonious mathematical structure within these frameworks. We provide a thorough example with graphical representations to support our findings, offering both numerical insights and visual confirmation of the established inequalities.

Analysis of linear continuous-time systems by the use of the conformable fractional calculus and Caputo

Analysis of linear continuous-time systems by the use of the conformable fractional calculus and Caputo

Existence results and behavior analysis for nonlinear age-dependent population dynamics problems with conformable fractional operator

In this paper, we investigate the existence of solutions and analyze the large-time behavior for Gurtin–Maccamy population model involving conformable fractional derivatives. As a preliminary step, we construct a generic structure of the solution associated with our proposed model by utilizing some basic properties and tools of conformable fractional calculus. We establish the existence of a unique solution of the given model with the given initial conditions. At last, by using the upper and lower solutions for the characteristic equation, we define the upper and lower boundaries for the obtained solution and describe the large-time behavior of the total population.

Conformable Fractional Calculus of Vector Valued Functions of Several Variables

Conformable Fractional Calculus of Vector Valued Functions of Several Variables

Density problem some of the fractional functional spaces via conformable fractional calculus and their application

Density problem some of the fractional functional spaces via conformable fractional calculus and their application

Fractional Leindler’s Inequalities via Conformable Calculus

In this paper, some fractional Leindler and Hardy-type inequalities and their reversed will be proved by using integration by parts and Hölder inequality on conformable fractional calculus. As a special case, some classical integral inequalities will be obtained. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new fractional inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type.

On Nodal Properties for Some Nonlinear Conformable Fractional Differential Equations

A class of conformable fractional differential equations \[ D_x^{\alpha} D_x^{\alpha} u(x) + \omega(x) f(u(x)) = 0 \quad \textrm{on $(0,1)$} \] is considered. We first give a sufficient condition for the existence of sign-changing solutions with the prescribed number of zeros to this problem. On the basis of this result, we turn to a specific case of the above problem and give a uniqueness theorem related to the function $\omega$. Essentially, the main methods using in this work are properties of conformable fractional calculus, the scaling argument and Prüfer-type substitutions.

Hermite-Hadamard-type inequalities for conformable integrals

In this study, some inequalities of Hermite-Hadamard type for integrals arising in conformable fractional calculus are presented. In fact, the obtained inequalities are not only valid for those integrals arising in conformable fractional calculus, but for more general integrals as well. Numerous known versions are recovered as special cases. We also illustrate our findings via applications to modified Bessel functions, special means, and midpoint approximations.

New integral operators for conformable fractional calculus with applications

In this paper, we establish new operators on fractional conformable calculus theory. Some properties of the introduced operators (such as: well definition, boundedness, semi group and commutativity) are proved and some applications on integral inequalities are presented. For our results, some recent study on operator theory and fractional calculus are deduced as some special cases.

Çarpımsal Uyumlu Kesirli Diferansiyel Denklemler

In this study, multiplicative conformable fractional differential equations are presented. Wronskian concept, linear dependence-independence concepts are defined on multiplicative conformable fractional calculus and some theorems and results are given among them. Finally, some examples are solved by giving some methods for finding general solutions of multiplicative conformable fractional differential equations.

Exponential stability and L2 gain analysis of uncertain fractional reset control systems

Abstract This paper considers the stability problem of a class of uncertain fractional reset control systems that undergo the ${L}_2$ gain performance improvement via the conformable fractional calculus. To remove the Zeno phenomenon in the system’s response, a new reset law based on the time regularization technique is designed. By developing a theory to design a new reset control, the stability of the proposed fractional order (FO) linear system with model uncertainties and bounded exogenous input is proved via an extended quadratic Lyapunov function and some efficient linear matrix inequalities. Likewise, the impulsive effect in the control input is removed by employing an appropriate low-pass filter (LPF). In the end, two numerical examples show that when the plant is modelled by the FO differential equations, by utilizing the constructive role of fractional reset controller, the robustness of FO reset system can also be attained.

Oscillation of Nonlinear Fractional Dynamic Equations with Forcing Term

In this paper, interval oscillation criteria for the nonlinear damped dynamic equations with forcing terms on time scales within conformable fractional derivatives are established. Our approach is determined from the implementation of generalized Riccati transformation, some properties of conformable time-scale fractional calculus, and certain mathematical inequalities. Also, we extend the study of oscillation to conformable fractional Euler-type dynamic equation. Examples are presented to emphasize the validity of the main theorems\enleadertwodots.

Conformable fractional Bohr Hamiltonian with Bonatsos and double-well sextic potentials

Using the conformable fractional calculus, a new formulation of the Bohr Hamiltonian is introduced. The conformable fractional energy spectra of free- and two- parameters anharmonic oscillator potentials are investigated. The energy eigenvalues and wave functions are calculated utilizing the finite-difference discretization method. It is proved that the conformable fractional spectra of the free-parameter Bonatsos potentials, close completely the gaps between the classical spectra of the vibrational dynamical symmetry, the models, and the critical point symmetry. The ground effective sextic potential, which generates both the ground state and the excited states is considered to have two degenerate minima. In this case, the conformable fractional spectra of sextic potentials show a change, as a function of barrier height, from -unstable energy level sequence to the spectrum of model and simultaneously provide new features. The shape coexistence phenomena in the ground band states are identified. The energy spectrum and shape coexistence with mixing phenomena in 96Mo nucleus are discussed in the framework of the conformable fractional Bohr Hamiltonian.

Inhomogeneous conformable abstract Cauchy problem

Abstract In this paper, we discuss the solution of the inhomogeneous conformable abstract Cauchy problem. The homogeneous problem is also studied. The analysis of conformable fractional calculus and fractional semigroups is also given. Existence, uniqueness and regularity of a mild solution for the conformable abstract Cauchy problem are studied. Applications illustrating our main abstract results are also given.

Generalizations of Hardy’s Type Inequalities via Conformable Calculus

In this paper, we derive some new fractional extensions of Hardy’s type inequalities. The corresponding reverse relations are also obtained by using the conformable fractional calculus from which the classical integral inequalities are deduced as special cases at α=1.

Conformable fractional isothermal gas spheres

The isothermal gas sphere is well known as a powerful tool to model many problems in astrophysics, physics, chemistry, and engineering. This singular differential equation has not an exact solution and solved only by numerical and approximate methods. In the present paper and within the framework of the Newtonian hydrostatic equilibrium, we have developed general analytical formulations for the fractional isothermal gas sphere. To obtain analytical expressions for mass, radius, and density, besides the fractional isothermal gas sphere, we used the conformable fractional calculus. Using the series expansion method, we obtained a general recurrences relation, which allows us to determine the series coefficients. The comparison of the series solution with the numerical ones for the fractional parameter α=1 is good for dimensionless parameter up to x ≃ 3.2, beyond this value, the series diverges. We applied a combination of Euler-Abel and Pade techniques to accelerate the series, therefore accelerated series converge to the numerical desired value. We analyzed some physical parameters of a typical model of the neutron stars such as the mass-radius relation, density, and pressure ratio for different models. We found that the current models of the conformable neutron stars had smaller volumes and masses than both stars in the context of modified Rienmann-Liouville derivatives as well as the integer one.

The effect of fractional calculus on the formation of quantum‐mechanical operators

In this paper, the deformation of the ordinary quantum mechanics is formulated based on the idea of conformable fractional calculus. Some properties of fractional calculus and fractional elementary functions are investigated. The fractional wave equation in 1 + 1 dimension and fractional version of the Lorentz transformation are discussed. Finally, the fractional quantum mechanics is formulated; infinite potential well problem, density of states for the ideal gas, and quantum harmonic oscillator problem are discussed.

A modification to the conformable fractional calculus with some applications

In the conformable fractional calculus, TαTβ≠TβTα and IαIβ≠IβIα, where Tα and Iα are conformable fractional differential and integral operators, respectively. Also, Tβ≠Tnα and Iβ≠Inα, where β=nα for some n∈N. In this work, we introduce a modification to the definition of conformable fractional derivative and conformable fractional integral. Our definition characterizes by the realization of the commutative property between the differential operators and the integral operators. The higher derivatives in the sense of the modified conformable fractional derivative are coinciding with the sequential derivatives. Finally, we present the solutions of the Cauchy-Euler and linear (with constant coefficients) fractional differential equations to show our new approach.