In this paper a new definitionn of fractional derivative and fractional integral in the sense ofConformable derivative type is presented. This form of definition shows that it is more compatible withclassical natural definition of derivative and is more convinient fractional derivative one. We will definethis for 0 ≤ α < 1 and n − 1 ≤ α < n and further, if α = 1 the definition coincides with the classicaldefinition of derivative of first order.

In this paper, we investigate stability analysis of fractional differential systems equipped with the Conformable fractional derivatives. Some stability conditions of fractional differential systems are proposed by applying the fractional exponential function and the fractional Laplace transform. More-over, we check the stability of Conformable fractional Lotka-Volterra system with the multi-step homotopy perturbation method to demonstrate the efficiency and effectiveness of the proposed procedure.

Initial-boundary value problems including space-time fractional PDEs have been used to model a wide range of problems in physics and engineering fields. In this paper, a non-self adjoint initial boundary value problem containing a third order fractional differential equation is considered. First, a spectral problem for this problem is presented. Then the eigenvalues and eigenfunctions of the main spectral problem are calculated. In order to calculate the roots of their characteristic equation, the asymptotic expansion of the roots is used. Finally, by suitable choice of these asymptotic expansions, related eigenfunctions and Mittag-Lefler functions, the analytic and numerical solutions to the main initial-boundary value problem are given.

In the present article, we utilize the Conformable Double Laplace Transform Method (CDLTM) to get the exact solutions of a wide class of Conformable fractional di, er-ential in mathematical physics. The results obtained show that the proposed method is e, cient, reliable and easy to be implemented on related linear problems in applied mathematics and physics. Moreover, the (CDLTM) has a small computational size as compared to other methods.

Our research is about the Sturm-Liouville equation which contains Conformable fractional derivatives of order $\alpha \in (0,1]$ in lieu of the ordinary derivatives. First, we present the eigenvalues, eigenfunctions, and nodal points, and the properties of nodal points are used for the reconstruction of an integral equation. Then, the Bernstein technique was utilized to solve the inverse problem, and the approximation of solving this problem was calculated. Finally, the numerical examples were introduced to explain the results. Moreover,  the analogy of this technique is shown in a numerical example with the Chebyshev interpolation technique .

DNA, or deoxyribonucleic acid, is found in every single cell and is the cell's primary information storage medium. DNA stores all an organism's genetic information, including the instructions it needs to grow, divide, and live. DNA is made up of four different building blocks called nucleotide bases: adenine (A), thymine (T), cytosine (C), and guanine (G). The genome is sequenced in vitro utilizing encoding strategies such as labelling one bond pair as 0 and the other as 1 to store digital information. In this study, the fractional differential order of double-chain DNA dynamical system was investigated, considering Atangana’s Conformable fractional derivative. The Conformable sub-equation method was applied to the system.  The analysis resulted in some interesting new exact solutions of the model. One-soliton kink solution, multiple-soliton solutions, and periodic-wave solutions are the three broad categories that may be used to describe the results. In order to better understanding the solutions found, we have visually investigated a few of them. Both solitary and anti-solitary waves of the DNA strands are seen, attesting to the nonlinear dynamics of the system. The gathered data might be used to conduct application evaluations and draw further scientific findings.

In this paper, we have obtained weighted versions of Ostrowski, Č ebysev and Grü ss type inequalities for Conformable fractional integrals which is given by Katugompola. By using the Katugampola definition for Conformable calculus, the present study confirms previous findings and contributes additional evidence that provide the bounds for more general functions.

We expand a new generalization of the two-dimensional differential trans-form method. The new generalization is based on the two-dimensional differ-ential transform method, fractional power series expansions, and Conformable fractional derivative. We use the new method for solving a nonlinear con-formable fractional partial differential equation and a system of Conformable fractional partial differential equation. Finally, numerical examples are pre-sented to illustrate the preciseness and effectiveness of the new technique.

This paper deals with the solution of a class of Volterra integral equations in the sense of the Conformable fractional derivative. For this goal, the well-organized Neumann method is developed and some theorems related to existence, uniqueness, and sufficient condition of convergence are presented. Some illustrative examples are provided to demonstrate the efficiency of the method in solving Conformable fractional Volterra integral equations.

In this paper, a novel Conformable fractional order (FO) sliding mode control technique is studied for a class of FO chaotic systems in the presence of uncertainties and disturbances. First, a novel FO nonlinear surface based on Conformable FO calculus is proposed to design the FO sliding mode controller. Then, asymptotic stability of the controller is derived by means of the Lyapunov direct method via Conformable FO operators. The stability analysis is performed in the sliding and reaching phase. In addition, the realization of reaching phase is guaranteed in finite time and the reaching time is calculated analytically. The proposed control approach has some superiorities. Reduction of the chattering phenomenon, high robustness against the uncertainty and external disturbance, and fast convergence speed are the main advantages of the proposed control scheme. Moreover, it has simple calculations because of using Conformable FO operators in the control design. The numerical simulations verify the efficiency of the proposed controller.