Analytical solution of local fractional Klein-Gordon equation for the generalized Hulthen potential

One dimensional Klein-Gordon (KG) equation is investigated in the domain of conformable fractional calculus for one dimensional scalar potential namely generalized Hulthen potential. The conformable fractional calculus is based on conformable fractional derivative which is the most natural definition in non integer order calculus. Fractional order differential equations can be solved analytically by means of this derivative operator. We obtained exact eigenvalue and eigenfunction solutions of local fractional KG equation and investigated the evolution of relativistic effects in correspondence with the fractional order.Comment: 16 pages, 3 figures, 3 table

Topological 1-soliton solutions to some conformable fractional partial differential equations

Topological 1-soliton solutions to various conformable fractional PDEs in both one and more dimensions are constructed by using simple hyperbolic function ansatz. Suitable traveling wave transformation reduces the fractional partial differential equations to ordinary ones. The next step of the procedure is to determine the power of the ansatz by substituting the it into the ordinary differential equation. Once the power is determined, if possible, the power determined form of the ansatz is substituted into the ordinary differential equation. Rearranging the resultant equation with respect to the powers of the ansatz and assuming the coefficients are zero leads an algebraic system of equations. The solution of this system gives the relation between the parameters used in the ansatz

qq-deformed conformable fractional Natural transform

In this paper, we develop a new deformation and generalization of the Natural integral transform based on the conformable fractional $q$-derivative. We obtain transformation of some deformed functions and apply the transform for solving linear differential equation with given initial conditions

On the nature of the conformable derivative and its applications to physics

The purpose of this work is to show that the Khalil and Katagampoula conformable derivatives are equivalent to the simple change of variables $x$ $\rightarrow$ $x^{\alpha }/\alpha ,$ where $\alpha$ 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, self-adjointness, Sturm-Liouville systems, and integral transforms. A third purpose of this work is to contribute to the physical interpretation when these derivatives are applied to physics and engineering. Quantum mechanics serves as the primary backdrop for this development.Comment: 44 pages, corrected preprin

Traveling Wave Solutions to Conformable Time Fractional RLW-class equations

The traveling wave solutions to some nonlinear conformable time fractional partial differential equations in RLW-class are set up by using sech and csch ansatzs. The conformable time fractional forms of the equal-width (EW), regularized long wave (RLW) and symmetric regularized long wave (sRLW) equations are considered in the study. By the assist of the simple traveling wave transformation, the equations are converted to some ordinary differential equations. Then, assuming these equations have solutions of forms of powers of sech and csch functions lead to determine the powers of the solutions if exist. The determination of the relation among the other parameters in the solutions follows the previous process. Finally, the solutions are expressed in some explicit forms

Stochastic solutions of Conformable fractional Cauchy problems

In this paper we give stochastic solutions of conformable fractional Cauchy problems. The stochastic solutions are obtained by running the processes corresponding to Cauchy problems with a nonlinear deterministic clock.Comment: 9 pages, submitted for Publicatio

Positive Green's functions for some fractional-order boundary value problems

We use the newly introduced conformable fractional derivative, which is different from the Caputo and Riemann-Liouville fractional derivatives, to reformulate several common boundary value problems, including those with conjugate, right-focal, and Lidstone conditions. With the fractional differential equation and fractional boundary conditions established, we find the corresponding Green's functions and prove their positivity under appropriate assumptions.Comment: 12 pages, preprin

Existence of positive solutions for a class of conformable fractional differential equations with integral boundary conditions and a parameter

In this paper, we study the existence of positive solutions for a class of conformable fractional differential equations with integral boundary conditions. By using the properties of the Green's function and the fixed point theorem in a cone, we obtain some existence results of positive solution. we also provide some examples to illustrate our results.Comment: Submitted. 11 page

Existence of Solution to a Local Fractional Nonlinear Differential Equation

We prove existence of solution to a local fractional nonlinear differential equation with initial condition. For that we introduce the notion of tube solution.Comment: This is a preprint of a paper whose final and definite form will be published in Journal of Computational and Applied Mathematics, ISSN: 0377-0427. Paper Submitted 04/Jul/2015; Revised 14/Dec/2015 and 03/Jan/2016; Accepted for publication 08/Jan/201

Solving Sequential Linear M fractional Differential Equations with Constants Coefficients

Fractional calculus is a powerful and effective tool for modelling nonlinear systems. The M derivative is the generalization of alternative fractional derivative. This M derivative obey the properties of integer calculus. In this paper, we present the method for solving M fractional sequential linear differential equations with constant coefficients for alpha is greater than or equal to 0 and beta is greater than 0. Existence and Uniqueness of the solutions for the nth order sequential linear M fractional differential equations are discussed in detail. We have present illustration for homogeneous and non homogeneous case.Comment: This article has 17 page