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    The analytical solutions for conformable integral equations and integro-differential equations by conformable Laplace transform
    (Springer, 2018) Ozkan, Ozan; Kurt, Ali
    In this article, existence theorem for conformable Laplace transform is expressed. Then by using basic properties of conformable Laplace transform such as convolution theorem, conformable Laplace transform of fractional derivative and fractional integral, authors obtained the exact solution of initial value problems for integral equations and integro-differential equations where the derivatives and integrals are in conformable sense. In the literature it is the first time that obtaining the solutions of integro differential equations, integral equations by means of conformable fractional derivative.
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    A New Method to Solve Time Fractional Diffusion Equations Arising in Chaos and Heat Conduction
    (Amer Scientific Publishers, 2018) Ozkan, Ozan; Kurt, Ali; Tasbozan, Orkun
    The goal of the present paper is to construct a method to obtain the solution of conformable fractional partial differential equations (CFPDEs). Since these systems can be transformed to partial differential equations (PDEs) by using wave transform, the reduced system can be solved by using differential transform method (DTM) solution methods. Based on this idea, we build an efficient solution procedure that reduces CFPDEs to PDEs via wave transform, then approximate the solution of obtained system by using Differential Transform Method (DTM) which is a special procedure for solving PDEs. As an example, we implement the method to time fractional Diffusion Equation (TFDE).
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    New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves
    (Pergamon-Elsevier Science Ltd, 2018) Tasbozan, Orkun; Senol, Mehmet; Kurt, Ali; Ozkan, Ozan
    In this paper, we present new exact solution sets of nonlinear conformable time-fractional coupled Drinfeld-Sokolov-Wilson equation which arise in shallow water flow models, when special assumptions are used to simplify the shallow water equations by means of Sine-Gordon expansion method. We also present an analytical approximate method namely perturbation-iteration algorithm (PIA) for the system. Basic definitions of fractional derivatives are described in the conformable sense. An example is given and the results are compared to exact solutions. The results show that the presented methods are powerful, reliable, simple to use and ready to apply to wide range of fractional partial differential equations.
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    On conformable double Laplace transform
    (Springer, 2018) Ozkan, Ozan; Kurt, Ali
    In this study authors introduce the conformable double Laplace transform which can be used to solve fractional partial differential equations that represents many physical and engineering models. In these models the derivatives and integrals are in the sense of newly defined conformable type. Then some properties of conformable double Laplace transform are expressed. Finally fractional heat equation and fractional telegraph equation which is used in various applications in science and engineering investigated as an application of this new transform.