On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations

In this paper, a new generalization of the Laplace transform, called the $(k,\psi )$-generalized Laplace transform, which plays an important role in solving many problem models, is introduced and its special properties are given. In addition, the previously defined integral transforms of some elementary functions and the relations between the new transform, the $(k,\psi )$-generalized Laplace transform, and other generalized Laplace transforms are given. Using the $(k,\psi )$-generalized Laplace transform, solutions to problems of fractional differential equations and fractional tempered differential equations are obtained as applications. Finally, examples are given to show that the $(k,\psi )$-generalized Laplace transform is useful for more general problems.