A novel generalized type of the Caputo fractional derivative: integral transforms, illustrative examples, and solution of fractional-order generalized differential equations

In this article, we introduce a novel generalized Caputo fractional derivative, using a special type of function known as the Wright function in the definition of the classical Caputo fractional derivative. We also apply the Fourier, Laplace, and Mellin integral transform methods, which are very useful popular mathematical tools in various scientific fields, to the new generalized fractional derivative. Moreover, as illustrative examples, we calculate the new generalized fractional derivative of constant, power, exponential, sine, and cosine functions. Furthermore, we obtain the solutions of the generalized motion, harmonic vibration, and Bessel differential equations defined by the new generalized fractional derivative using the Fourier, Laplace, and Mellin integral transform methods. Finally, we obtain approximate behavior graphs for both the classical Caputo fractional derivative and the generalized Caputo fractional derivative using some specific data and present these graphs comparatively.