On the solutions of generalized Cauchy differential equations and diffusion equations with k-Hilfer-Prabhakar derivative

In this article, natural transform of k-Prabhakar integral, k-Prabhakar derivative, k-Hilfer-Prabhakar fractional derivative (k-HPFD) are calculated. In addition, we also obtain the natural transform of regularized versions of k-Prabhakar integral, k-Prabhakar derivative, k-HPFD. Finally, we solve various k-Hilfer-Prabhakar type Cauchy equations via operations of natural and Fourier transforms. The diffusion equations play a key role in oceanography and all models of hydrodynamics. Our new generalized solutions of k-HPFD type Cauchy problems and diffusion models may be used to explore fluid mechanics, ocean engineering, and wave phenomena and so on. The solutions of Cauchy equations and diffusion models considered with k-HPFD operator and its regularized version are computed in a shape of generalized Mittag-Leffler form by subsequent operations of integral transforms.