Applications of generalized formable transform with \(\Psi \)-Hilfer–Prabhakar derivatives

This paper introduces the \(\Psi \) $\mathrm{\Psi }$ -formable integral transform, discusses the several essential properties and results—Convolution, \(\Psi \) $\mathrm{\Psi }$ -formable transform of tth derivative, \(\Psi \) $\mathrm{\Psi }$ -Riemann Liouville fractional integration and differentiation, \(\Psi \) $\mathrm{\Psi }$ -Caputo fractional differentiation, \(\Psi \) $\mathrm{\Psi }$ -Hilfer fractional differentiation, \(\Psi \) $\mathrm{\Psi }$ -Prabhakar fractional integration and differentiation, and \(\Psi \) $\mathrm{\Psi }$ -Hilfer–Prabhakar fractional derivatives. Next, we use the Fourier integral and \(\Psi \) $\mathrm{\Psi }$ -Modifiable conversions to solve some Cauchy-type fractional differential equations using the generalized three-parameter Mittag–Leffler function and \(\Psi \) $\mathrm{\Psi }$ -Hilfer–Prabhakar fractional derivatives.