ESSENTIAL CRITERIA FOR PROPORTIONAL FRACTIONAL GENERALIZED HUKUHARA DIFFERENTIABILITY FOR GOURSAT PROBLEMS WITH APPLICATION IN MHD COUPLE STRESS FLUID FLOWING THROUGH A CHANNEL EMBEDDED IN A POROUS MEDIUM

A novel category of fractional derivatives with an exponential kernel, the generalized proportional fractional (G-P-F) derivative has several uses in practical issues. This operator is currently employed for the first time with this sort of fluid circulation. Initially, we address Goursat problems with generalized proportional fractional Caputo’s type generalized Hukuhara [Formula: see text] differentiability. Two distinct kinds of generalized proportional fractional Caputo type [Formula: see text]-differentiability and the second-order convoluted derivatives in Goursat equations illustrate complexity for solving Goursat problems. Furthermore, this method’s capacity to create representations that more properly depict memory-effect mechanisms is one of its main advantages to transform Goursat problems into fuzzy integrals for analyzing similar structures in order to appropriately handle the convoluted derivatives and the two distinct kinds of generalized proportional fractional Caputo’s type [Formula: see text]-differentiability. Meanwhile, the idea of fixed-point analysis is employed to address whether generalized proportional fractional fuzzy Goursat problems possess unique results as for [Formula: see text] and [Formula: see text] differentiability. Inspired by these considerations, the idea of G-P-F derivatives is presented in this study for simulating coupled stress fluid (CSF) while taking into account the simultaneous impacts of mass transport and heat. According to the impact of external pressure, the magnetohydrodynamic (MHD) flow of CSF is examined in a conduit comprising porous media. The lateral plate moves continuously, whereas the reverse plate stays still, which causes the CSF to migrate. The implicit finite difference approach is used to computationally address the non-dimensional G-P-F mathematical model of CSF, which is defined in the Laplace transform sense with an exponential kernel. The outcomes are shown schematically to show how different factors affect the temporal, concentration, and velocity aspects. As the proportionality index increases, the MHD coupling pressure substance CSF in a conduit containing porous medium exhibits decreasing flow, temperature, and concentration characteristics. According to visual outcomes, the G-P-F approaches CSF circulation in the path are significantly precise compared to both fractional and classical solutions. Furthermore, in contrast to the fractional framework, the G-P-F model offers a more precise description of memory impacts in the fluctuation of CSF. Lastly, the Sherwood number, Nusselt number, and skin friction are calculated and shown in tabulated style.