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In this paper, a new approach to find exact solutions is carried out for a generalized unsteady magnetohydrodynamic transport of a rate-type fluid near an unbounded upright plate, which is analyzed for ramped-wall temperature and velocity with constant concentration. The vertical plate is suspended in a porous medium and encounters the effects of radiation. An innovative definition of the time-fractional operator in power-law-kernel form is implemented to hypothesize the constitutive mass, energy, and momentum equations. The Laplace integral transformation technique is applied on a dimensionless form of governing partial differential equations by introducing some non-dimensional suitable parameters to establish the exact expressions in terms of special functions for ramped velocity, temperature, and constant-concentration fields. In order to validate the problem, the absence of the mass Grashof parameter led to the investigated solutions obtaining good agreement in existing literature. Additionally, several system parameters were used, such as as magnetic value M, Prandtl value P r , Maxwell parameter λ , dimensionless time τ , Schmidt number " S c ", fractional parameter α , andMass and Thermal Grashof numbers G m and G r , respectively, to examine their impacts on velocity, wall temperature, and constant concentration. Results are also discussed in detail and demonstrated graphically via Mathcad-15 software. A comprehensive comparative study between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently. [ABSTRACT FROM AUTHOR]

A boundary value problem associated with filtering governed by differential equations with time dependent coefficients and exhibiting a weak nonlinearity is solved. Namely, the boundary value problem is split into two independent boundary problems, i.e., that of Goursat for the concentration of a sorbate and that absorbed by a sorbent. Then each of the formulated problems is solved separately applying the Riemann transformation. [ABSTRACT FROM AUTHOR]