In the paper, we consider a type of Cattaneo equation with time fractional derivative without singular kernel based on fourth-order compact finite difference (CFD) in the space directions. In case of two dimensional, two alternating direction implicit (ADI) methods are proposed to split the equation into two separate one dimensional equations. The time fractional derivation is described in the Caputo-Fabrizio’s sense with scheme of order O(τ2). The solvability, unconditional stability and H1 norm convergence of the scheme are proved. Numerical results confirm the theoretical results and the effectiveness of the proposed scheme.

A fractional Cattaneo model from the generalized Cattaneo model with two fractional derivatives of different orders is considered for studying the thermoelastic response for a multilayer elliptic ring membrane with source function. The solution is obtained by applying an integral transform technique analogous to Vodicka's approach considering series expansion functions in terms of an eigenfunction to the generalized fractional Cattaneo-type heat conduction equation within an elliptic coordinates system. The analytical expressions of displacement and stress components employing Airy's stress function approach are investigated. The results are obtained as a series solution in terms of Mathieu functions and hold convergence test. The effects of fractional parameters on the temperature fields and their thermal stresses are also discussed. The findings are depicted graphically for different kinds of surface temperature gradients, and it is distinguished that the higher the fractional-order parameter, the higher the thermal response. Lastly, the generalized theory of thermoelasticity predicts an instantaneous response, but the fractional theory, which is currently under consideration, predicts a delayed response to physical stimuli, which is something that can be seen occurring in nature. This delayed response can be explained by the fact that fractional theories are currently being considered. This gives credibility to the motivation behind this topic of study in the research.

Heat and mass transfer effects in three-dimensional mixed convection flow of Eyring Powell fluid over an exponentially stretching surface with convective boundary conditions are inspected. Cattaneo-Christov Heat Flux model is a modified version of the classical Fourier's law that takes into account the interesting aspect of thermal relaxation time. First-order chemical reaction effects are also taken into account. Similarity transformations are invoked to reduce the leading boundary layer partial differential equations into the ordinary differential equations. The nonlinear, coupled ordinary differential with boundary conditions has been analyzed numerically by using the Finite Element Method.

This study investigates the combined heat mass transport features in steady Magneto-Hydro-Dynamic (MHD) viscoelastic fluid ow through stretching walls of channel. The channel walls were considered porous. The analysis of heat transport was carried out with the help of Cattaneo-Christov heat diffusion formula and generalized Fick's theory was developed for the study of mass transport. The system of partial differential expressions was changed into an ordinary differential set by introducing suitable variables. The homotopic scheme was employed for solving the resultant equations and then, validity of the results was veri ed by various graphs. Moreover, an extensive analysis was performed on the influence of involved constraints on liquid velocity, concentration, and temperature pro les. It was observed that the normal component of velocity decreased by increasing Reynolds number or the viscoelastic constraint. Both temperature and concentration pro les were enhanced by increasing combined parameter and Reynolds number. The presence of thermal relaxation number and concentration relaxation number decreased temperature and concentration pro les, respectively.

The induced magnetic field is used to control the fluid motion and heat transfer in a variety of applications, such as in MHD devices, microfluidics, electrically conducting fluids in channels and in circular pipes, and clinical applications such as drug delivery and cooling of nuclear reactors. Henceforth this investigation aims to elucidate the behavior of viscoelastic (second-grade fluid) ternary nanofluid flow through a permeable stretching sheet with an induced magnetic field. The stretching surface is subjected to the Cattaneo-Christov heat and mass flux model to investigate heat and mass transfer properties. Solutions of reduced governing equations are obtained numerically via the shooting method and computed using the bvp-4c algorithm. The impacts of diverse active parameters such as porous medium, magnetic parameter, reciprocal magnetic Prandtl parameter, stretching parameter, HSS parameter, and relaxation time parameter for heat and mass flux are studied graphically.  In addition, the values of significant engineering factors are calculated and comparative analysis is presented through bar graphs. It is seen that regular heat sink/source promotes thermal distribution and relaxation time for mass flux enhances the mass transfer rate between fluid flow and solid surface.

In this paper, the time invariant impressions of Cattaneo-Christov heat and mass flux theories were implemented to overcome the initial instant disturbances throughout whole medium. The motion of three-dimensional, incompressible, magnetized viscous fluid flow was induced by the oscillatory disk. Porous media was used to saturate the rotating disk. Similarity transformations were accomplished to normalize the flow problem. Successive Over Relaxation (SOR) technique was implemented to discuss the new findings of normalized non-linear resulting system. It is perceived that increase in porosity parameter results in decrease of oscillatory velocity pro les. The characterization of porous media is useful in geothermal and petroleum reservoirs. Time varying oscillatory curves for concentration and temperature decay for varying concentration and thermal relaxation time parameters, respectively. Moreover, the interesting nature of phase-log shift is also observed in temperature and concentration pro les. Three-dimensional flow features are also labeled for velocity, temperature and concentration  fields.

This work performs a comparative investigation into Casson-Williamson fluid flow over a heated porous stretchy sheet. The energy and mass transfer equations are modeled by Cattaneo-Christov theory. The governing flow models are altered into an Ordinary Differential equation (ODE) model through proper transformations. The Homotopy Analysis Method (HAM) scheme is applied to determine the series solutions. The responses of diverse flow variables to fluid speed, fluid warmness, liquid concentration, skin friction coefficient, local Nusselt number, local Sherwood number, local entropy generation number, and Bejan number are analyzed through graphs and charts. It was found that the fluid speed would subside following an increase in values of the magnetic , field, porosity, Casson fluid, Williamson fluid, and injection/suction parameters. The fluid warmness is increased due to high-level radiation, convective heating, and heat generation/absorption parameters and it suppresses the previous highs when enriching the convective cooling parameter. The chemical reaction parameter causes an increase in the thickness of the solutal boundary layer. The larger skin friction coefficient occurs in Casson fluid than Williamson fluid. The local entropy generation is attenuated upon increase in the Casson and Williamson parameters and it aggravates when the Biot number rises. The Bejan number is elevated when the Reynolds, Brinkman, and Biot numbers experience an increase.

This study presents the characteristics of Cattaneo-Christov heat ux model for the boundary layer ow of Burgers' uid model. Instead of simple Fourier's law of heat conduction, we presented the Cattaneo-Christov model to analyze the thermal relaxation properties when the heat source/sink was present in the system. Mathematical modeling of the laws of momentum and energy were presented under the order analysis approach. It was revealed that the term \ B2 0u= " was for the hydro-magnetic rheology of the Newtonian model whereas the generalized magnetic  eld term was for the Burgers' model, which was incorporated in the current analysis. Suitable transformations were utilized for the conversion of partial di erential system into coupled nonlinear set of ordinary di erential equations, which was tackled analytically through homotopy analysis technique. The plots of various physical quantities are presented, showing the dynamics of the considered analysis. Streamlines for Burgers' and Newtonian models are presented, which show their di erence in rheology. Numerical values for skin friction and surface heat transfer rate are presented in tables.

During the homogeneous-heterogeneous autocatalytic chemical reaction in the dynamics of micropolar fluid, relaxation of heat transfer is inevitable; hence Cattaneo-Christov heat flux model is investigated in this report. In this study, radiative heat flux through an optically thick medium is treated as nonlinear due to the fact that thermal radiation at low heat energy is distinctly different from that of high heat energy, hence classical approach of using Taylor series for simplification is ignored and implicit differentiation is used leading to temperature parameter. Uniqueness of the present analysis is the consideration of cubic autocatalytic chemical reaction between the homogeneous bulk fluid and two species of catalyst at the wall. Application of similarity analysis enabled us to recast the flow equations into a set of coupled nonlinear ODEs. The resulting equations along with the appropriate conditions are solved computationally. Graphical illustrations of the effect of pertinent parameters on momentum, heat and mass boundary layers are presented and discussed. The concentration of the homogeneous bulk fluid with microstructures and catalyst at the surface decreases and increases with diffusion ratio, respectively. Buoyancy has a decreasing effect on temperature distribution.

In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation (S.D.E.). So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.