This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.
The combined impact of a linear chemical reaction and Lorentz force on heat and mass transfer in a third-grade fluid with the Darcy–Forchheimer relation over an inclined, exponentially stretching surface embedded in a porous medium is investigated. The proposed process is mathematically expressed in terms of nonlinear and coupled partial differential equations, with the symmetry of the conditions normal to the surface. To solve the mathematical model of the proposed phenomenon, the partial differential equations are first reduced to ordinary differential equations; then, MATLAB built-in Numerical Solver bvp4c is used to obtain the numerical results of these equations. The influence of all the pertinent parameters that appeared in the flow model on the unknown material properties of interest is depicted in the forms of tables and graphs. The physical attitude of the unknown variables is discussed with physical reasoning. From the numerical solutions, it is inferred that, as Lorentz force parameter is increased, the velocity of the fluid decreases, but fluid temperature and mass concentration increase. Thisis due to the fact that Lorentz force retards the motion of fluid, and the increasing resistive force causes the rise in the temperature of the fluid. It is also noted that, owing to an increasein the magnitude of chemical reaction parameter , the velocity profile and the mass concentration decline as well, but the fluid temperature increases in a reasonable manner. It is noted that, by augmenting the values of the local inertial coefficient and the permeability parameter , the velocity field decreases, the temperature field increases, and mass concentration also increases with reasonable difference. Increasing values of Prandtl number results in a decrease in the profiles of velocity and temperature. All the numerical results are computed at the angle of inclination . The current results are compared with the available results in the existing literature for thisspecial case, and there is good agreement between them that shows the validation of the present study. All the numerical results show asymptotic behavior by satisfying the given boundary conditions.
The current study aims to investigate the thermal-diffusion and diffusion-thermo effects on heat and mass transfer in third-grade fluid with Darcy–Forchheimer relation impact over an exponentially inclined stretching sheet embedded in a porous medium. The proposed mechanism in terms non-linear and coupled partial differential equations is reduced to set of ordinary differential equations by employing an appropriate similarity variable formulation. The reduced form of equations is solved by using the MATLAB built-in numerical solver bvp4c. The numerical results for unknown physical properties such as velocity profile, temperature field, and mass concentration along with their gradients such as the skin friction, the rate of heat transfer, and the rate of mass transfer at angle of inclination α=π/6 are obtained under the impact of material parameters that appear in the flow model. The solutions are displayed in forms of graphs as well as tables and are discussed with physical reasoning. From the demonstration of the graphical results, it is inferred that thermal-diffusion parameter Sr velocity, temperature, and concentration profiles are augmented. For the increasing magnitude of the diffusion-thermo parameter Df the fluid velocity and fluid temperature rise but the opposite trend in mass concentration is noted. The current results are compared with the available results in the existing literature, and there is good agreement between them that shows the validation of the present study.
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