Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describe... more Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated using the Navier–Stokes equations (NSEs) with time-fractional derivatives of order β∈(0,1). In Hγ,℘, we prove the existence and uniqueness of local and global mild solutions by using fuzzy techniques. Meanwhile, we provide a local moderate solution in Banach space. We further show that classical solutions to such equations exist and are regular in Banach space.
The current study aims to investigate the thermal-diffusion and diffusion-thermo effects on heat ... more 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 reasonin...
The combined impact of a linear chemical reaction and Lorentz force on heat and mass transfer in ... more 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 f...
In this article, we investigated the existence and uniqueness of mild solutions for fractional-or... more In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form 0 c D I γ x I = α x I + P I , x I + A I W I , I ∈ 0 , T , x I 0 = x 0 , in which γ ∈ 0 , 1 , E 1 is the fuzzy metric space and I = 0 , T is a real line interval. With the help of few conditions on functions P : I × E 1 × E 1 ⟶ E 1 , W I is control and it belongs to E 1 , A ∈ F I , L E 1 , and α stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.
This paper concerns with the existence and uniqueness of fuzzy fractional evolution equation with... more This paper concerns with the existence and uniqueness of fuzzy fractional evolution equation with uncertainty involves function of form cDαx(t)=f(t,x(t),Dβx(t)),Iαx(0)=x0,x′(0)=x1, where 1
This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy... more 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.
Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describe... more Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated using the Navier–Stokes equations (NSEs) with time-fractional derivatives of order β∈(0,1). In Hγ,℘, we prove the existence and uniqueness of local and global mild solutions by using fuzzy techniques. Meanwhile, we provide a local moderate solution in Banach space. We further show that classical solutions to such equations exist and are regular in Banach space.
The current study aims to investigate the thermal-diffusion and diffusion-thermo effects on heat ... more 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 reasonin...
The combined impact of a linear chemical reaction and Lorentz force on heat and mass transfer in ... more 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 f...
In this article, we investigated the existence and uniqueness of mild solutions for fractional-or... more In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form 0 c D I γ x I = α x I + P I , x I + A I W I , I ∈ 0 , T , x I 0 = x 0 , in which γ ∈ 0 , 1 , E 1 is the fuzzy metric space and I = 0 , T is a real line interval. With the help of few conditions on functions P : I × E 1 × E 1 ⟶ E 1 , W I is control and it belongs to E 1 , A ∈ F I , L E 1 , and α stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.
This paper concerns with the existence and uniqueness of fuzzy fractional evolution equation with... more This paper concerns with the existence and uniqueness of fuzzy fractional evolution equation with uncertainty involves function of form cDαx(t)=f(t,x(t),Dβx(t)),Iαx(0)=x0,x′(0)=x1, where 1
This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy... more 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.
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