Caputo Time-fractional Derivative Research Articles

Casson Model of MHD Flow of SA-Based Hybrid Nanofluid Using Caputo Time-Fractional Models

In this paper MHD flow of Casson hybrid nanofluids are investigated with Caputo time-fractional derivative. Alumina (Al) and copper (Cu) are used as nanoparticles in this study with heat, mass transfer and MHD flow over a vertical channel in a porous medium. The problem is modeled using Caputo fractional derivatives and thermophysical properties of hybrid nanoparticles. The influence of concerned parameters is investigated physically and graphically on the heat, concentration and flow. The effect of volume fraction on thermal conductivity of hybrid nanofluids is observed.

Analysis of advective–diffusive transport phenomena modelled via non-singular Mittag-Leffler kernel

In this study, a linear advection–diffusion equation described by Atangana–Baleanu derivative with non-singular Mittag-Leffler kernel is considered. The Cauchy, Dirichlet and source problems are formulated on the half-line. The main motivation of this work is to find the fundamental solutions of prescribed problems. For this purpose, Laplace transform method with respect to time t and sine/cosine-Fourier transform methods with respect to spatial coordinate x are applied. It is remarkable that the obtained results are quite similar to the existing fundamental solutions of advection–diffusion equation with time-Caputo fractional derivative. Although the results are mathematically similar in both formulations, the AB derivative is a non-singular operator and provides a significant advantage in the computational processes. Therefore, it is preferable to replace the Caputo derivative in modelling such diffusive transports.

Spectral methods for the time-fractional Navier–Stokes equation

In this paper, the L1 Fourier spectral method is considered to solve the time-fractional Navier–Stokes equation with periodic boundary condition. The Fourier spectral method is employed for spatial approximation, and the L1 finite difference scheme is used to discrete the Caputo time fractional derivative. Analysis of stability and convergence are accomplished as well, leading to the conclusion that our numerical method is unconditionally stable, and the solution converges to the exact one with order O(τ2−α+N−s), where τ and N are the time step size and polynomial degree, respectively. The numerical example is provided to testify the effectiveness of our scheme, from the results of which, it turns out that the L1 Fourier spectral method is effective for solving the time-fractional Navier–Stokes equation.

Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations

In this paper, a reduced spline (RS) method based on a proper orthogonal decomposition (POD) technique for numerical solution of time fractional sub-diffusion equations is investigated. Combining POD with polynomial and non-polynomial spline methods yield a new model with lower dimensions and sufficiently high accuracy, so that the amount of computations and the calculation time decrease in comparison with usual spline methods. Although, the classical L1 scheme on uniform meshes is one of the most successful methods for approximation of the Caputo fractional derivative in time, due to the fact that the solution of time fractional sub-diffusion equations typically has a singularity at the origin, a weak convergence will conclude for such scheme. To overcome this difficulty, we use the L1 scheme on graded meshes in time, such that considered time steps are very small near the origin which compensate the singularity of the solution. Also, the error estimates between the POD approximate solution of the RS scheme and the exact solution of fractional sub-diffusion equation are established for two types of uniform and graded meshes in time. Numerical examples are given to illustrate the feasibility and efficiency of the proposed method.

Optimal chemotherapy and immunotherapy schedules for a cancer‐obesity model with Caputo time fractional derivative

This work presents a new mathematical model to depict the effect of obesity on cancerous tumor growth when chemotherapy and immunotherapy have been administered. We consider an optimal control problem to destroy the tumor population and minimize the drug dose over a finite time interval. The constraint is a model including tumor cells, immune cells, fat cells, and chemotherapeutic and immunotherapeutic drug concentrations with the Caputo time fractional derivative. We investigate the existence and stability of the equilibrium points, namely, tumor‐free equilibrium and coexisting equilibrium, analytically. We discretize the cancer‐obesity model using the L1 method. Simulation results of the proposed model are presented to compare three different treatment strategies: chemotherapy, immunotherapy, and their combination. In addition, we investigate the effect of the differentiation order α and the value of the decay rate of the amount of chemotherapeutic drug to the value of the cost functional. We find out the optimal treatment schedule in case of chemotherapy and immunotherapy.

Regularity of Solutions to Space–Time Fractional Wave Equations: A PDE Approach

Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.

A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients

A high-order compact finite difference method is proposed for solving a class of time-fractional sub-diffusion equations. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ (0, 1). The Caputo time-fractional derivative is discretized by a (3−α) th-order numerical formula (called the L2 formula here) which is constructed by piecewise quadratic interpolating polynomials but does not require any sub-stepping scheme for the approximation at the first-time level. The variable coefficient spatial differential operator is approximated by a fourth-order compact finite difference operator. By developing a technique of discrete energy analysis, a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all α ∈ (0, 1). The optimal error estimate is obtained in the L2 norm and shows that the proposed method has the temporal (3−α) th-order accuracy and the spatial fourth-order accuracy. Further approximations are also considered for enlarging the applicability of the method while preserving its high-order accuracy. Applications are given to three model problems, and numerical results are presented to demonstrate the theoretical analysis results.

MHD mixed convection Poiseuille flow in a porous medium: New trends of Caputo time fractional derivatives in heat transfer problems⋆

This paper discusses new trends of fractional derivatives in heat transfer problems. More exactly, in this work magnetohydrodynamic (MHD) mixed convection Poiseuille flow of electrically conducting, an incompressible viscous fluid with memory, in a vertical channel filled with porous medium is studied under the influence of an oscillating pressure gradient. The vertical channel is taken in stationary state with non-uniform walls temperature. The problem is formulated in terms of fractional differential equations with Caputo time fractional derivatives. The closed forms of the non-dimensional temperature, velocity, Nusselt numbers and skin friction coefficients on the walls are determined by employing the Laplace transform method. The solutions are presented in terms of the time-fractional derivative of the Wright function, Robotnov and Hartley F -function and Lorenzo-Hartley R -function. Similar solutions for ordinary fluid, corresponding to the fractional parameter equal to one, are obtained as a particular case of the fractional problem. The influences of the fractional parameter $ \alpha$ , Peclet number Pe and Reynolds number Re on the heat and momentum transfer are studied. It is found that the heat transfer can be enhanced in the fluid with memory. Fluids described with a fractional model flow faster/slower than the ordinary fluid, depending on the Reynolds number/Peclet number.

Subordination in a Class of Generalized Time-Fractional Diffusion-Wave Equations

Abstract Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

The quenching of solutions to time–space fractional Kawarada problems

Quenching solutions to a Kawarada problem with a Caputo time-fractional derivative and a fractional Laplacian are considered. The solutions to such problems may only exist locally in time when quenching occurs. Quenching and non-quenching solutions are shown to remain positive and be monotonically increasing in time under minor restrictions. Conditions for quenching to occur are demonstrated and shown to depend on the domain size.

Heat transfer and second order slip effect on MHD flow of fractional Maxwell fluid in a porous medium

This work explores the effect of second order slip on magnetohydrodynamic (MHD) flow of a fractional Maxwell fluid on a moving plate and a comparison between two numerical algorithms (Tzou and Stehfest’s algorithm). The problem is formulated for Maxwell fluid with Caputo-time fractional derivatives. Semi analytical solutions are derived using Laplace transform. The solution is obtained using numerical technique of Laplace transform Stehfest’s algorithm. To insight the physical behavior of the fluid, a graphical illustration is made. The influence of fractional parameter, the magnetic force, the porosity parameter, the slip parameters are analyzed and discussed. A tabular and graphical comparisons with Tzou’s algorithm are made for the sake of validation of the present results.

The Critical Exponent(s) for the Semilinear Fractional Diffusive Equation

In this paper we show that there exist two different critical exponents for global small data solutions to the semilinear fractional diffusive equation with Caputo fractional derivative in time. The second critical exponent appears if the second data is assumed to be zero. This peculiarity is related to the fact that the order of the equation is fractional. To prove our result, we first derive Lr-Lq linear estimates for the solution to the inhomogeneous linear Cauchy problem and then we apply a contraction argument.

An efficient finite element method for the two-dimensional nonlinear time–space fractional Schrödinger equation on an irregular convex domain

In this paper, we introduce an efficient Galerkin finite element method for solving the two-dimensional nonlinear time–space fractional Schrödinger equation, with time-dependent potential function, defined on an irregular convex domain. The Caputo time fractional derivative is discretized by the finite difference scheme, while the space fractional derivative defined on the irregular convex domain is dealt with the unstructured mesh finite element method. Given the nonlinear property of the considered equation, an efficient linearized strategy is used. To testify the efficiency of the proposed numerical scheme, two numerical examples are conducted with error and convergence analysis.

A Crank-Nicolson-type compact difference method and its extrapolation for time fractional Cattaneo convection-diffusion equations with smooth solutions

A high-order Crank-Nicolson-type compact difference method is proposed for a class of time fractional Cattaneo convection-diffusion equations with smooth solutions. The convection coefficient of the equation may be spatially variable. A suitable transformation is adopted to transform the original equation into a reaction-diffusion equation, which is then discretized by a fourth-order compact difference approximation for the spatial derivative and by a second-order Crank-Nicolson-type difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability of the method and its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. A Richardson extrapolation algorithm, including its rigorous convergence analysis, is presented. This extrapolation algorithm improves the temporal accuracy of the computed solution to the third order. An application of the proposed method to the non-smooth solution which has a weak singularity at the initial time is also discussed by introducing a correction term. Numerical results demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

Spectral method for solving the time fractional Boussinesq equation

In this paper, Fourier spectral approximation for the time fractional Boussinesq equation with periodic boundary condition is considered. The space is discretized by the Fourier spectral method and the Crank–Nicolson scheme is used to discretize the Caputo time fractional derivative. Stability and convergence analysis of the numerical method are proven. Some numerical examples are included to testify the effectiveness of our given method. Based on the presented numerical results, the Fourier spectral method is shown to be effective for solving the time fractional Boussinesq equation.

A renovated Buongiorno’s model for unsteady Sisko nanofluid with fractional Cattaneo heat flux

A renovated Buongiorno’s model with fractional differential equation is proposed to investigate the heat and mass transfer characteristics of Sisko nanofluid over a continuously moving flat plate. The fractional Cattaneo heat flux is introduced to describe the anomalous heat transport of Sisko nanofluid considering the influences of Brownian diffusion and thermophoresis. The governing boundary layer equations of continuity, momentum, energy and concentration are reduced by dimensionless variable and solved numerically. The quantities of physical interest are graphically presented and discussed in detail. It is found that the renovated model with Caputo time fractional derivatives is more capable to explain the abnormal thermal conductivity enhancement, which is also able to describe the influence of memory on the nanofluid behavior. Results indicate that the heat and mass transfer ability of Sisko nanofluid is reduced by the temperature fractional derivative parameter in both cases of shear thinning and shear thickening. Moreover, the rise of temperature fractional derivative parameter results in a significant increase of the average Nusselt number and a slight decrease of the average Sherwood number.

Entropy Generation in Different Types of Fractionalized Nanofluids

The study of classical nanofluid is limited to partial differential equations with integer-order neglecting memory effect. Fractionalized nanofluids, modeled by partial differential equations with Caputo time-fractional derivative, have the capability to address the memory effect. This article deals with the flow and entropy generation of electrically conducting different types of fractionalized nanofluids passing over an infinite vertical plate embedded in porous medium. The governing equations are transformed into dimensionless form, and then, a time-fractional model is generated using the Caputo approach. Two different nanoparticles (molybdenum disulfide and graphene oxide) are dispersed in three different base fluids (water, kerosene oil and methanol). The problem is solved for the exact solutions using the Laplace transformation technique. The impacts of fractional parameter $$\alpha $$ and volume fraction of nanoparticles $$\varphi $$ on velocity profile, entropy generation, Bejan number and the rate of heat transfer are exhibited in tabular form. Finally, the graphs are plotted for different types of nanoparticles and base fluids and discussed physically. Moreover, from present solutions, the well-known published results are recovered to validate the obtained results.

On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives

Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron’s method, unique existence for the Cauchy–Dirichlet and Cauchy–Neumann problems are proved.

A new Caputo time fractional model for heat transfer enhancement of water based graphene nanofluid: An application to solar energy

Abstract In this article the idea of Caputo time fractional derivatives is applied to MHD mixed convection Poiseuille flow of nanofluids with graphene nanoparticles in a vertical channel. The applications of nanofluids in solar energy are argued for various solar thermal systems. It is argued in the article that using nanofluids is an alternate source to produce solar energy in thermal engineering and solar energy devices in industries. The problem is modelled in terms of PDE’s with initial and boundary conditions and solved analytically via Laplace transform method. The obtained solutions for velocity, temperature and concentration are expressed in terms of Wright’s function. These solutions are significantly controlled by the variations of parameters including thermal Grashof number, Solutal Grashof number and nanoparticles volume fraction. Expressions for skin-friction, Nusselt and Sherwood numbers are also determined on left and right walls of the vertical channel with important numerical results in tabular form. It is found that rate of heat transfer increases with increasing nanoparticles volume fraction and Caputo time fractional parameters.

Insight into the Natural Convection Flow Through a Vertical Cylinder Using Caputo Time-Fractional Derivatives

The variation in temperature distribution with time for the case of the fractional model which models the flow of fluid through a vertical cylinder is considered. This article provides an insight into the natural convective flow of a viscous fluid through a vertical heated cylinder using the fractional differential equation with Caputo derivatives. Analytical solutions for temperature and velocity functions were obtained using Laplace transform and finite Hankel integral transform methods. Stehfest’s algorithm was used to obtain the inverse Laplace transforms. Numerical simulations and graphical illustrations were carried out in order to analyze the influence of the time-fractional derivative on the transport phenomenon. The significant difference between the fractional fluid flow and ordinary fluid at various time ( $$ t $$ ) is unraveled. At the initial time, the flow of fractional fluid is faster than the ordinary fluid.