Caputo Time-fractional Derivative Research Articles

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.

Initial-boundary Value Problems for Fractional Diffusion Equations with Time-Dependent Coefficients

Abstract We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is attached. We prove the unique existence of weak and regular solutions.

A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation

In the present work, orthogonal spline collocation (OSC) method with convergence order O(τ3−α + hr+ 1) is proposed for the two-dimensional (2D) fourth-order fractional reaction-diffusion equation, where τ, h, r, and α are the time-step size, space size, polynomial degree of space, and the order of the time-fractional derivative (0 < α < 1), respectively. The method is based on applying a high-order finite difference method (FDM) to approximate the time Caputo fractional derivative and employing OSC method to approximate the spatial fourth-order derivative. Using the argument developed recently by Lv and Xu (SIAM J. Sci. Comput. 38, A2699–A2724, 2016) and mathematical induction method, the optimal error estimates of proposed fully discrete OSC method are proved in detail. Then, the theoretical analysis is validated by a number of numerical experiments. To the best of our knowledge, this is the first proof on the error estimates of high-order numerical method with convergence order O(τ3−α + hr+ 1) for the 2D fourth-order fractional equation.

Analysis of unsteady natural convective radiating gas flow in a vertical channel by employing the Caputo time-fractional derivative

In this paper the exact solution of the unsteady natural convection radiating flow in an open ended vertical channel is studied. The channel is stationary with non-uniform temperature. The governing equations are fractional differential equations with the Caputo time-fractional derivative. Closed form analytical solutions for the temperature and velocity fields are obtained by using the Laplace transform technique. These solutions are expressed with the Wright function, the Robotnov and Hartley function. The effects of the fractional order and physical parameters on temperature and fluid velocity are presented graphically.

A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements

The consideration of diffusion processes in magnetic resonance imaging (MRI) signal attenuation is classically described by the Bloch-Torrey equation. However, many recent works highlight the distinct deviation in MRI signal decay due to anomalous diffusion, which motivates the fractional order generalization of the Bloch-Torrey equation.In this work, we study the two-dimensional multi-term time and space fractional diffusion equation generalized from the time and space fractional Bloch-Torrey equation. By using the Galerkin finite element method with a structured mesh consisting of rectangular elements to discretize in space and the L1 approximation of the Caputo fractional derivative in time, a fully discrete numerical scheme is derived. A rigorous analysis of stability and error estimation is provided. Numerical experiments in the square and L-shaped domains are performed to give an insight into the efficiency and reliability of our method. Then the scheme is applied to solve the multi-term time and space fractional Bloch-Torrey equation, which shows that the extra time derivative terms impact the relaxation process.

Heat transfer analysis of fractional second-grade fluid subject to Newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: A comparison

The present study is a comparative analysis of unsteady flows of a second-grade fluid with Newtonian heating and time-fractional derivatives, namely, the Caputo fractional derivative (singular kernel) and the Caputo-Fabrizio fractional derivative (non-singular kernel). A physical model for second-grade fluids is developed with fractional derivatives. The expressions for temperature and velocity fields in dimensionless form as well as rates of heat transfer are determined by means of the Laplace transform technique. Solutions for ordinary cases corresponding to integer order derivatives are also obtained. Numerical computations for a comparison between the solutions of the problem with the Caputo time-fractional derivative, problem with Caputo-Fabrizio time-fractional derivative and of the ordinary fluid problem were made. The influence of some flow parameters and fractional parameter $\alpha$ on temperature field as well as velocity field was presented graphically and in tabular forms.

Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space

AbstractThe numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain into a bounded computational domain and an unbounded exterior domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary conditions. The original problem defined on the unbounded domain is thus reduced to an initial boundary value problem on a bounded computational domain. A finite difference and L1 approximation are applied for the space variables and the Caputo time-fractional derivative, respectively. Two numerical examples demonstrate the performance of the proposed method.

Two Approaches to Obtaining the Space-Time Fractional Advection-Diffusion Equation

Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically.

A second order box-type scheme for fractional sub-diffusion equation with spatially variable coefficient under Neumann boundary conditions

In the present work, a box-type difference scheme with convergence order O(tau^{2}+h^{2}) is proposed for the fractional sub-diffusion equation with spatially variable coefficient under Neumann boundary conditions. Here h, τ are space and temporal step length, respectively. The method is based on applying the L2-1_{sigma} formula to approximate the time Caputo fractional derivative and introducing the auxiliary variable. By virtue of the special properties of the L2-1_{sigma} formula and the mathematical induction method, the unconditional stability and convergence for our scheme are proved by the discrete energy method. Numerical examples are given to verify the theoretical analysis and efficiency of the box-type scheme.

Fractional Diffusion in a Solid with Mass Absorption

The space-time-fractional diffusion equation with the Caputo time-fractional derivative and Riesz fractional Laplacian is considered in the case of axial symmetry. Mass absorption (mass release) is described by a source term proportional to concentration. The integral transform technique is used. Different particular cases of the solution are studied. The numerical results are illustrated graphically.

Time fractional cable equation and applications in neurophysiology

We propose an extension of the cable equation by introducing a Caputo time fractional derivative. The fundamental solutions of the most common boundary problems are derived analytically via Laplace Transform, and result be written in terms of known special functions. This generalization could be useful to describe anomalous diffusion phenomena with leakage as signal conduction in spiny dendrites. The presented solutions are computed in Matlab and plotted.

Multi-dimensional [formula omitted]-fractional diffusion–wave equation and some properties of its fundamental solution

In this paper, a multi-dimensional α-fractional diffusion–wave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order α/2 and the Riesz space-fractional derivative of the order α so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the α-fractional diffusion–wave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the α-fractional diffusion–wave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity.

A Novel Unstructured Mesh Finite Element Method for Solving the Time-Space Fractional Wave Equation on a Two-Dimensional Irregular Convex Domain

Abstract Most existing research on applying the finite element method to discretize space fractional operators is studied on regular domains using either uniform structured triangular meshes, or quadrilateral meshes. Since many practical problems involve irregular convex domains, such as the human brain or heart, which are difficult to partition well with a structured mesh, the existing finite element method using the structured mesh is less efficient. Research on the finite element method using a completely unstructured mesh on an irregular domain is of great significance. In this paper, a novel unstructured mesh finite element method is developed for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. The novel unstructured mesh Galerkin finite element method is used to discretize in space and the Crank-Nicholson scheme is used to discretize the Caputo time fractional derivative. The implementation of the unstructured mesh Crank-Nicholson Galerkin method (CNGM) is detailed and the stability and convergence of the numerical scheme are analyzed. Numerical examples are presented to verify the theoretical analysis. To highlight the ability of the proposed unstructured mesh Galerkin finite element method, a comparison of the unstructured mesh with the structured mesh in the implementation of the numerical scheme is conducted. The proposed numerical method using an unstructured mesh is shown to be more effective and feasible for practical applications involving irregular convex domains.

Mellin integral transform approach to analyze the multidimensional diffusion-wave equations

In this paper, a family of the multidimensional time- and space-fractional diffusion-wave equations with the Caputo time-fractional derivative of the order β, 0 < β ⩽ 2 and the fractional Laplacian (−Δ)α2 with 1 < α ⩽ 2 is considered. A representation of the first fundamental solution to this equation is deduced in form of a Mellin–Barnes integral by employing the technique of the Mellin integral transform. The Mellin–Barnes representation is used to derive some new identities for the fundamental solutions in different dimensions and to identify already known and some new particular cases of the fundamental solution that have especially simple closed form.