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Fractional calculus has been widely used to study the flow of viscoelastic fluids recently, and fractional differential equations have attracted a lot of attention. However, the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena. In this paper, the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically. Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow. The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed. Based on the L1-approximation formula of Caputo fractional derivatives, the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented, and the numerical solutions are derived. In order to test the correctness and availability of numerical schemes, two numerical examples are established to give the exact solutions. The comparisons between the numerical solutions and the exact solutions have been made, and their high consistency indicates that the present numerical methods are effective. Then, this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions, and discusses the effects of the weight coefficient ϖ(α) in distributed order time fractional derivatives, the order α(r, t) in variable fractional order derivatives, the relaxation time λ, and the frequency ω of the periodic pressure gradient on the fluid flow velocity. Finally, the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

In this study, we analytically and numerically investigate the non-Fourier heat conduction behavior within a finite medium based on the time fractional dual-phase-lag model. Firstly, the time fractional dual-phase-lag model and the corresponding fractional heat conduction equation for short-pulse laser heating is built. Laplace and Fourier cosine transforms are performed to derive the semi-analytical expression of temperature distribution in the Laplace domain. Then, by the L1 approximation for the Caputo derivative, the finite difference algorithm is developed for the short-pulse laser heating problem. The solvability, stability, and convergence of this algorithm are also examined. Meanwhile, the efficiency and accuracy of this method have been verified by using three numerical examples. Finally, based on numerical analysis, we study the non-Fourier heat conduction behavior and discuss the effect of variability of parameters, such as fractional parameter and the ratio between the relaxation and retardation times, on the temperature distribution graphically. We believe that this analysis, besides benefiting the laser heating applications, will also provide a deep theoretical insight to interpret the anomalous heat transport process.

Laser technology has been widely used in biomedical therapies and external surgeries. To promote its applications, a good thermal model is required to analyze the temperature distribution within the living tissues. In this paper, we develop a time-space fractional hyperbolic bioheat transfer model to study the non-Fourier bioheat transfer process within the living biological tissues during laser irradiation. Based on the the L 1 approximation for the Caputo time fractional derivative and the central difference scheme for the Riesz fractional derivative, the finite difference algorithm is developed for the laser irradiation problem. The efficiency and accuracy of this method have been verified by using three numerical examples. In view of the thermophysical properties of the biological tissues, the effect of time and space fractional parameters, phase lag time and blood perfusion rate on temperature distribution within living biological tissues have been analyzed and shown graphically.

This work investigates the unsteady electroosmotic slip flow of viscoelastic fluid through a parallel plate micro-channel under combined influence of electroosmotic and pressure gradient forcings with asymmetric zeta potentials at the walls. The generalized second grade fluid with fractional derivative was used for the constitutive equation. The Navier slip model with different slip coefficients at both walls was also considered. By employing the Debye-Huckel linearization and the Laplace and sin-cos-Fourier transforms, the analytical solutions for the velocity distribution are derived. And the finite difference method for this problem was also given. Finally, the influence of pertinent parameters on the generation of flow is presented graphically.

This work numerically investigates the space fractional Navier–Stokes equations obtained through replacing Laplacian operator in Navier–Stokes equations by Riesz fractional derivatives. The pressure-driven flow between two parallel plates is solved with the finite difference method of fractional differential equations. The influence of pertinent parameters on the characteristics of fluid flow is analyzed with graphics. Further, the Levenberg–Marquardt algorithm is also proposed to estimate model parameters.

To describe the time-dependent creep behavior of viscoelastic material, fractional constitutive relation models which are represented by the fractional element networks are studied. Three sets of creep experimental data for polymer and rock are employed to demonstrate the effectiveness of these fractional derivative models. The corresponding constrained problem of nonlinear optimization is solved with an interior-point algorithm to obtain best fitting parameters of these fractional derivative models. The comparison results of measured values and calculated values versus time are displayed through graphics. The results demonstrate that the fractional PoyntingThomson model is optimal in simulating the creep behavior of viscoelastic materials. And it also shows that the interior-point method is effective in the inverse problem to estimate parameters of fractional viscoelastic models.

In this paper, we consider the two-dimensional non-linear fractional reaction-subdiffusion equation. A novel compact numerical method which is second-order temporal accuracy and fourth-order spatial accuracy is derived. The stability and convergence of the compact numerical method have been discussed rigorously by means of the Fourier method. Finally, numerical examples are presented to show the effectiveness and the high-order accuracy of the compact numerical method.

To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomalous heat conduction under the Neumann boundary condition in a semi-infinity medium is investigated. Exact solutions are obtained in series form of the H-function by using the Laplace transform method. Finally, numerical examples are presented graphically when different kinds of surface temperature gradient are given. The effects of fractional parameters are also discussed.