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This study proposes a time-stepping [Formula: see text] scheme to approximate the linear fractional dynamical systems based on nonuniform mesh. The developed numerical scheme is unconditionally stable, and exhibits second-order accuracy when the suitable graded mesh is used. A posteriori error estimation is derived for the obtained numerical scheme and the corresponding adaptive algorithm is devised. Finally, two numerical examples are provided to demonstrate the effectiveness of our approach and verify the theoretical results.

The helmet plays an important role in protection of pilot's head and enhances the pilot's capabilities and performance significantly with the use of mounted devices such as the Night Vision Goggle (NVG). However, the use of helmet-mounted devices might increase the risk of injury due to the increased helmet weight and change in the centre of gravity of head. In this study, four helmets with different combinations of mounted devices were modelled in a validated human head-neck multi-body model to analyse their effects on the pilot's neck injury during simulated ejection. The probability of neck injury was evaluated and predicted using the

In this paper, a numerical method is proposed to solve the time-space fractional diffusion equation with Robin fractional derivative boundary condition. Under the weak regularity assumptions of solution, we present a numerical scheme based on the L1 method for time discretization on graded mesh and the Grunwald-Letnikov formula for spatial discretization on uniform mesh. And a fast scheme for the considered problem is constructed based on the exponential summation approximation of the kernel function t−α. Meanwhile, a detailed analysis of stability and convergence is given. Then, the extrapolation method is applied to the space direction to make it reach the second-order accuracy. Finally, numerical experiments show that the proposed method is effective.

In this paper, we propose a space-time finite element method for the distributed-order time fractional reaction diffusion equations (DOTFRDEs). First, using the composite trapezoidal rule, composite Simpson's rule and Gauss-Legendre quadrature rule to discretize the distributed-order derivative and employing the finite element method both in space and time, three fully discrete finite element schemes for DOTFRDEs are developed. Second, for the obtained numerical schemes, the existence, uniqueness and stability are discussed. Third, under the hypothesis about the singular behavior of exact solution near t = 0 , the convergence of these numerical schemes are investigated in detail based on the graded time mesh. Forth, in order to reduce the storage requirement and computational cost of these numerical schemes, an efficient sum-of-exponentials approximation for the kernel t − α , α ∈ ( 0 , 1 ) is introduced, and the developed space-time finite element schemes are improved. At last, some numerical tests are given to verify the rationality and effectiveness of our method.

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions, which is discretized by the space–time finite element method to propagate solutions. We develop a multigrid-reduction-in-time (MGRIT) algorithm with time-dependent time-grid propagators and provide its two-level convergence theory under the assumptions of the stability and simultaneous diagonalizability on time-grid propagators. Numerical results show that the proposed method possesses the saturation error order, theoretical results of the two-level variant deliver good predictions for our model problems, and significant speedups of the MGRIT can be achieved when compared to the two-level variant with F-relaxation (an equivalent version of the parareal algorithm) and the sequential time-stepping approach.

In this paper, we study the following time-fractional Feynman-Kac equation $$\begin{aligned} {_\sigma ^CD_t^{\alpha }G(x,t)}-\Delta G(x,t)=f(x,t),~~~ 0 0. \end{aligned}$$ As is well known, the optimal rate of convergence $$\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) $$ with $$\sigma =0$$ on graded meshes has been proved in [Stynes et al., SIAM J. Numer. Anal. 55, 1057–1079 (2017)] by L1 scheme. However, there are still some significant differences when $$\sigma >0$$ . More concretely, it shall drop down to the $$\mathcal {O}\left( \tau ^{\min \{1,~r\alpha \}}\right) $$ by the implicit L1 scheme. This motivates us to design the implicit-explicit L1 scheme, which recovers a convergence rate $$\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) $$ on graded meshes. Finally, numerical experiments are given to illustrate theoretical results.

In this paper, we propose a space–time finite element method for the multi-term time–space fractional diffusion equation. First, some fractional derivative spaces are listed, some properties of the fractional derivatives and fractional derivative spaces are given, some definitions and properties of finite element spaces are introduced, and a space–time finite element fully discrete scheme for the considered problem is developed. Second, the existence, uniqueness and stability of the obtained numerical scheme are discussed. Third, under the hypothesis about singular behavior of exact solution near t = 0 , the convergence is investigated in detail based on the suitable graded time mesh. At last, some numerical tests are given to verify the rationality and effectiveness of our method.

This paper considers the nonlinear stochastic fractional integro-differential equations (SFIDEs) under the non-Lipschitz conditions, which are general and include many stochastic (fractional) integro-differential equations discussed in literature. An important connection between SFIDEs and stochastic Volterra integral equations (SVIEs) is derived in detail by the Fubini theorem. Using the Euler–Maruyama (EM) approximation, we prove the existence, uniqueness and stability results of the solution to SFIDEs. Moreover, it is shown that the modified EM solution of SFIDEs shares strong first-order sharp convergence. The numerical examples are performed to show the accuracy and effectiveness of the numerical scheme and verify the correctness of our theoretical analysis.

In this paper, a class of fractional advection–dispersion equations with Caputo tempered fractional derivative are considered numerically. An efficient algorithm for the evaluation of Caputo tempered fractional derivative is proposed to sharply reduce the computational work and storage, and this is of great significance for large-scale problems. Based on the nonsmooth regularity assumptions, a semi-discrete form is obtained by finite difference method in time, and its stability and convergence are investigated. Then by finite element method, we derive the corresponding fully discrete scheme and discuss its convergence. At last, some numerical examples, based on different domains, are presented to demonstrate effectiveness of numerical schemes and confirm the theoretical analysis.

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when \begin{document}$ \alpha = 1 $\end{document}. Finally, several numerical tests are reported for verification of the theoretical findings.

In this paper, we develop an h-p version of finite element method for one-dimensional fractional differential equation ?0Dx?u+Au=f(x)$-_{0}D_{x}^{\alpha }u+Au=f(x)$ with Dirichlet boundary conditio...

In this paper, we consider the Galerkin finite element method for the two-dimensional space and time fractional Bloch–Torrey equation (2D-STFBTE). Utilizing the L 2 − 1 σ formula to discretize temporal Caputo derivative, we obtain the local truncation error with O ( τ 3 − α ) in the temporal direction, where α is the order of time fractional derivative. Furthermore, the semi-discrete form for the problem is given and, the stability and convergence of the semi-discrete variational formulation are rigorously proved in L 2 norm and fractional norm respectively. Then, we derive a fully discrete scheme for the 2D-STFBTE and investigate its convergence theoretically. Finally, extensive numerical results based on linear piecewise polynomials verify the theoretical results and show the effectiveness of the proposed scheme.

This paper proposes a numerical approximation combining finite difference method in time and finite element method in space to solve the two-dimensional time-space fractional Bloch-Torrey equation. Unlike the existing works which focus on the decoupled model, the proposed numerical scheme is established and discussed based on the coupled equations which is from the separation of real and imaginary parts of the original fractional Bloch-Torrey model. In addition, in order to get fast estimation to Caputo fractional derivative and reduce the storage of numerical scheme, an efficient sum-of-exponentials approximation for the kernel t − α , α ∈ ( 0 , 1 ) is adopted. Under the non-uniform time mesh, the stability and convergence are discussed for the semi-discrete scheme, and the error estimate is investigated for the fully discrete scheme in detail. Finally, several numerical tests are provided to verify the correctness of the obtained theoretical results and the effectiveness of our method.

In this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the...

In this paper, we study the HIV infection model based on fractional derivative with particular focus on the degree of T-cell depletion that can be caused by viral cytopathicity. The arbitrary order of the fractional derivatives gives an additional degree of freedom to fit more realistic levels of CD4+ cell depletion seen in many AIDS patients. We propose an implicit numerical scheme for the fractional-order HIV model using a finite difference approximation of the Caputo derivative. The fractional system has two equilibrium points, namely the uninfected equilibrium point and the infected equilibrium point. We investigate the stability of both equilibrium points. Further we examine the dynamical behavior of the system by finding a bifurcation point based on the viral death rate and the number of new virions produced by infected CD4+ T-cells to investigate the influence of the fractional derivative on the HIV dynamics. Finally numerical simulations are carried out to illustrate the analytical results.

In this paper, a class of distributed-order time fractional diffusion equations (DOFDEs) on bounded domains is considered. By L1 method in temporal direction, a semi-discrete variational formulation of DOFDEs is obtained firstly. The stability and convergence of this semi-discrete scheme are discussed, and the corresponding fully discrete finite element scheme is investigated. To improve the convergence rate in time, the weighted and shifted Grunwald difference method is used. By this method, another finite element scheme for DOFDEs is obtained, and the corresponding stability and convergence are considered. And then, as a supplement, a higher order finite difference scheme of the Caputo fractional derivative is developed. By this scheme, an approach is suggested to improve the time convergence rate of our methods. Finally, some numerical examples are given for verification of our theoretical analysis.

Objective to study the changes of physical status under the condition of lacking food and water on oxygen-deficient plateau. Methods 2 volunteers survived for 72 h with little water, pilot biscuit and emergency oxygen supply on the high altitude plateau by themselves, using sleeping bags and chemical heating bags to keep out the cold at night. The volunteers were asked not to eat biscuit, drink water and use oxygen unless extremely needed, and the amount of biscuit and water the volunteer took were recorded. The volunteers survived at 5237 m altitude for 18 h first, and then walked down to 4742 m altitude and survived for 54 h continuously. The body composition indexes and physiological parameters were monitored. Results Under the condition of lacking food and water on oxygen-deficient plateau for 72 h, the physical status of the 2 volunteers changed to be feeble, but they could walk and do some light work normally. Their lips were dry and cracked, but no serious changes occurred except fever. The total amounts of biscuit and water were 60 g and 650 ml for one volunteer, and 94 g and 950 ml for the other. For both volunteers, the fat percentage declined, while the muscle percentage rose in the whole survival time. Their body temperatures rose in the second day, and the heart rates and blood pressures did the same in the second and the third days. The oxygen saturations declined significantly after 6 h and recovered after 2 days for one volunteer, while declined slightly in the all 3 days for the other. Conclusion The human physical status would decline under the condition of lacking food and water on oxygen-deficient plateau which was higher than 4700 m. With a little emergency oxygen supply and not much more activities, young people could still sustain for 72 h without serious states.

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order $$0

Due to the successful applications in engineering, physics, biology, finance, etc., there has been substantial interest in fractional diffusion equations over the past few decades, and literatures on developing and analyzing efficient and accurate numerical methods for reliably simulating such equations are vast and fast growing. This paper gives a concise overview on finite element methods for these equations, which are divided into time fractional, space fractional and time-space fractional diffusion equations. Besides, we also involve some relevant topics on the regularity theory, the well-posedness, and the fast algorithm.

In this work, we develop a finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction–diffusion equation (2D-DOTSFRDE) with low regularity solution at the initial time. A fast evaluation of the distributed-order time fractional derivative based on graded time mesh is obtained by substituting the weak singular kernel for the sum-of-exponentials. The stability and convergence of the developed semi-discrete scheme to 2D-DOTSFRDE are discussed. For the spatial approximation, the finite element method is employed. The convergence of the corresponding fully discrete scheme is investigated. Finally, some numerical tests are given to verify the obtained theoretical results and to demonstrate the effectiveness of the method.

In this paper, a class of two-dimensional space and time fractional Bloch-Torrey equations (2D-STFBTEs) are considered. Some definitions and properties of fractional derivative spaces are presented. By finite difference method and Galerkin finite element method, a semi-discrete variational formulation for 2D-STFBTEs is obtained. The stability and convergence of the semi-discrete form are discussed. Then, a fully discrete scheme of 2D-STFBTEs is derived and the convergence is investigated. Finally, some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis.

The backward Euler and Crank-Nicolson-Galerkin fully-discrete approximate schemes for two-dimensional space-fractional advection-dispersion equations are established. Firstly, we prove that the corresponding variational problem has a unique solution, and the proposed fully-discrete schemes are unconditionally stable, whose solutions are all unique. Secondly, the optimal error estimates are derived by use of properties of projection operator and fractional derivatives. Finally, numerical examples demonstrate effectiveness of numerical schemes and confirm the theoretical analysis.

In this paper, a two-dimensional fractional FitzHugh-Nagumo monodomain model (2D-FFHNMM) with zero Dirichlet boundary condition is considered. The model consists of a coupled two-dimensional space fractional nonlinear reaction-diffusion model (2D-SFNRDM) and an ordinary differential equation. The 2D-SFNRDM and ordinary differential equation are decoupled at each time step. A new Crank-Nicolson alternating direction implicit (ADI) Galerkin finite element method for the 2D-SFNRDM is developed. The stability and convergence of the numerical method are discussed. Finally, some numerical examples on 2D-SFNRDM and 2D-FFHNMM are given for verification of our theoretical analysis.

It is well accepted that fractional partial differential equations (FPDE) can be used to model many processes for which the normal partial differential equations (PDE) fail to describe precisely. Numerical approaches seem to be promising alternatives when exact solution of FPDE is difficult to derive. However, numerical solution of FPDE encounters new challenges brought in by the fractional order derivatives. In this paper we consider the 2D time dependent fractional diffusion–wave equation (FDWE) with Caputo derivative in temporal direction. We discretize the fractional order derivative with finite difference method (FDM) and present a moving least squares (MLS) meshless approximation in spatial directions which can be used to handle more complex problem domain. The convergence and stability properties of semi-discretized scheme related to time are theoretically analyzed. Finally, we conduct several numerical experiments to test our method for both regular and irregular node point distribution on rectangular and circular domain. The results indicate that the proposed method is accurate and efficient.

In this article, a class of two-dimensional Riesz space fractional diffusion equations is considered. Some fractional spaces are established and some equivalences between fractional derivative spaces and fractional Sobolev space are presented. By the Galerkin finite element method and backward difference method, a fully discrete scheme is obtained. According to Lax–Milgram theorem, the existence and uniqueness of the solution to the fully discrete scheme are investigated. The stability and convergence of the scheme are also derived. Finally, some numerical examples are given for verification of our theoretical analysis.

To explore the impact of man-seat interfaces on long-time-sitting fatigue and comfort, a simulating flight facility in which can fix three different seats(A, B and C) was set up. 4 volunteers carried out simulated flight operation in it for 8 h continuously. 8 objective indicators were monitored, and the subjective fatigue and discomfort feelings were investigated too. The results showed 5 objective indicators were significant to long-time-sitting fatigue. Visual sensitivity declined along with the increase of fatigue. The fatigue feelings of neck, back, waist, buttocks and thighs all had an increasing tendency with the extending of sitting time. The comfort of seat A was superior to that of seat B, and seat C was superior to both A and B. Adjustable lumbar pad could effectively improve comfort, and a modeling designed man-seat interface could do that significantly.

To investigate the protective effects and related mechanisms of soybean isoflavone (SI) on human umbilical vein endothelial cells (HUVECs) injury induced by H₂O₂ and lipopolysaccharide (LPS).H₂O₂ and LPS were used to induce HUVECs injury in vitro. Nine experimental groups were examined: control group, H₂O₂ (2 mmol/L for 4 h), LPS (2 mmol/L for 4 h), H₂O₂+low dose SI (1 mg/ml), H₂O₂+moderate dose SI (2.5 mg/ml), H₂O₂+high dose SI (5 mg/ml), LPS+low dose SI (1 mg/ml), LPS+moderate dose SI (2.5 mg/ml), LPS+high dose SI (5 mg/ml). The survival ratio of HUVECs was detected with MTT assay. The cultured cells were loaded by Fura-2/AM and the change of [Ca²⁺] in HUVECs was measured by fluorospectrophotometry. The contents of malondialdehyde (MDA), superoxide dismutase (SOD), reduced glutathione (GSH-Px) were measured by the commercial kits. The levels of tissue plasminogen activator IL-6 in the supematant were measured by enzyme linked immunosorbent assay (ELISA) kits. Apoptosis rate of the HUVECs was analyzed by flow cytometry.H₂O₂ and LPS significantly decreased HUVECs viability, increased the contents of MDA, IL-6 and decreased the contents of SOD and GSH-Px, and increased the apoptosis rate [(37.8 ± 1.8)% and (38.9 ± 1.1)%]. Co-treatment with SI could reduce MDA and IL-6 while increase SOD and GSH-Px and reduce apoptosis in a dose-dependent manner.The findings demonstrate that soybean isoflavone could attenuate H₂O₂ and LPS induced injury in human umbilical vein endothelial cells through protecting mitochondrial function, improving antioxygenic activity, and suppressing the mobilization of cytosolic calcium.

In this paper, the implicit difference scheme is presented to solve a class of space fractional advection-diffusion equations with variable coefficients. The stability and convergence of this method is discussed respectively. Finally, we give several numerical examples to confirm our theoretical analysis.

In this paper, a class of multi-term time fractional advection diffusion equations (MTFADEs) is considered. By finite difference method in temporal direction and finite element method in spatial direction, two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained. The stability and convergence of these numerical schemes are discussed. Next, a V-cycle multigrid method is proposed to solve the resulting linear systems. The convergence of the multigrid method is investigated. Finally, some numerical examples are given for verification of our theoretical analysis.