Finite element approximation for a modified anomalous subdiffusion equation

In this paper, we consider a modified anomalous subdiffusion equation (MASFE) for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. Firstly, a semi-discrete approximation for the MASFE is proposed. The stability and convergence of the semi-discrete approximation are discussed. Secondly, a finite element approximation for the MASFE is derived. The stability and convergence of the finite element approximation are investigated, respectively. Finally, some numerical examples are presented to demonstrate the effectiveness of theoretical analysis.     

A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients

The aim of this paper is to present a new numerical method for solving a wide class of fractional partial differential equations (FPDEs) such as wave-diffusion equations, modified anomalous fractional sub-diffusion equations, time-fractional telegraph equations. The proposed method is based on the Fourier series expansion along the spatial coordinate which transforms the original equation into a sequence of multi-term fractional ordinary differential equations (ODEs). These fractional equations are solved by the use of a new efficient numerical technique – the backward substitution method. The numerical examples confirm the high accuracy and efficiency of the proposed numerical scheme in solving FPDEs with variable in time coefficients.     

A low-Mach number method for the numerical simulation of complex flows

A new numerical procedure which considers a modification to the artificial acoustic stiffness correction method (AASCM) is here presented, to perform simulations of low Mach number flows with the compressible Navier–Stokes equations. An extra term is added to the energy fluxes instead of using an energy source correction term as in the original model. This new scheme re-scales the speed of sound to values similar to the flow velocity, enabling the use of larger time steps and leading to a more stable numerical method. The new method is validated performing Large Eddy Simulations on test problems. The effect of a crucial numerical parameter alpha is evaluated as well as the robustness of the method to variations of the Mach number. Numerical results are compared to the existing experimental data showing that the new method achieves good agreement increasing the time-step, and therefore accelerating the computation for low-Mach convective flows.     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

带非线性源项的Riesz回火分数阶扩散方程的预估校正方法

本文针对带非线性源项的Riesz回火分数阶扩散方程,利用预估校正方法离散时间偏导数,并用修正的二阶Lubich回火差分算子逼近Riesz空间回火的分数阶偏导数,构造出一类新的数值格式.给出了数值格式在一定条件下的稳定性与收敛性分析,且该格式的时间与空间收敛阶均为二阶.数值试验表明数值方法是有效的.     

Horizontal water flow in unsaturated porous media using a fractional integral method with an adaptive time step

Predicting the horizontal groundwater flow in unsaturated porous media is a challenge in many areas of science and engineering. The governing equation associated with this phenomenon is a nonlinear partial differential equation known as the Richards equation. However, the numerical results obtained using this equation can differ substantially from the experimental results. In order to overcome this difficulty, a new version of the Richards equation was proposed recently that considers a time derivative of fractional order. In this study, we present a numerical method for solving this fractional Richards equation. Our method comprises an adaptive time marching scheme that uses Picard iterations to solve the corresponding nonlinear equations. A computational code was implemented for the proposed method using the Scilab programming language. We performed numerical simulations of the anomalous diffusion of water in a white siliceous brick and showed that the numerical results were consistent with the available experimental data.     

A backward euler orthogonal spline collocation method for the time‐fractional Fokker–Planck equation

We formulate and analyze a novel numerical method for solving a time‐fractional Fokker–Planck equation which models an anomalous subdiffusion process. In this method, orthogonal spline collocation is used for the spatial discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence of the method are considered, and the theoretical results are supported by numerical examples, which also exhibit superconvergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1534–1550, 2015     

Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.     

Resonant Generation of Finite-Amplitude Waves by Flow Past Topography on a β-Plane

The forced Korteweg-de Vries (fKdV) equation is the generic equation for resonant flow past an obstacle. However, for flow past topography on a β-plane, the case when the upstream flow is uniform is anomalous in that there is no quadratic nonlinear term in the fKdV equation. Here we show that in this important case an alternative theory is required and obtain a new evolution equation, which has some similarities to the fKdV equation with two significant differences. These are that a small-amplitude topography now produces finite-amplitude waves and the flow response is limited by a wave breakdown characterized by an incipient flow reversal. Various numerical solutions are described.     

Solution of a new model of fractional telegraph point reactor kinetics using differential transformation method

A new model of fractional telegraph point reactor kinetics FTPRK is introduced to approximate the time dependent Boltzmann transport equation considering new terms that contain time derivative of the reactivity and fractional integral of the neutron density. Caputo fractional derivatives and fractional Leibniz rule are used for such derivation. Cattaneoequation is applied to overcome the flaw of infinite neutron velocity and to describe the anomalous transport. Effect of the new term on the neutron behaviour is discussed. The new model is applied to both TRIGA reactor and to commercial pressured water reactor of a Three Mile Island type reactor, TMI-type PWR. Results for step, ramp and sinusoidal excess reactivities with thermal hydraulic feedback are presented and discussed for different values of anomalous sub-diffusion exponent, the fractional order, 0 < µ ≤ 1. To maintain the reactor safe at start-up after insertion of step reactivity and based on the concept of prompt jump approximation, the FTPRK model is simplified and solved analytically by Mittag–Liffler function. Physical interpretations of the fractional order µ and relaxation time τ and their effects on the behaviour of the neutron population are discussed. Also, the effect of a small perturbation in the geometric buckling on the neutron behaviour is discussed for finite reactor core. The new model is solved numerically using the fractional order multi-step differential transform method MDTM. The MDTM constitutes an easy algorithm based on Taylor's formula and Caputo fractional derivative. Two theorems with their proofs are introduced to solve the fractional system. Two major disadvantages of the method about the choice of the fractional order values and the step size length are addressed. We present a procedure which enables us to solve the system with appropriate values of fraction orders.     

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two‐dimensional linear and nonlinear time‐fractional modified anomalous subdiffusion equations and time‐fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.     

A stabilized implicit fractional-step method for the time-dependent Navier–Stokes equations using equal-order pairs

A stabilized implicit fractional-step method for numerical solutions of the time-dependent Navier–Stokes equations is presented in this paper. The time advancement is decomposed into a sequence of two steps: the first step has the structure of the linear elliptic problem; the second step can be seen as the generalized Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. On the other hand, a locally stabilized term is added in the second step of the schemes. It allows one to enhance the numerical stability and efficiency by using the equal-order pairs. Convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method. Some numerical experiments are also used to demonstrate the efficiency of this new method.     

Long time behavior of non-Fickian delay reaction-diffusion equations

In this paper, we investigate the long time behavior of non-Fickian delay reaction-diffusion equations. These kinds of Volterra integro-differential equations are derived by combining a time memory term in the flux and a delay parameter in the reaction term. Energy estimates, dissipativity, asymptotic stability, and contractivity of the problems are obtained. Moreover, we prove that the numerical method discussed in the present paper has the ability to preserve stability and contractivity of the underlying systems. Some confirmations of these are illustrated by using the numerical method on two biological models.     

Implicit numerical approximation scheme for the fractional Fokker-Planck equation

In this paper, we consider an anomalous subdiffusion process, governed by fractional Fokker-Planck equation. An effective numerical method for approximating Fokker-Planck equation in a bounded domain is presented. The stability and convergence of the numerical method are analyzed. Some numerical examples are presented to show the application of the present technique. The numerical results exhibit the good performance of our theoretical analysis.     

Passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term

In this paper, the problem of passivity analysis is investigated for neutral type neural networks with Markovian jumping parameters and time delay in the leakage term. The delay is assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. By constructing proper Lyapunov–Krasovskii functional, new delay-dependent passivity conditions are derived in terms of linear matrix inequalities (LMIs). Moreover, it is well known that the passivity behavior of neural networks is very sensitive to the time delay in the leakage term. Finally, three numerical examples are given to show the effectiveness and less conservatism of the proposed method.     

HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY

In this paper, efficient numerical schemes are proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.     

Numerical method with high order accuracy for solving a anomalous subdiffusion equation

In this paper, a numerical method with second order temporal accuracy and fourth order spatial accuracy is developed to solve a anomalous subdiffusion equation; by Fourier analysis, the convergence, stability and solvability of the numerical method are analyzed; the theoretical results are strongly supported by the numerical experiment.     

UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS

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A meshless method based on the dual reciprocity method for one‐dimensional stochastic partial differential equations

This article describes a new meshless method based on the dual reciprocity method (DRM) for the numerical solution of one‐dimensional stochastic heat and advection–diffusion equations. First, the time derivative is approximated by the time–stepping method to transforming the original stochastic partial differential equations (SPDEs) into elliptic SPDEs. The resulting elliptic SPDEs have been approximated with the new method, which is a combination of radial basis functions (RBFs) method and the DRM method. We have used inverse multiquadrics (IMQ) and generalized IMQ (GIMQ) RBFs, to approximate functions in the presented method. The noise term has been approximated at the source points, at each time step. The developed formulation is verified in two test problems with investigating the convergence and accuracy of numerical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 292–306, 2016     

Multiresolution schemes for strongly degenerate parabolic equations in one space dimension

An adaptive finite volume method for one‐dimensional strongly degenerate parabolic equations is presented. Using an explicit conservative numerical scheme with a third‐order Runge‐Kutta method for the time discretization, a third‐order ENO interpolation for the convective term, and adding a conservative discretization for the diffusive term, we apply the multiresolution method combining two fundamental concepts: the switch between central interpolation or exact computing of numerical flux and a thresholded wavelet transform applied to cell averages of the solution to control the switch. Applications to mathematical models of sedimentation‐consolidation processes and traffic flow with driver reaction, which involve different types of boundary conditions, illustrate the computational efficiency of the new method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007