An advanced meshless method for time fractional diffusion equation

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of fractional partial differential equations.

Velocity-jump models with crowding effects

Velocity jump processes are discrete random walk models that have many applications including the study of biological and ecological collective motion. In particular, velocity jump models are often used to represent a type of persistent motion, known as a “run and tumble”, which is exhibited by some isolated bacteria cells. All previous velocity jump processes are non-interacting, which means that crowding effects and agent-to-agent interactions are neglected. By neglecting these agent-to-agent interactions, traditional velocity jump models are only applicable to very dilute systems. Our work is motivated by the fact that many applications in cell biology, such as wound healing, cancer invasion and development, often involve tissues that are densely packed with cells where cell-to-cell contact and crowding effects can be important. To describe these kinds of high cell density problems using a velocity jump process we introduce three different classes of crowding interactions into a one-dimensional model. Simulation data and averaging arguments lead to a suite of continuum descriptions of the interacting velocity jump processes. We show that the resulting systems of hyperbolic partial differential equations predict the mean behavior of the stochastic simulations very well.

Wound healing angiogenesis : the clinical implications of a simple mathematical model

Nonhealing wounds are a major burden for health care systems worldwide. In addition, a patient who suffers from this type of wound usually has a reduced quality of life. While the wound healing process is undoubtedly complex, in this paper we develop a deterministic mathematical model, formulated as a system of partial differential equations, that focusses on an important aspect of successful healing: oxygen supply to the wound bed by a combination of diffusion from the surrounding unwounded tissue and delivery from newly formed blood vessels. While the model equations can be solved numerically, the emphasis here is on the use of asymptotic methods to establish conditions under which new blood vessel growth can be initiated and wound-bed angiogenesis can progress. These conditions are given in terms of key model parameters including the rate of oxygen supply and its rate of consumption in the wound. We use our model to discuss the clinical use of treatments such as hyperbaric oxygen therapy, wound bed debridement, and revascularisation therapy that have the potential to initiate healing in chronic, stalled wounds.

Analysis of conical-cylinder shells fundamental characteristics for structural health diagnostics

This paper describes the formulation for the free vibration of joined conical-cylindrical shells with uniform thickness using the transfer of influence coefficient for identification of structural characteristics. These characteristics are importance for structural health monitoring to develop model. This method was developed based on successive transmission of dynamic influence coefficients, which were defined as the relationships between the displacement and the force vectors at arbitrary nodal circles of the system. The two edges of the shell having arbitrary boundary conditions are supported by several elastic springs with meridional/axial, circumferential, radial and rotational stiffness, respectively. The governing equations of vibration of a conical shell, including a cylindrical shell, are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the transfer matrix of a single component has been determined, the entire structure matrix is obtained by the product of each component matrix and the joining matrix. The natural frequencies and the modes of vibration were calculated numerically for joined conical-cylindrical shells. The validity of the present method is demonstrated through simple numerical examples, and through comparison with the results of previous researchers.

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

Double diffusive magneto-convection fluid flow in a strong cross magnetic field with uniform surface heat and mass flux

In this study, magnetohydrodynamic natural convection boundary layer flow of an electrically conducting and viscous incompressible fluid along a heated vertical flat plate with uniform heat and mass flux in the presence of strong cross magnetic field has been investigated. For smooth integrations the boundary layer equations are transformed in to a convenient dimensionless form by using stream function formulation as well as the free variable formulation. The nonsimilar parabolic partial differential equations are integrated numerically for Pr ≪1 that is appropriate for liquid metals against the local Hartmann parameter ξ . Further, asymptotic solutions are obtained near the leading edge using regular perturbation method for smaller values of ξ . Solutions for values of ξ ≫ 1 are also obtained by employing the matched asymptotic technique. The results obtained for small, large and all ξ regimes are examined in terms of shear stress, τw, rate of heat transfer, qw, and rate of mass transfer, mw, for important physical parameter. Attention has been given to the influence of Schmidt number, Sc, buoyancy ratio parameter, N and local Hartmann parameter, ξ on velocity, temperature and concentration distributions and noted that velocity and temperature of the fluid achieve their asymptotic profiles for Sc ≥ 10:0.

Mixed convection flow of micropolar fluid in an open ended arc-shape cavity

Numerical simulations for mixed convection of micropolar fluid in an open ended arc-shape cavity have been carried out in this study. Computation is performed using the Alternate Direct Implicit (ADI) method together with the Successive Over Relaxation (SOR) technique for the solution of governing partial differential equations. The flow phenomenon is examined for a range of values of Rayleigh number, 102 ≤ Ra ≤ 106, Prandtl number, 7 ≤ Pr ≤ 50, and Reynolds number, 10 ≤ Re ≤ 100. The study is mainly focused on how the micropolar fluid parameters affect the fluid properties in the flow domain. It was found that despite the reduction of flow in the core region, the heat transfer rate increases, whereas the skin friction and microrotation decrease with the increase in the vortex viscosity parameter, Δ.

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].

Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

An implicit RBF meshless method for a fractal mobile/immobile transport model

Fractional differential equation is used to describe a fractal model of mobile/immobile transport with a power law memory function. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. In this paper, we firstly propose a finite difference method to discretize the time variable and obtain a semi-discrete scheme. Then we discuss its stability and convergence. Secondly we consider a meshless method based on radial basis functions (RBF) to discretize the space variable. By contrast to conventional FDM and FEM, the meshless method is demonstrated to have distinct advantages: calculations can be performed independent of a mesh, it is more accurate and it can be used to solve complex problems. Finally the convergence order is verified from a numerical example is presented to describe the fractal model of mobile/immobile transport process with different problem domains. The numerical results indicate that the present meshless approach is very effective for modeling and simulating of fractional differential equations, and it has good potential in development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations.

Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

A variable-stepsize Jacobian-free exponential integrator for simulating transport in heterogeneous porous media : application to wood drying

A Jacobian-free variable-stepsize method is developed for the numerical integration of the large, stiff systems of differential equations encountered when simulating transport in heterogeneous porous media. Our method utilises the exponential Rosenbrock-Euler method, which is explicit in nature and requires a matrix-vector product involving the exponential of the Jacobian matrix at each step of the integration process. These products can be approximated using Krylov subspace methods, which permit a large integration stepsize to be utilised without having to precondition the iterations. This means that our method is truly "Jacobian-free" - the Jacobian need never be formed or factored during the simulation. We assess the performance of the new algorithm for simulating the drying of softwood. Numerical experiments conducted for both low and high temperature drying demonstrates that the new approach outperforms (in terms of accuracy and efficiency) existing simulation codes that utilise the backward Euler method via a preconditioned Newton-Krylov strategy.

Performance assessment of exponential Rosenbrock method for large systems of ODE

This paper studies time integration methods for large stiff systems of ordinary differential equations (ODEs) of the form u'(t) = g(u(t)). For such problems, implicit methods generally outperform explicit methods, since the time step is usually less restricted by stability constraints. Recently, however, explicit so-called exponential integrators have become popular for stiff problems due to their favourable stability properties. These methods use matrix-vector products involving exponential-like functions of the Jacobian matrix, which can be approximated using Krylov subspace methods that require only matrix-vector products with the Jacobian. In this paper, we implement exponential integrators of second, third and fourth order and demonstrate that they are competitive with well-established approaches based on the backward differentiation formulas and a preconditioned Newton-Krylov solution strategy.