An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations

In this work, we present a new implicit numerical scheme for fractional subdiffusion equations. In this approach we use the Keller Box method [1] to spatially discretise the fractional subdiffusion equation and we use a modified L1 scheme (ML1), similar to the L1 scheme originally developed by Oldham and Spanier [2], to approximate the fractional derivative. The stability of the proposed method was investigated by using Von-Neumann stability analysis. We have proved the method is unconditionally stable when $0<{\lambda}_q <\min(\frac{1}{\mu_0},2^\gamma)$ and $0<\gamma \le 1$, and demonstrated the method is also stable numerically in the case $\frac{1}{\mu_0}<{\lambda}_q \le 2^\gamma$ and $\log_3{2} \le \gamma \le 1$. The accuracy and convergence of the scheme was also investigated and found to be of order $O(\Delta t^{1+\gamma})$ in time and $O(\Delta x^2)$ in space. To confirm the accuracy and stability of the proposed method we provide three examples with one including a linear reaction term

The Implicit Keller Box method for the one dimensional time fractional diffusion equation

Abstract There are a number of physical situations that can be modeled by fractional partial differential equations. In this paper, we discuss a numerical scheme based o

Numerical investigation of two models of nonlinear fractional reaction subdiffusion equations

We consider new numerical schemes to solve two different systems of nonlinear fractional reaction subdiffusion equations. These systems of equations model the reversible reaction A+B⇌C in the presence of anomalous subdiffusion. The first model is based on the Henry \& Wearne [1] model where the reaction term is added to the subdiffusion equation. The second model is based on the model by Angstmann, Donnelly \& Henry [2] which involves a modified fractional differential operator. For both models the Keller Box method [3] along with a modified L1 scheme (ML1), adapted from the Oldham and Spanier L1 scheme [4], are used to approximate the spatial and fractional derivatives respectively. Numerical prediction of both models were compared for a number of examples given the same initial and boundary conditions and the same anomalous exponents. From the results, we see similar short time behaviour for both models predicted. However for long times the solution of the second model remains positive whilst the Henry \& Wearne based–model predictions may become negative

Numerical solution methods for fractional partial differential equations

Fractional partial differential equations have been developed in many different fields such as physics, finance, fluid mechanics, viscoelasticity, engineering and biology. These models are used to describe anomalous diffusion. The main feature of these equations is their nonlocal property, due to the fractional derivative, which makes their solution challenging. However, analytic solutions of the fractional partial differential equations either do not exist or involve special functions, such as the Fox (H-function) function (Mathai & Saxena 1978) and the Mittag-Leffler function (Podlubny 1998) which are diffcult to evaluate. Consequently, numerical techniques are required to find the solution of fractional partial differential equations. This thesis can be considered as two parts, the first part considers the approximation of the Riemann-Liouville fractional derivative and the second part develops numerical techniques for the solution of linear and nonlinear fractional partial differential equations where the fractional derivative is defied as a Riemann-Liouville derivative. In the first part we modify the L1 scheme, developed initially by Oldham & Spanier (1974), to develop the three schemes which will be defined as the C1, C2 and C3 schemes. The accuracy of each method is considered. Then the memory effect of the fractional derivative due to nonlocal property is discussed. Methods of reduction of the computation L1 scheme are proposed using regression approximations. In the second part of this study, we consider numerical solution schemes for linear fractional partial differential equations. Here the numerical approximation schemes are developed using an approximation of the fractional derivative and a spatial discretization scheme. In this thesis the L1, C1, C2, C3 fractional derivative approximation schemes, developed in the first part of the thesis, are used in conjunction with either the Centred-finite difference scheme, the Dufort-Frankel scheme or the Keller Box scheme. The stability of these numerical schemes are investigated via the technique of the Fourier analysis (Von Neumann stability analysis). The convergence of each the numerical schemes is also discussed. Numerical tests were used to conform the accuracy and stability of each proposed method. In the last part of the thesis numerical schemes are developed to handle nonlinear partial differential equations and systems of nonlinear fractional partial differential equations. We considered two models of a reversible reaction in the presence of anomalous subdiffusion. The Centred-finite difference scheme and the Keller Box methods are used to spatially discretise the spatial domain in these schemes. Here the L1 scheme and a modification of the L1 scheme are used to approximate the fractional derivative. The accuracy of the methods are discussed and the convergence of the scheme are demonstrated by numerical experiments. We also give numerical examples to illustrate the e�ciency of the proposed scheme

Computation of Eyring-Powell micropolar convective boundary layer flow from an inverted non-isothermal cone : thermal polymer coating simulation

Thermal coating of components with non-Newtonian materials is a rich area of chemical and process mechanical engineering. Many different rheological characteristics can be simulated for such coatings with a variety of different mathematical models. In this work we study the steady-state coating flow and heat transfer of a non-Newtonian liquid (polymer) on an inverted isothermal cone with variable wall temperature. The Eringen micropolar and three-parameter Eyring-Powell models are combined to simulate microstructural and shear characteristics of the polymer. The governing partial differential conservation equations and wall and free stream boundary conditions are rendered into dimensionless form and solved computationally with the KellerBox finite difference method (FDM). Validation with earlier Newtonian solutions from the literature is also included. Graphical and tabulated results are presented to study the variations of fluid velocity, micro-rotation (angular velocity), temperature, skin friction, wall couple stress (micro-rotation gradient) and wall heat transfer rate. With increasing values of the first Eyring-Powell parameter temperatures are elevated, micro-rotation is suppressed and velocities are enhanced near the cone surface but reduced further into the boundary layer. Increasing values of the second Eyring-Powell parameter induce strong flow deceleration, decrease temperatures but enhance micro-rotation values. An increase in non-isothermal power law index suppresses velocities, temperatures and micro-rotations i.e. all transport characteristics are maximum for the isothermal case (n =0). Increasing Eringen vortex viscosity parameter significantly enhances temperatures and also micro-rotations. The present numerical simulations find applications in thermal polymer coating operations and industrial deposition techniques and provide a useful benchmark for more general computational fluid dynamics (CFD) simulations

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

Brownian motion: a paradigm of soft matter and biological physics

This is a pedagogical introduction to Brownian motion on the occasion of the 100th anniversary of Einstein's 1905 paper on the subject. After briefly reviewing Einstein's work in its contemporary context, we pursue some lines of further developments and applications in soft condensed matter and biology. Over the last century Brownian motion became promoted from an odd curiosity of marginal scientific interest to a guiding theme pervading all of the modern (live) sciences.Comment: 30 pages, revie

Diffusion of Lipids and Proteins in Complex Membranes

Integral membrane proteins are tiny factories with big responsibilities in signaling and transport. These proteins are constantly looking for oligomerization partners and favorable lipid environments to perform their functions that are critical for our health. The search processes are driven by thermally-agitated lateral diffusion. Cellular membranes are crowded and highly heterogeneous entities. Their structure is assumed to couple to the dynamics of molecules within the membrane, rendering diffusion therein complex too. Clarifying this connection can help us to grasp how cells regulate dynamic processes by locally varying their membrane properties, and how this further affects protein function. Unfortunately, despite persistent experimental work, our understanding of this structure–dynamics–function coupling remains poor.In this Thesis, we present our findings on how protein crowding and lipid packing affect the lateral dynamics of lipids and proteins in membranes and monolayers. We have employed molecular dynamics simulations using both atomistic and coarsegrained models to resolve how the rate and nature of diffusion are affected by these two factors. We also advanced the related methodology, which turned out to be beneficial for studying lipid membranes that are crowded with proteins.We find that crowding and packing slow down lipid and protein diffusion and extend the anomalous diffusion regime. We demonstrate that models used to predict diffusion coefficients of lipids and proteins struggle in such conditions. Finally, we observe that protein crowding effects non-Gaussian diffusion that does not follow the diffusion mechanism observed for protein-free bilayers, nor any other known mechanism.Our observations help us understand the dynamics in crowded membranes, and hence shed light on the kinetics of numerous membrane-mediated phenomena. The findings suggest that normal diffusion is likely absent in the membranes of living cells, where the motion of each lipid and protein is heavily affected by its heterogeneous surroundings. The results also pave the way towards understanding central processes in the utterly complex plasma membranes of living cells. Here, the possible future applications lie in pharmaceuticals that affect protein function by disturbing the formation of functional protein–protein or protein–lipid units by perturbing the dynamic properties of the membranes and monolayers