Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by a Caputo fractional derivative, and the second order space derivative by a symmetric fractional derivative. First, a method of separating variables expresses the analytical solution of the TSS-FDE in terms of the Mittag--Leffler function. Second, we propose two numerical methods to approximate the Caputo time fractional derivative: the finite difference method; and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

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.

Differential expression of chemokine and matrix re-modelling genes is associated with contrasting schistosome-induced hepatopathology in murine models

The pathological outcomes of schistosomiasis are largely dependent on the molecular and cellular mechanisms of the host immune response. In this study, we investigated the contribution of variations in host gene expression to the contrasting hepatic pathology observed between two inbred mouse strains following Schistosoma japonicum infection. Whole genome microarray analysis was employed in conjunction with histological and immunohistochemical analysis to define and compare the hepatic gene expression profiles and cellular composition associated with the hepatopathology observed in S. japonicum-infected BALB/c and CBA mice. We show that the transcriptional profiles differ significantly between the two mouse strains with high statistical confidence. We identified specific genes correlating with the more severe pathology associated with CBA mice, as well as genes which may confer the milder degree of pathology associated with BALB/c mice. In BALB/c mice, neutrophil genes exhibited striking increases in expression, which coincided with the significantly greater accumulation of neutrophils at granulomatous regions seen in histological sections of hepatic tissue. In contrast, up-regulated expression of the eosinophil chemokine CCL24 in CBA mice paralleled the cellular influx of eosinophils to the hepatic granulomas. Additionally, there was greater down-regulation of genes involved in metabolic processes in CBA mice, reflecting the more pronounced hepatic damage in these mice. Profibrotic genes showed similar levels of expression in both mouse strains, as did genes associated with Th1 and Th2 responses. However, imbalances in expression of matrix metalloproteinases (e.g. MMP12, MMP13) and tissue inhibitors of metalloproteinases (TIMP1) may contribute to the contrasting pathology observed in the two strains. Overall, these results provide a more complete picture of the molecular and cellular mechanisms which govern the pathological outcome of hepatic schistosomiasis. This improved understanding of the immunopathogenesis in the murine model schistosomiasis provides the basis for a better appreciation of the complexities associated with chronic human schistosomiasis.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

Differential subcellular and extracellular localisations of proteins required for insulin-like growth factor- and extracellular matrix-induced signalling events in breast cancer progression

Background: Cancer metastasis is the main contributor to breast cancer fatalities as women with the metastatic disease have poorer survival outcomes than women with localised breast cancers. There is an urgent need to develop appropriate prognostic methods to stratify patients based on the propensities of their cancers to metastasise. The insulin-like growth factor (IGF)-I:IGF binding protein (IGFBP):vitronectin complexes have been shown to stimulate changes in gene expression favouring increased breast cancer cell survival and a migratory phenotype. We therefore investigated the prognostic potential of these IGF- and extracellular matrix (ECM) interaction-induced proteins in the early identification of breast cancers with a propensity to metastasise using patient-derived tissue microarrays. Methods: Semiquantitative immunohistochemistry analyses were performed to compare the extracellular and subcellular distribution of IGF- and ECM-induced signalling proteins among matched normal, primary cancer and metastatic cancer formalin-fixed paraffin-embedded breast tissue samples. Results: The IGF- and ECM-induced signalling proteins were differentially expressed between subcellular and extracellular localisations. Vitronectin and IGFBP-5 immunoreactivity was lower while β1 integrin immunoreactivity was higher in the stroma surrounding metastatic cancer tissues, as compared to normal breast and primary cancer stromal tissues. Similarly, immunoreactive stratifin was found to be increased in the stroma of primary as well as metastatic breast tissues. Immunoreactive fibronectin and β1 integrin was found to be highly expressed at the leading edge of tumours. Based on the immunoreactivity it was apparent that the cell signalling proteins AKT1 and ERK1/2 shuffled from the nucleus to the cytoplasm with tumour progression. Conclusion: This is the first in-depth, compartmentalised analysis of the distribution of IGF- and ECM-induced signalling proteins in metastatic breast cancers. This study has provided insights into the changing pattern of cellular localisation and expression of IGF- and ECM-induced signalling proteins in different stages of breast cancer. The differential distribution of these biomarkers could provide important prognostic and predictive indicators that may assist the clinical management of breast disease, namely in the early identification of cancers with a propensity to metastasise, and/or recur following adjuvant therapy.

GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion

The efficient computation of matrix function vector products has become an important area of research in recent times, driven in particular by two important applications: the numerical solution of fractional partial differential equations and the integration of large systems of ordinary differential equations. In this work we consider a problem that combines these two applications, in the form of a numerical solution algorithm for fractional reaction diffusion equations that after spatial discretisation, is advanced in time using the exponential Euler method. We focus on the efficient implementation of the algorithm on Graphics Processing Units (GPU), as we wish to make use of the increased computational power available with this hardware. We compute the matrix function vector products using the contour integration method in [N. Hale, N. Higham, and L. Trefethen. Computing Aα, log(A), and related matrix functions by contour integrals. SIAM J. Numer. Anal., 46(5):2505–2523, 2008]. Multiple levels of preconditioning are applied to reduce the GPU memory footprint and to further accelerate convergence. We also derive an error bound for the convergence of the contour integral method that allows us to pre-determine the appropriate number of quadrature points. Results are presented that demonstrate the effectiveness of the method for large two-dimensional problems, showing a speedup of more than an order of magnitude compared to a CPU-only implementation.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.