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Processes described by the advection-diffusion-reaction (ADR) equation often arise in heterogeneous media. Of particular interest is the case of layered or otherwise composite media exhibiting many layers or other intricate geometric complexity. Numerical models for processes in these types of media are generally accompanied by a significant computational cost due to the large number of grid nodes required to capture the detail of the heterogeneous domain. An approach addressing this issue involves obtaining approximate solutions on a coarse grid comprised of a small number of nodes and reconstructing a solution on a fine grid comprising a large number of nodes. In this paper, we discuss a projection-based framework for coarse-to-fine-grid approximation methods, which is used to derive a dual-grid mapping method. We initially utilise an interpolation-based mapping, and then consider a new discretisation-based adaptive mapping defined through the solution of coupled problems on subsets of the fine grid. With our new adaptive dual-grid mapping method, we are able to accurately approximate complex solutions on very coarse grids and reconstruct much finer scale approximations on fine grids. Code implementing the developed methods is provided and also applied to several test cases, where accuracy and significantly reduced computational cost are demonstrated in comparison to a numerical solution applied directly on the fine grid. [ABSTRACT FROM AUTHOR]

In this paper, we consider the perceived effectiveness of using industry-based transdisciplinary, complex problems as a vehicle for final year undergraduate mathematics students to apply their knowledge and skills in realistic settings, preparing them for the world of work. The discussion centres on a recently designed, implemented and evaluated model for a capstone subject incorporating a work integrated learning project in a Bachelor of Mathematics degree. To provide background for the discussion, we present the motivation for introducing such learning activities into the programme as well as the design of the model itself. We also report on the outcomes of implementation of the Work Integrated Learning experiences, discussing participant perspectives and describing lessons learnt over the two years. We conclude with a set of identified characteristics of capstone experiences in mathematics, and with a call to re-conceptualize the ways design teams collaborate to support academics. [ABSTRACT FROM AUTHOR]

The purpose of this research is to establish the generalised fractional Bloch-Torrey equation for better simulating anomalous diffusion in heterogeneous biological tissues. The introduction of the distributed-order time fractional derivative allows for an improved interpretation of the complex diffusion behaviours with multi-scale effects. The use of variable coefficients in the model increases its applicability for describing the spatial heterogeneity evident in the cellular structures. The proposed distributed-order time and space fractional Bloch-Torrey equation is discretised in time and space by the L 2-1 σ formula and the finite element method, respectively. The stability and convergence analyses of these numerical methods are provided. To further improve the computational efficiency, a reduced-order extrapolation scheme is developed. We verify the effectiveness of the proposed methods by numerical examples. Moreover, the coupled fractional dynamic system solution behaviour is explored on a human brain-like domain divided into the white matter and grey matter regions. Compared with the model having constant coefficients, solution behaviours suggest that variable diffusion coefficients offer an effective way to differentiate the distinct diffusion phenomena evolving in different tissue micro-environments. Furthermore, to evaluate the impacts of the weight function in the distributed-order operator, we choose three types of beta distributions with the same mean but different values of the variance. The results indicate that the larger value of the variance leads to a more remarkable fluctuation and a slower decay of the transverse magnetisation. This generalised distributed-order fractional model may provide further insights into capturing anomalous diffusion in heterogeneous media. • A distributed-order time-space FBTE with variable coefficients is developed. • The fractional dynamic system is solved by the L 2 − 1 σ formula and FEM. • The stability and convergence of numerical methods are provided. • An improved fast algorithm reduces the computational cost. • This research provides more insights into capturing the diffusion complexity. [ABSTRACT FROM AUTHOR]

Realistic virtual models of leaf surfaces are important for several applications in the plant sciences, such as modelling agrichemical spray droplet movement and spreading on the surface. In this context, the virtual surfaces are required to be smooth enough to facilitate the use of the mathematical equations that govern the motion of the droplet. Although an effective approach is to apply discrete smoothing D[sup 2]-spline algorithms to reconstruct the leaf surfaces from three-dimensional scanned data, difficulties arise when dealing with wheat (Triticum aestivum L.) leaves, which tendtotwist and bend. To overcome this topological difficulty, we develop a parameterisation technique that rotates and translates the original data, allowing the surface to be fitted using the discrete smoothing D[sup 2]-spline methods in the new parameter space. Our algorithm uses finite element methods to represent the surface as a linear combination of compactly supported shape functions. Numerical results confirm that the parameterisation, along with the use of discrete smoothing D[sup 2]-spline techniques, produces realistic virtual representations of wheat leaves. [ABSTRACT FROM AUTHOR]

The foliage of a plant performs vital functions. As such, leaf models are required to be developed for modeling the plant architecture from a set of scattered data captured using a scanning device. The leaf model can be used for purely visual purposes or as part of a further model, such as a fluid movement model or biological process. For these reasons, an accurate mathematical representation of the surface and boundary is required. This paper compares three approaches for fitting a continuously differentiable surface through a set of scanned data points from a leaf surface, with a technique already used for reconstructing leaf surfaces. The techniques which will be considered are discrete smoothing D²-splines [R. Arcangéli, M. C. Lopez de Silanes, and J. J. Torrens, Multidimensional Minimising Splines, Springer, New York, 2004], the thin plate spline finite element smoother [S. Roberts, M. Hegland, and I. Altas, SIAM J. Numer. Anal., 1 (2003), pp. 208-234], and the radial basis function Clough-Tocher method [M. N. Oqielat, I. W. Turner, and J. A. Belward, Appl. Math. Model., 33 (2009), pp. 2582-2595]. Numerical results show that discrete smoothing D²-splines produce reconstructed leaf surfaces which better represent the original physical leaf. [ABSTRACT FROM AUTHOR]

Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data. Author summary: Heterogeneity is a dominant factor in the behaviour of many biological and biophysical processes, and is often a primary source of the variability evident in experimental data. Despite this, it is relatively rare for mathematical models of biological systems to incorporate variability in model parameters as a source of noise. Therefore, methods for analysing whether model parameters and sources of variability are identifiable from commonly reported experimental data are relatively underdeveloped. As we demonstrate, such identifiability analysis is vital for model selection, experimental design, and gaining biological insights. In this work, we develop a fast, approximate framework for model calibration and identifiability analysis of mathematical models that incorporate biological heterogeneity through random parameters. Our method is highly flexible, and can be employed in both frequentist and Bayesian inference paradigms. Compared to alternative approaches, our approach is computationally efficient, with a computational cost comparable to analysis of standard models that neglect parameter variability. [ABSTRACT FROM AUTHOR]

Abstract: Surface interpolation finds application in many aspects of science and technology. Two specific areas of interest are surface reconstruction techniques for plant architecture and approximating cell face fluxes in the finite volume discretisation strategy for solving partial differential equations numerically. An important requirement of both applications is accurate local gradient estimation. In surface reconstruction this gradient information is used to increase the accuracy of the local interpolant, while in the finite volume framework accurate gradient information is essential to ensure second order spatial accuracy of the discretisation. In this work two different least squares strategies for approximating these local gradients are investigated and the errors associated with each analysed. It is shown that although the two strategies appear different, they produce the same least squares error. Some carefully chosen case studies are used to elucidate this finding. [Copyright &y& Elsevier]

A mesoscopic drying model that enables the drying simulation of quartersawn and flatsawn wood sections consisting of several growth rings is presented. The procedure to generate the virtual board description directly from real sample images is also described. This virtual structure accommodates the prominent sample features, including its geometrical and physical properties, together with the density and structural variation across the growth rings. We give a synopsis of the sophisticated techniques developed specifically to generate this virtual description and exhibit the final computational meshes produced by the software for quartersawn and flatsawn sections. Low temperature drying simulations are then performed for both heterogenous and homogeneous model variants using these virtual descriptions and comparisons are made of the resulting MC field evolution. A highlight of these comparisons is that the heterogeneous model captures realistic drying effects, including the fast drying of earlywood and the late removal of liquid water in latewood. In comparing the drying of quartersawn and flatsawn boards we conclude that the effect of the heterogeneous nature of the MC fields is diminished somewhat when considering the flatsawn section over the quartersawn section. [ABSTRACT FROM AUTHOR]

An unstructured mesh finite volume discretisation method for simulating diffusion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume finite-element method and it retains the local continuity of the flux at the control volume faces. A least squares function reconstruction technique together with a new flux decomposition strategy is used to obtain an accurate flux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it significantly increases the accuracy of the previous lineai- scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality. [ABSTRACT FROM AUTHOR]

• Implicit reconstruction can be accurate and efficient for thin leaf surfaces. • Whole plant reconstructions are handled naturally, including clustering of leaves. • Reconstructed leaf surfaces have continuous mean curvature. Thin surfaces, such as the leaves of a plant, pose a significant challenge for implicit surface reconstruction techniques, which typically assume a closed, orientable surface. We show that by approximately interpolating a point cloud of the surface (augmented with off-surface points) and restricting the evaluation of the interpolant to a tight domain around the point cloud, we need only require an orientable surface for the reconstruction. We use polyharmonic smoothing splines to fit approximate interpolants to noisy data, and a partition of unity method with an octree-like strategy for choosing subdomains. This method enables us to interpolate an N -point dataset in O (N) operations. We present results for point clouds of capsicum and tomato plants, scanned with a handheld device. An important outcome of the work is that sufficiently smooth leaf surfaces are generated that are amenable for droplet spreading simulations. [ABSTRACT FROM AUTHOR]

• A new time-space variable-order fractional Bloch-Torrey equation is developed. • Some appropriate correction terms are introduced to deal with non-smooth solutions. • A finite volume method is employed for solving the fractional dynamical system. • This fractional model can better capture the diffusion in heterogeneous media. A new generalised two-dimensional time and space variable-order fractional Bloch-Torrey equation is developed in this study. The variable-order Riesz fractional derivative and variable diffusion coefficient are introduced to simulate diffusion phenomena in heterogeneous, irregularly shaped biological tissues. The fractional Bloch-Torrey equation is discretised by the weighted and shifted Grünwald-Letnikov formula with respect to time and by finite volume method with respect to space. Additionally, to improve the accuracy of the numerical method for dealing with non-smooth solutions, some appropriate correction terms are introduced in the time approximation. Numerical examples on different irregular domains with various non-smooth solutions are explored to verify the effectiveness of the presented numerical scheme. Furthermore, we also solve the coupled variable-order fractional Bloch-Torrey equation on a human brain-like domain which is composed of white matter and grey matter. The solution behaviour of this model is compared with that of the constant-order fractional model, and the transverse magnetisation in magnetic resonance imaging on different biological micro-environments are graphically analysed. Results suggest that incorporation of the non-local property and spatial heterogeneity in the model by use of fractional operators can lead to a better capability for capturing the complexities of diffusion phenomena in biological tissues. This research may provide a basis for further research on the application of fractional calculus to clinical research and medical imaging. [ABSTRACT FROM AUTHOR]

• New semi-analytical method for solving the homogenization boundary value problem. • Computes the effective diffusivity for block locally-isotropic heterogeneous media. • New method can capture highly complex heterogeneous geometries. • Faster than a standard finite volume method. • Potential to speed up coarse-scale simulations of diffusion in heterogeneous media. Direct numerical simulation of diffusion through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to homogenize the model by replacing the spatially-varying fine-scale diffusivity with an effective diffusivity calculated from the solution of an appropriate boundary value problem. In this paper, we present a new semi-analytical method for solving this boundary value problem and computing the effective diffusivity for pixellated, locally-isotropic, heterogeneous media. We compare our new solution method to a standard finite volume method and show that equivalent accuracy can be achieved in less computational time for several standard test cases. We also demonstrate how the new solution method can be applied to complex heterogeneous geometries represented by a two-dimensional grid of rectangular blocks. These results indicate that our new semi-analytical method has the potential to significantly speed up simulations of diffusion in heterogeneous media. [ABSTRACT FROM AUTHOR]

This paper aims to estimate the parameters of the time-fractional Black-Scholes (TFBS) partial differential equation with the Caputo fractional derivative by using the real option prices of the S &P 500 index options. First, the numerical solution is obtained by developing a high-order scheme with order ( 3 - α ) for the time discretisation. Some theoretical analyses such as stability and convergence are presented in order to verify the efficiency and accuracy of the proposed scheme. Secondly, we employ a modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) to identify the fractional order α and implied volatility σ of the TFBS equation, and explore the financial meanings of α under extreme stock market conditions such as the Covid-19 and the 2008 global financial crisis. We analyse the values of α and compare the mean squared errors of both the TFBS model and the BS model. Our empirical results show that α may be regarded as a market fluctuation indicator for classifying financial environments, and the TFBS model is more capable of fitting real option data compared with the BS model, especially for put options during the economic downturn. In addition, we find and discuss an interesting relation between α and σ from both the TFBS model and the BS model in three expressions, which could be an open problem for further research. [ABSTRACT FROM AUTHOR]

Diffusion-weighted imaging is an in vivo , non-invasive medical diagnosis technique that uses the Brownian motion of water molecules to generate contrast in the image and therefore reveals exquisite details about the complex structures and adjunctive information of its surrounding biological environment. Recent work highlights that the diffusion-induced magnetic resonance imaging signal loss deviates from the classic monoexponential decay. To investigate the underlying mechanism of this deviated signal decay, diffusion is re-examined through the Bloch–Torrey equation by using fractional calculus with respect to both time and space. In this study, we explore the influence of the complex geometrical structure on the diffusion process. An effective implicit alternating direction method implemented on approximate irregular domains is proposed to solve the two-dimensional time–space Riesz fractional partial differential equation with Dirichlet boundary conditions. This scheme is proved to be unconditionally stable and convergent. Numerical examples are given to support our analysis. We then applied the proposed numerical scheme with some decoupling techniques to investigate the magnetisation evolution governed by the time–space fractional Bloch–Torrey equations on irregular domains. [ABSTRACT FROM AUTHOR]

Free surface flow past a two-dimensional semi-infinite curved plate is considered, with emphasis given to solving the shape of the resulting wave train that appears downstream on the surface of the fluid. This flow configuration can be interpreted as applying near the stern of a wide blunt ship. For steady flow in a fluid of finite depth, we apply the Wiener-Hopf technique to solve a linearised problem, valid for small perturbations of the uniform stream. Weakly nonlinear results obtained by a forced KdV equation are also presented, as are numerical solutions of the fully nonlinear problem, computed using a conformal mapping and a boundary integral technique. By considering different families of shapes for the semi-infinite plate, it is shown how the amplitude of the waves can be minimised. For plates that increase in height as a function of the direction of flow, reach a local maximum, and then point slightly downwards at the point at which the free surface detaches, it appears the downstream wavetrain can be eliminated entirely [ABSTRACT FROM AUTHOR]

In the wound healing process, the cell movement associated with chemotaxis generally outweighs the movement associated with random motion, leading to advection dominated mathematical models of wound healing. The equations in these models must be solved with care, but often inappropriate approaches are adopted. Two one-dimensional test problems arising from advection-dominated models of wound healing are solved using four algorithms—MATLAB's inbuilt routine pdepe.m, the Numerical Algorithms Group routine d03pcf.f, and two finite volume methods. The first finite volume method is based on a first-order up winding treatment of chemotaxis terms and the second on a flux limiting approach. The first test problem admits an analytic solution which can be used to validate the numerical results by analyzing two measures of the error for each method: the average absolute difference and a mass balance error. These criteria as well as the visual comparison between the numerical methods and the exact solution lead us to conclude that flux limiting is the best approach to solving advection-dominated wound healing problems numerically in one dimension. The second test problem is a coupled nonlinear three species model of wound healing angiogenesis. Measurement of the mass balance error for this test problem further confirms our hypothesis that flux limiting is the most appropriate method for solving advection-dominated governing equations in wound healing models.We also consider two two-dimensional test problems arising from wound healing, one that admits an analytic solution and a more complicated problem of blood vessels growth into a devascularized wound bed. The results from the two-dimensional test problems also demonstrate that the flux limiting treatment of advective terms is ideal for an advection-dominated problem. [ABSTRACT FROM AUTHOR]

• A time-fractional Black-Scholes model with smooth payoff function (TFBSM-APF) is considered. • A high-order numerical scheme is constructed for solving TFBSM-APF. • Convergence and stability analyses of the proposed schemes are presented. • Numerical examples are given. • The theoretical analysis and numerical approach can be extended to solve other similar time-fractional models. The purpose of this paper is to investigate a high order numerical method for solving time-fractional Black-Scholes equation in which the fractional operator is defined by the Caputo fractional derivative. The proposed space-time spectral method employs the Jacobi polynomials for the temporal discretisation and Fourier-like basis functions for the spatial discretisation. The stability and convergence of the numerical scheme are analyzed. Two numerical examples are considered to validate the accuracy and illustrate the practicability of the proposed method. The results agree with the theoretical analysis and this approach can be applied in dealing with option pricing models with smooth payoff functions. [ABSTRACT FROM AUTHOR]

In this paper, a second order accurate time and space difference method is proposed to solve the nonlinear two-sided space-fractional diffusion equation with variable coefficients. We combine the alternating direction implicit method and Bi-conjugate gradient stabilized iterative solver to develop an efficient numerical method for a two dimensional, nonlinear, two-sided space-fractional diffusion equation with variable coefficients. Furthermore, the proposed method is proved to be uniquely solvable, stable and convergent. Finally, some numerical results verify the accuracy and efficiency of our computational approach. [ABSTRACT FROM AUTHOR]

The cable equation plays a significant role in many areas of electrophysiology and in modeling neuronal dynamics. In recent years, considerable attention has been devoted to distributed-order differential equations because they appear to be more effective for modeling complex processes. In this work, a finite difference/Legendre spectral method is presented for the numerical simulation of the two-dimensional (2D) distributed-order time-fractional cable equation, where the finite difference method is employed in the temporal discretization and Legendre spectral method is adopted in the spatial discretization. The midpoint quadrature rule is used to approximate the distributed-order, such that the considered equation could be transformed into a multi-term fractional equation. The stability and convergence analysis of the proposed scheme is established, which illustrates that the numerical solution converges to the exact solution with order O (τ 2 + σ 2 + N − s) , where τ , σ , N are the time step size, the step length in the approximation of the distributed-order and the polynomial degree, respectively. Furthermore, to demonstrate the versatility and applicability of our method, we provide numerical results that show good agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

Burgers' equation is a well-studied model in applied mathematics with connections to the Navier–Stokes equations in one spatial direction and traffic flow, for example. Following on from previous work, we analyse solutions to Burgers' equation in the complex plane, concentrating on the dynamics of the complex singularities and their relationship to the solution on the real line. For an initial condition with a simple pole in each of the upper- and lower-half planes, we apply formal asymptotics in the small- and large-time limits in order to characterise the initial and later motion of the singularities. The small-time limit highlights how infinitely many singularities are born at t = 0 and how they orientate themselves to lie increasingly close to anti-Stokes lines in the far field of the inner problem. This inner problem also reveals whether or not the closest singularity to the real axis moves toward the axis or away. For intermediate times, we use the exact solution, apply method of steepest descents, and implement the AAA approximation to track the complex singularities. Connections are made between the motion of the closest singularity to the real axis and the steepness of the solution on the real line. While Burgers' equation is integrable (and has an exact solution), we deliberately apply a mix of techniques in our analysis in an attempt to develop methodology that can be applied to other nonlinear partial differential equations that do not. • Burgers' equation is an integrable nonlinear partial differential equation. • Solutions to Burgers' equation have a rich structure of simple poles. • We observe infinitely many poles born at the singularity of the initial condition. • These poles tend to align themselves close to anti-Stokes lines. • The pole closest to the real axis is connected to the steepness of the solution. [ABSTRACT FROM AUTHOR]

The efficient computation of matrix function vector products has recently become an important area of research, 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 discretization, is advanced in time using the exponential Euler method. We focus on the efficient implementation of the algorithm on graphics processing units (GPUs), 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. J. Higham, and L. N. Trefethen, SIAM J. Numer. Anal., 46 (2008), pp. 2505-2523]. 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 predetermine 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. [ABSTRACT FROM AUTHOR]

Chronic wounds are a significant socioeconomic problem for governments worldwide. Approximately 15% of people who suffer from diabetes will experience a lower-limb ulcer at some stage of their lives, and 24% of these wounds will ultimately result in amputation of the lower limb. Hyperbaric Oxygen Therapy (HBOT) has been shown to aid the healing of chronic wounds; however, the causal reasons for the improved healing remain unclear and hence current HBOT protocols remain empirical. Here we develop a three-species mathematical model of wound healing that is used to simulate the application of hyperbaric oxygen therapy in the treatment of wounds. Based on our modelling, we predict that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds. Furthermore, treatment should continue until healing is complete, and HBOT will not stimulate healing under all circumstances, leading us to conclude that finding the right protocol for an individual patient is crucial if HBOT is to be effective. We provide constraints that depend on the model parameters for the range of HBOT protocols that will stimulate healing. More specifically, we predict that patients with a poor arterial supply of oxygen, high consumption of oxygen by the wound tissue, chronically hypoxic wounds, and/or a dysfunctional endothelial cell response to oxygen are at risk of nonresponsiveness to HBOT. The work of this paper can, in some way, highlight which patients are most likely to respond well to HBOT (for example, those with a good arterial supply), and thus has the potential to assist in improving both the success rate and hence the cost-effectiveness of this therapy. [ABSTRACT FROM AUTHOR]