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We are an active university Mathematics Department with a strong teaching and research reputation. We offer students the chance to study at undergraduate or postgraduate level on degree programmes leading to: BSc in Mathematics, BSc/BA joint courses in Mathematics or Applied Statistics and a wide range of other subjects. We have an active research group focusing on Computational Applied Mathematics, with research students studying for the degrees of MPhil and PhD, postdoctoral workers and associated collaborators from across the world.

### Recent Submissions

• #### Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries.

(Elsevier, 2018-07-12)
In this work we discuss the connection between classical and fractional viscoelastic Maxwell models, presenting the basic theory supporting these constitutive equations, and establishing some background on the admissibility of the fractional Maxwell model. We then develop a numerical method for the solution of two coupled fractional differential equations (one for the velocity and the other for the stress), that appear in the pure tangential annular ow of fractional viscoelastic fluids. The numerical method is based on finite differences, with the approximation of fractional derivatives of the velocity and stress being inspired by the method proposed by Sun and Wu for the fractional diffusion-wave equation [ Z.Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics 56 (2006) 193-209]. We prove solvability, study numerical convergence of the method, and also discuss the applicability of this method for simulating the rheological response of complex fluids in a real concentric cylinder rheometer. By imposing a torsional step-strain, we observe the different rates of stress relaxation obtained with different values of \alpha and \beta (the fractional order exponents that regulate the viscoelastic response of the complex fluids).
• #### Mathematical models of DNA methylation dynamics: Implications for health and ageing

(Elsevier, 2018)
DNA methylation status is a key epigenetic process which has been intimately associated with gene regulation. In recent years growing evidence has associated DNA methylation status with a variety of diseases including cancer, Alzheimers disease and cardiovascular disease. Moreover, changes to DNA methylation have also recently been implicated in the ageing process. The factors which underpin DNA methylation are complex, and remain to be fully elucidated. Over the years mathematical modelling has helped to shed light on the dynamics of this important molecular system. Although the existing models have contributed significantly to our overall understanding of DNA methylation, they fall-short of fully capturing the dynamics of this process. In this paper we develop a linear and nonlinear model which captures more fully the dynamics of the key intracellular events which characterise DNA methylation. In particular the outcomes of our linear model result in gene promoter specific methylation levels which are more biologically plausible than those revealed by previous mathematical models. In addition, our non-linear model predicts DNA methylation promoter bistability which is commonly observed experimentally. The findings from our models have implications for our current understanding of how changes to the dynamics which underpin DNA methylation affect ageing and health.
• #### A high-order scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equation

(Elsevier, 2018-10-05)
A new high-order finite difference scheme to approximate the Caputo fractional derivative $\frac{1}{2} \big ( \, _{0}^{C}D^{\alpha}_{t}f(t_{k})+ \, _{0}^{C}D^{\alpha}_{t}f(t_{k-1}) \big ), k=1, 2, \dots, N,$ with the convergence order $O(\Delta t^{4-\alpha}), \, \alpha\in(1,2)$ is obtained when $f^{\prime \prime \prime} (t_{0})=0$, where $\Delta t$ denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order $O(\Delta t^{4-\alpha} + h^2)$, where $h$ denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
• #### A Posteriori Analysis for Space-Time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain

(ECP sciences, 2018)
This paper presents an a posteriori error analysis for the discontinuous in time space-time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains [25]. Using a Cl ement-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coe cients but posed on a cylindrical domain. We formulate a discontinuous in time space{time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of [19] for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso [36], proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.
• #### Error estimates of high-order numerical methods for solving time fractional partial differential equations

(De Gruyter, 2018-07-12)
Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3- \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.
• #### A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

(Springer, 2018-05-02)
This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in L∞(L2) for the velocity error.
• #### Malliavin Calculus for the stochastic Cahn- Hilliard/Allen-Cahn equation with unbounded noise diffusion

(Elsevier, 2018-05-08)
The stochastic partial di erential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of Cahn-Hilliard and Allen-Cahn type operators with a multiplicative, white, space-time noise of unbounded di usion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution u. In dimension one, according to the regularity result in [5], u admits continuous paths a.s. Using this property, and inspired by a method proposed in [8], we construct a modi ed approximating sequence for u, which properly treats the new second order Allen-Cahn operator. Under a localization argument, we prove that the Malliavin derivative of u exists locally, and that the law of u is absolutely continuous, establishing thus that a density exists.
• #### Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis

(Springer, 2017-12-31)
This book presents new developments in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grades students in mathematics, engineering, physics, economics, and biology.
• #### Units of the group algebra of the group $C_n\times D_6$ over any finite field of characteristic $3$

(International Electronic Journal of Algebra, 2018)
In this paper, we establish the structure of the unit group of the group algebra ${\FF}_{3^t}(C_n\times D_6)$ for $n \geq 1$.
• #### Constructions for Self-Dual Codes Induced from Group Rings

(Elsevier, 2018-02-03)
In this work, we establish a strong connection between group rings and self-dual codes. We prove that a group ring element corresponds to a self-dual code if and only if it is a unitary unit. We also show that the double-circulant and four-circulant constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the well-known methods, for self-dual codes. We establish the relevance of these new constructions by finding many extremal binary self-dual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary self-dual codes of length 68.
• #### A higher order numerical method for time fractional partial differential equations with nonsmooth data

(Elsevier, 2018-01-02)
Gao et al. (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu (2016), where $k$ is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ uniformly with respect to the time variable $t$. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ uniformly with respect to the time variable $t$. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ as in Gao \et \cite{gaosunzha} (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ for any fixed $t_{n}>0$ for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
• #### A novel high-order algorithm for the numerical estimation of fractional differential equations

(Elsevier, 2018-01-09)
This paper uses polynomial interpolation to design a novel high-order algorithm for the numerical estimation of fractional differential equations. The Riemann-Liouville fractional derivative is expressed by using the Hadamard finite-part integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented and it is established that the convergence order of the algorithm is O(h4−a). Asymptotic expansion of the error for the presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm.
• #### Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes

(Springer, 2017-11-15)
We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in [13] by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasi-G codes and give a construction of these codes.
• #### A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

(Cambridge University Press, 2017-10-09)
We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted vs.\ repelled by a single chemical substance. The production vs.\ destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model we investigate the variational structures, in particular we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy-Littlewood-Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.
• #### Some time stepping methods for fractional diffusion problems with nonsmooth data

(De Gruyter, 2017-09-02)
We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293(2015), 201-217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the Riemann-Liouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1$ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1$ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
• #### An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data

(Society for Industrial and Applied Mathematics, 2018-01-11)
We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197-221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has convergence rate $O(k^{2-\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
• #### Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noises

(Springer, 2018-02)
Fourier spectral methods for solving stochastic space fractional partial differential equations driven by special additive noises in one-dimensional case are introduced and analyzed. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The space-time noise is approximated by the piecewise constant functions in the time direction and by some appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. For the linear problem, we obtain the precise error estimates in the $L_{2}$ norm and find the relations between the error bounds and the fractional powers. For the nonlinear problem, we introduce the numerical algorithms and MATLAB codes based on the FFT transforms. Our numerical algorithms can be adapted easily to solve other stochastic space fractional partial differential equations with multiplicative noises. Numerical examples for the semilinear stochastic space fractional partial differential equations are given.
• #### An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

(De Gruyter, 2017-09)
In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
• #### A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes

(Taylor & Francis, 2017-10-09)
We consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth, Li et al. \lq \lq Finite difference methods with non-uniform meshes for nonlinear fractional differential equations\rq\rq, Journal of Computational Physics, 316(2016), 614-631, obtained the error estimates of finite difference methods with non-uniform meshes. However the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with non-uniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.
• #### Detailed error analysis for a fractional adams method with graded meshes

(Springer, 2017-09-21)
We consider a fractional Adams method for solving the nonlinear fractional differential equation $\, ^{C}_{0}D^{\alpha}_{t} y(t) = f(t, y(t)), \, \alpha >0$, equipped with the initial conditions $y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots, \lceil \alpha \rceil -1$. Here $\alpha$ may be an arbitrary positive number and $\lceil \alpha \rceil$ denotes the smallest integer no less than $\alpha$ and the differential operator is the Caputo derivative. Under the assumption $\, ^{C}_{0}D^{\alpha}_{t} y \in C^{2}[0, T]$, Diethelm et al. \cite[Theorem 3.2]{dieforfre} introduced a fractional Adams method with the uniform meshes $t_{n}= T (n/N), n=0, 1, 2, \dots, N$ and proved that this method has the optimal convergence order uniformly in $t_{n}$, that is $O(N^{-2})$ if $\alpha > 1$ and $O(N^{-1-\alpha})$ if $\alpha \leq 1$. They also showed that if $\, ^{C}_{0}D^{\alpha}_{t} y(t) \notin C^{2}[0, T]$, the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well known that for $y \in C^{m} [0, T]$ for some $m \in \mathbb{N}$ and $0 < \alpha 1$, we show that the optimal convergence order of this method can be recovered uniformly in $t_{n}$ even if $\, ^{C}_{0}D^{\alpha}_{t} y$ behaves as $t^{\sigma}, 0< \sigma <1$. Numerical examples are given to show that the numerical results are consistent with the theoretical results.