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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

• Codes over a ring of order 32 with two Gray maps

We describe a ring of order 32 and prove that it is a local Frobenius ring. We study codes over this ring and we give two distinct non-equivalent linear orthogonality-preserving Gray maps to the binary space. Self-dual codes are studied over this ring as well as the binary self-dual codes that are the Gray images of those codes. Specifically, we show that the image of a self-dual code over this ring is a binary self-dual code with an automorphism consisting of 2n transpositions for the first map and n transpositions for the second map. We relate the shadows of binary codes to additive codes over the ring. As Gray images of codes over the ring, binary self-dual [ 70 , 35 , 12 ] codes with 91 distinct weight enumerators are constructed for the first time in the literature.
• Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order α∈(0,1), and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of O(Δtα) in the mean square norm, where Δt denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory.
• BDF2 ADI orthogonal spline collocation method for the fractional integro-differential equations of parabolic type in three dimensions

In this paper, we are concerned with constructing a fast and an efficient alternating direction implicit (ADI) scheme for the fractional parabolic integro-differential equations (FPIDE) with a weakly singular kernel in three dimensions (3D). Our constructed scheme is based on a second-order backward differentiation formula (BDF2) for temporal discretization, orthogonal spline collocation (OSC) method for spatial discretization and a second-order fractional quadrature rule proposed by Lubich for the Riemann-Liouville fractional integral. The stability and convergence of the constructed numerical scheme are derived. Finally, some numerical examples are given to illustrate the accuracy and validity of the BDF2 ADI OSC method. Based on the obtained results, the numerical results are in line with the theoretical ones.
• Spatial discretization for stochastic semilinear superdiffusion driven by fractionally integrated multiplicative space-time white noise

We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space-time white noise. The white noise is characterized by its properties of being white in both space and time and the time fractional derivative is considered in the Caputo sense with an order $\alpha \in (1, 2)$. A spatial discretization scheme is introduced by approximating the space-time white noise with the Euler method in the spatial direction and approximating the second-order space derivative with the central difference scheme. By using the Green functions, we obtain both exact and approximate solutions for the proposed problem. The regularities of both the exact and approximate solutions are studied and the optimal error estimates that depend on the smoothness of the initial values are established. This paper builds upon the research presented in Mathematics. 2021. 9, 1917, where we originally focused on error estimates in the context of subdiffusion with $\alpha \in (0, 1)$. We extend our investigation to the spatial approximation of stochastic superdiffusion with $\alpha \in (1, 2)$ and place particular emphasis on refining our understanding of the superdiffusion phenomenon by analyzing the error estimates associated with the time derivative at the initial point.
• High-order schemes based on extrapolation for semilinear fractional differential equation

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α∈(1, 2). The error has the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯) at any fixed time tN=T, N∈Z+, where di, i=3, 4, … and di∗, i=2, 3, … denote some suitable constants and τ=T/N denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α∈(1, 2) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.
• Unconditionally stable and convergent difference scheme for superdiffusion with extrapolation

Approximating the Hadamard finite-part integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the Riemann-Liouville fractional derivative of order α∈(1, 2) and the error is shown to have the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯) at any fixed time, where τ denotes the step size and dl, l=3, 4, ⋯ and dl∗, l=2, 3, ⋯ are some suitable constants. Applying the proposed scheme in temporal direction and the central difference scheme in spatial direction, a new finite difference method is developed for approximating the time fractional wave equation. The proposed method is unconditionally stable, convergent with order O(τ3-α), α∈(1, 2) and the error has the asymptotic expansion. Richardson extrapolation is applied to improve the accuracy of the numerical method. The convergence orders are O(τ4-α) and O(τ2(3-α)), α∈(1, 2), respectively, after first two extrapolations. Numerical examples are presented to show that the numerical results are consistent with the theoretical findings.
• Quotients of the Highwater algebra and its cover

Primitive axial algebras of Monster type are a class of non-associative algebras with a strong link to finite (especially simple) groups. The motivating example is the Griess algebra, with the Monster as its automorphism group. A crucial step towards the understanding of such algebras is the explicit description of the 2-generated symmetric objects. Recent work of Yabe, and Franchi and Mainardis shows that any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra H, or its characteristic 5 cover Ĥ. In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of H and Ĥ. We proceed in a unified way, by defining a cover of H in all characteristics. This cover has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic 5.
• The weight enumerators of singly-even self-dual [88,44,14] codes and new binary self-dual [68,34,12] and [88,44,14] codes

In this work, we focus on constructing binary self-dual [68, 34, 12] and [88, 44, 14] codes with new parameters in their weight enumerators. For this purpose, we present a new bordered matrix construction for self-dual codes which is derived as a modification of two known bordered matrix constructions. We provide the necessary conditions for the new construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. We also construct the possible weight enumerators for singly-even self-dual [88, 44, 14] codes and their shadows as this has not been done in the literature yet. We employ the modified bordered matrix together with the well-known neighbour method to construct binary self-dual codes that could not be obtained from the other, known bordered matrix constructions. Many of the codes turn out to have parameters in their weight enumerators that were not known in the literature before.
• Numerical Methods for Stochastic Allen-Cahn Equation and Stochastic Subdiffusion and Superdiffusion

In this Thesis, we consider the numerical solution of stochastic partial differential equations with particular interest on the Ɛ-dependent Allen-Cahn equation, and the stochastic time fractional partial differential equations in both subdiffusion and superdiffusion cases.
• Strong approximation of stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise

Recently, Kovács et al. considered a Mittag‐Leffler Euler integrator for a stochastic semilinear Volterra integral‐differential equation with additive noise and proved the strong convergence error estimates [see SIAM J. Numer. Anal. 58(1) 2020, pp. 66‐85]. In this article, we shall consider the Mittag‐Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise. The mild solutions of our models involve four different Mittag‐Leffler functions. We first consider the existence, uniqueness and the regularities of the solutions. We then introduce the full discretization schemes for solving the problems. The temporal discretization is based on the Mittag‐Leffler integrators and the spatial discretization is based on the spectral method. The optimal strong convergence error estimates are proved under the reasonable assumptions for the semilinear term and for the regularity of the noise. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
• High Order Approximations of Solutions to Initial Value Problems for Linear Fractional Integro-Differential Equations

We consider a general class of linear integro-differential equations with Caputo fractional derivatives and weakly singular kernels. First, the underlying initial value problem is reformulated as an integral equation and the possible singular behavior of its exact solution is determined. After that, using a suitable smoothing transformation and spline collocation techniques, the numerical solution of the problem is discussed. Optimal convergence estimates are derived and a superconvergence result of the proposed method is established. The obtained theoretical results are supported by numerical experiments.
• Social Sustainability in the Oil and Gas Industry in the Niger Delta Region

Social sustainability is a critical component of the Sustainable Development Goals and net zero targets. This research establishes the major principles and main elements of social sustainability, its theoretical and practical underpinnings, investigates its implementation and provides a framework for its implementation in the oil and gas industry in the Niger Delta region. Qualitative case study was employed for the study and thematic analysis was conducted on multiple sources of data. Using the oil and gas industry in the Niger Delta for empirical analysis, the research explored social sustainability through a synthesis of its major principles and elements alongside the socioeconomic characteristics of the region. The research established that social sustainability is a multidimensional concept and its major principle is the wellbeing of society within and across generations. Due to its multidimensional nature, the elements of social sustainability need to be derived on a case-by-case basis for effective implementation of the concept. In the oil and gas industry in the Niger Delta, equity and social justice, partnership, employment and human capacity development, and social services and infrastructure emerged as the major elements of social sustainability. Stakeholder theory was used to frame the research and consistent with stakeholder theory, the adoption of partnership with the host communities suggests a partnership with resources to gain legitimacy. Novel approaches to social sustainability implementation such as the GMoU pioneered by Chevron, and new models of partnership based on the concept of value creation in stakeholder theory, are recommended. The research contributes to an emerging body of knowledge by showing the application and manifestations of a global concept at the local level. The findings have practical and theoretical value for practitioners and researchers of social sustainability.
• A posteriori error analysis of space-time discontinuous Galerkin methods for the ε-stochastic Allen-Cahn equation

In this work, we apply an \textit{a posteriori} error analysis for the space-time, discontinuous in time, Galerkin scheme which has been proposed in \cite{AIMA} for the $\eps$-dependent stochastic Allen-Cahn equation with mild noise $\dot{W}^\eps$ tending to rough as $\eps\rightarrow 0$. Our results are derived under low regularity since the noise even smooth in space, is assumed only one-time continuously differentiable in time, according to the minimum regularity properties of \cite{Fun99}. We prove \textit{a posteriori} error estimates for the $m$-dimensional problem, $m\leq 4$ for a general class of space-time finite element spaces. The \textit{a posteriori} bound is growing only polynomially in $\eps^{-1}$ if the step length $h$ is bounded by a positive power of $\eps$. This agrees with the restriction posed so far in the \textit{a priori} error analysis of continuous finite element schemes for the $\eps$-dependent deterministic Allen-Cahn or deterministic and stochastic Cahn-Hilliard equation. As an application we examine tensorial elements where the discrete solution is approximated by polynomial functions of separated space and time variables; the \textit{a posteriori} estimates there involve dimensions, and the space, time discretization parameters. We then consider the special case of the mild noise $\dot{W}^\eps$ as defined in \cite{weber1} through the convolution of a Gaussian process with a proper mollifying kernel, which is then numerically constructed. Finally, we provide some useful insights for the numerical algorithm, and present for the first time some numerical experiments of the scheme for both one and two-dimensional problems in various cases of interest, and compare with the deterministic ones.
• Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order α∈(1, 2). The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results.
• Mutation-Based Algebraic Artificial Bee Colony Algorithm for Computing the Distance of Linear Codes

Finding the minimum distance of linear codes is a non-deterministic polynomial-time-hard problem and different approaches are used in the literature to solve this problem.&#x0D; Although, some of the methods focus on finding the true distances by using exact algorithms, some of them focus on optimization algorithms to find the lower or upper bounds of the distance. In this study,&#x0D; we focus on the latter approach. We first give the swarm intelligence background of artificial bee colony algorithm, we explain the algebraic approach of such algorithm and call it the algebraic artificial bee colony algorithm (A-ABC). Moreover, we develop the A-ABC algorithm by integrating it with the algebraic differential mutation operator. We call the developed algorithm the mutation-based algebraic artificial bee colony algorithm (MBA-ABC). We apply both; the A-ABC and MBA-ABC algorithms to the problem of finding the minimum distance of linear codes. The achieved results indicate that the MBA-ABC algorithm has a superior performance when compared with the A-ABC algorithm when finding the minimum distance of Bose, Chaudhuri, and Hocquenghem (BCH) codes (a special type of linear codes).
• From forbidden configurations to a classification of some axial algebras of Monster type

Ivanov introduced the shape of a Majorana algebra as a record of the 2-generated subalgebras arising in that algebra. As a broad generalisation of this concept and to free it from the ambient algebra, we introduce the concept of an axet and shapes on an axet. A shape can be viewed as an algebra version of a group amalgam. Just like an amalgam, a shape leads to a unique algebra completion which may be non-trivial or it may collapse. Then for a natural family of shapes of generalised Monster type we classify all completion algebras and discover that a great majority of them collapse, confirming the observations made in an earlier paper [12].
• L1 scheme for solving an inverse problem subject to a fractional diffusion equation

This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle < 𝜋∕2, that is the resolvent set of this operator contains {𝑧 ∈ ℂ ⧵ {0} ∶ |Arg 𝑧| < 𝜃} for some 𝜋∕2 < 𝜃 < 𝜋. The relationship between the time fractional order 𝛼 ∈ (0, 1) and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as 𝛼 approaches 1. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results.
• Group matrix ring codes and constructions of self-dual codes

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring Mk(R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring Mk(R) are one sided ideals in the group matrix ring Mk(R)G and the corresponding codes over the ring R are Gk-codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes.
• Higher moments for the Stochastic Cahn - Hilliard Equation with multiplicative Fourier noise

We consider in dimensions $d=1,2,3$ the $\eps$-dependent stochastic Cahn-Hilliard equation with a multiplicative and sufficiently regular in space infinite dimensional Fourier noise with strength of order $\mathcal{O}(\eps^\gamma)$, $\gamma>0$. The initial condition is non-layered and independent from $\eps$. Under general assumptions on the noise diffusion $\sigma$, we prove moment estimates in $H^1$ (and in $L^\infty$ when $d=1$). Higher $H^2$ regularity $p$-moment estimates are derived when $\sigma$ is bounded, yielding as well space H\"older and $L^\infty$ bounds for $d=2,3$, and path a.s. continuity in space. All appearing constants are expressed in terms of the small positive parameter $\eps$. As in the deterministic case, in $H^1$, $H^2$, the bounds admit a negative polynomial order in $\eps$. Finally, assuming layered initial data of initial energy uniformly bounded in $\eps$, as proposed by X.F. Chen in \cite{chenjdg}, we use our $H^1$ $2$d-moment estimate and prove the stochastic solution's convergence to $\pm 1$ as $\eps\rightarrow 0$ a.s., when the noise diffusion has a linear growth.