• Numerical approximation of the Stochastic Cahn-Hilliard Equation near the Sharp Interface Limit

      Antonopoulou, Dimitra; Banas, Lubomir; Nurnberg, Robert; Prohl, Andreas; University of Chester; University of Bielefeld; Imperial College London; University of Tuebingen
      Abstract. We consider the stochastic Cahn-Hilliard equation with additive noise term that scales with the interfacial width parameter ε. We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where ε only enters polynomially; the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For γ sufficiently large, convergence in probability of iterates towards the deterministic Hele-Shaw/Mullins-Sekerka problem in the sharp-interface limit ε → 0 is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ γ) on the geometric evolution in the sharp-interface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) Mullins-Sekerka problem. The computational results indicate that the limit for γ ≥ 1 is the deterministic problem, and for γ = 0 we obtain agreement with a (new) stochastic version of the Mullins-Sekerka problem.
    • A Numerical Feasibility Study of Kinetic Energy Harvesting from Lower Limb Prosthetics

      Jia, Yu; Wei, Xueyong; Pu, Jie; Xie, Pengheng; Wen, Tao; Wang, Congsi; Lian, Peiyuan; Xue, Song; Shi, Yu; Aston University; University of Chester; Xidian University; Xi'an Jiaotong University (MDPI, 2019-10-10)
      With the advancement trend of lower limb prosthetics headed towards bionics (active ankle and knee) and smart prosthetics (gait and condition monitoring), there is an increasing integration of various sensors (micro-electromechanical system (MEMS) accelerometers, gyroscopes, magnetometers, strain gauges, pressure sensors, etc.), microcontrollers and wireless systems, and power drives including motors and actuators. All of these active elements require electrical power. However, inclusion of a heavy and bulky battery risks to undo the lightweight advancements achieved by the strong and flexible composite materials in the past decades. Kinetic energy harvesting holds the promise to recharge a small on-board battery in order to sustain the active systems without sacrificing weight and size. However, careful design is required in order not to over-burden the user from parasitic effects. This paper presents a feasibility study using measured gait data and numerical simulation in order to predict the available recoverable power. The numerical simulations suggest that, depending on the axis, up to 10s mW average electrical power is recoverable for a walking gait and up to 100s mW average electrical power is achievable during a running gait. This takes into account parasitic losses and only capturing a fraction of the gait cycle to not adversely burden the user. The predicted recoverable power levels are ample to self-sustain wireless communication and smart sensing functionalities to support smart prosthetics, as well as extend the battery life for active actuators in bionic systems. The results here serve as a theoretical foundation to design and develop towards regenerative smart bionic prosthetics.
    • Numerical investigation of D-bifurcations for a stochastic delay logistic equation

      Ford, Neville J.; Norton, Stewart J. (World Scientific, 2005)
    • Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHugh-Nagumo equation

      Ford, Neville J.; Lima, Pedro M.; Lumb, Patricia M.; University of Chester, Instituto Superior Tecnico, University of Lisbon, University of Chester (Elsevier, 2016-09-08)
      In this work we introduce and analyse a stochastic functional equation, which contains both delayed and advanced arguments. This equation results from adding a stochastic term to the discrete FitzHugh-Nagumo equation which arises in mathematical models of nerve conduction. A numerical method is introduced to compute approximate solutions and some numerical experiments are carried out to investigate their dynamical behaviour and compare them with the solutions of the corresponding deterministic equation.
    • Numerical methods for a nonuniquely solvable Volterra integral equation

      Diogo, Teresa; Ford, Neville J.; Lima, Pedro M.; Valtchev, Svilen (2003)
    • Numerical methods for a Volterra integral equation with non-smooth solutions

      Diogo, Teresa; Ford, Neville J.; Lima, Pedro M.; Valtchev, Svilen (Elsevier Science, 2006-05-01)
      This article discusses the numerical treatment of a singular Volterra integral equation with an infinite set of solutions.
    • Numerical methods for deterministic and stochastic fractional partial differential equations

      Yan, Yubin; Khan, Monzorul (University of Chester, 2020-03)
      In this thesis we will explore the numerical methods for solving deterministic and stochastic space and time fractional partial differential equations. Firstly we consider Fourier spectral methods for solving some linear stochastic space fractional partial differential equations perturbed by space-time white noises in one dimensional case. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space fractional partial differential equations. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. Secondly we consider Fourier spectral methods for solving stochastic space fractional partial differential equation driven by special additive noises in one dimensional case. 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 appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equation is then solved by using Fourier spectral methods. Thirdly, we will consider the discontinuous Galerkin time stepping methods for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in t of degree at most q−1, q ≥ 1, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given. Finally, we consider error estimates for the modified L1 scheme for solving time fractional partial differential equation. Jin et al. (2016, An analysis of the L1 scheme for the subdiffifusion equation with nonsmooth data, IMA J. of Number. Anal., 36, 197-221) ii established the O(k) convergence rate for the L1 scheme for both smooth and nonsmooth initial data. We introduce a modified L1 scheme and prove that the convergence rate is O(k2−α=), 0 < α < 1 for both smooth and nonsmooth initial data. We first write the time fractional partial differential equations as a Volterra integral equation which is then approximated by using the convolution quadrature with some special generating functions. A Laplace transform method is used to prove the error estimates for the homogeneous time fractional partial differential equation for both smooth and nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
    • Numerical methods for solving space fractional partial differential equations by using Hadamard finite-part integral approach

      Yan, Yubin; Wang, Yanyong; Hu, Ye; University of Chester; Lvliang University (Springer, 2019-07-26)
      We introduce a novel numerical method for solving two-sided space fractional partial differential equation in two dimensional case. The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order $O(h^{3- \alpha})$, where $h$ is the space step size and $\alpha\in (1, 2)$ is the order of Riemann-Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equation. We obtained the error estimates with the convergence orders $O(\tau +h^{3-\alpha}+ h^{\beta})$, where $\tau$ is the time step size and $\beta >0$ is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equation constructed by using the standard shifted Gr\"unwald-Letnikov formula or higher order Lubich'e methods which require the solution of the equation satisfies the homogeneous Dirichlet boundary condition in order to get the first order convergence, the numerical method for solving space fractional partial differential equation constructed by using Hadamard finite-part integral approach does not require the solution of the equation satisfies the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained by using the Hadamard finite-part integral approach for solving space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained by using the numerical methods constructed with the standard shifted Gr\"unwald-Letnikov formula or Lubich's higer order approximation schemes.
    • Numerical modelling of qualitative behaviour of solutions to convolution integral equations

      Diogo, Teresa; Ford, Judith M.; Ford, Neville J.; Lima, Pedro M.; Instituto Superior Técnico ; University of Chester ; University of Chester ; Instituto Superior Técnico (University of Chester, 2006)
      We consider the qualitative behaviour of solutions to linear integral equations of the form where the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley-Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.
    • Numerical modelling of qualitative behaviour of solutions to convolution integral equations

      Ford, Neville J.; Diogo, Teresa; Ford, Judith M.; Lima, Pedro M.; University of Chester ; Instituto Superior Tecnico, Lisbon ; University of Chester ; Instituto Superior Tecnico, Lisbon (Elsevier, 2007-08-15)
    • Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

      Morgado, Maria L.; Rebelo, Magda S.; Ferras, Luis L.; Ford, Neville J.; Universidade de Tras-os-Montes e Alto Douro; Universidade NOVA de Lisboa; University of Minho; University of Chester (Elsevier, 2016-11-09)
      In this work we present a new numerical method for the solution of the distributed order time fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed.
    • Numerical solution methods for distributed order differential equations

      Diethelm, Kai; Ford, Neville J. (Institute of Mathematics & Informatics, Bulgarian Academy of Sciences, 2005)
      This article discusses a basic framework for the numerical solution of distributed order differential equations.
    • The numerical solution of forward–backward differential equations: Decomposition and related issues

      Ford, Neville J.; Lumb, Patricia M.; Lima, Pedro M.; Teodoro, M. Filomena; University of Chester : University of Chester : Instituto Superior Técnico, Lisbon : Instituto Politécnico de Setúbal (Elsevier, 2010-09-01)
      This journal article discusses the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions.
    • The numerical solution of fractional and distributed order differential equations

      Ford, Neville J.; Edwards, John T.; Connolly, Joseph A. (University of Liverpool (University College Chester), 2004-12)
      Fractional Calculus can be thought of as a generalisation of conventional calculus in the sense that it extends the concept of a derivative (integral) to include non-integer orders. Effective mathematical modelling using Fractional Differential Equations (FDEs) requires the development of reliable flexible numerical methods. The thesis begins by reviewing a selection of numerical methods for the solution of Single-term and Multi-term FDEs. We then present: 1. a graphical technique for comparing the efficiency of numerical methods. We use this to compare Single-term and Multi-term methods and give recommendations for which method is best for any given FDE. 2. a new method for the solution of a non-linear Multi-term Fractional Dif¬ferential Equation. 3. a sequence of methods for the numerical solution of a Distributed Order Differential Equation. 4. a discussion of the problems associated with producing a computer program for obtaining the optimum numerical method for any given FDE.
    • Numerical Solution of Fractional Differential Equations and their Application to Physics and Engineering

      Morgado, Luisa; Ford, Neville; Ferras, Luis L. (University of Chester, 2018-12-03)
      This dissertation presents new numerical methods for the solution of fractional differential equations of single and distributed order that find application in the different fields of physics and engineering. We start by presenting the relationship between fractional derivatives and processes like anomalous diffusion, and, we then develop new numerical methods for the solution of the time-fractional diffusion equations. The first numerical method is developed for the solution of the fractional diffusion equations with Neumann boundary conditions and the diffusivity parameter depending on the space variable. The method is based on finite differences, and, we prove its convergence (convergence order of O(Δx² + Δt²<sup>-α</sup>), 0 < α < 1) and stability. We also present a brief description of the application of such boundary conditions and fractional model to real world problems (heat flux in human skin). A discussion on the common substitution of the classical derivative by a fractional derivative is also performed, using as an example the temperature equation. Numerical methods for the solution of fractional differential equations are more difficult to develop when compared to the classical integer-order case, and, this is due to potential singularities of the solution and to the nonlocal properties of the fractional differential operators that lead to numerical methods that are computationally demanding. We then study a more complex type of equations: distributed order fractional differential equations where we intend to overcome the second problem on the numerical approximation of fractional differential equations mentioned above. These equations allow the modelling of more complex anomalous diffusion processes, and can be viewed as a continuous sum of weighted fractional derivatives. Since the numerical solution of distributed order fractional differential equations based on finite differences is very time consuming, we develop a new numerical method for the solution of the distributed order fractional differential equations based on Chebyshev polynomials and present for the first time a detailed study on the convergence of the method. The third numerical method proposed in this thesis aims to overcome both problems on the numerical approximation of fractional differential equations. We start by solving the problem of potential singularities in the solution by presenting a method based on a non-polynomial approximation of the solution. We use the method of lines for the numerical approximation of the fractional diffusion equation, by proceeding in two separate steps: first, spatial derivatives are approximated using finite differences; second, the resulting system of semi-discrete ordinary differential equations in the initial value variable is integrated in time with a non-polynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a non-polynomial approximation in the first sub-interval which contains the origin in time (the point where the solution may be singular) and a polynomial approximation in the remaining intervals. This way we obtain a method that allows a faster numerical solution of fractional differential equations (than the method obtained with non-polynomial approximation) and also takes into account the potential singularity of the solution. The thesis ends with the main conclusions and a discussion on the main topics presented along the text, together with a proposal of future work.
    • The numerical solution of fractional differential equations: Speed versus accuracy

      Ford, Neville J.; Simpson, A. Charles (Manchester Centre for Computational Mathematics, 2003-05-23)
      This paper discusses the development of efficient algorithms for a certain fractional differential equation.
    • The numerical solution of fractional differential equations: Speed versus accuracy

      Ford, Neville J.; Simpson, A. Charles (Springer, 2001-04)
      This article discusses the development of efficient algorithms for a certain fractional differential equation.
    • The numerical solution of linear multi-term fractional differential equations: Systems of equations

      Edwards, John T.; Ford, Neville J.; Simpson, A. Charles (Elsevier, 2002-11-15)
      This article discusses how the numerical approximation of a linear multi-term fractional differential equation can be calculated by the reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity.
    • Numerical solution of multi-order fractional differential equations

      Diethelm, Kai; Ford, Neville J. (Elsevier, 2004)
    • Numerical solution of the Bagley Torvik equation

      Diethelm, Kai; Ford, Neville J. (Springer, 2002-09)