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Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHughNagumo equationIn 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 FitzHughNagumo 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 Volterra integral equation with nonsmooth solutionsThis 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 equationsIn 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 spacetime 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 spacetime 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 spacetime 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, 197221) 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 finitepart integral approachWe introduce a novel numerical method for solving twosided space fractional partial differential equation in two dimensional case. The approximation of the space fractional RiemannLiouville derivative is based on the approximation of the Hadamard finitepart 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 RiemannLiouville 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\"unwaldLetnikov 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 finitepart 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 finitepart integral approach for solving space fractional partial differential equation with nonhomogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained by using the numerical methods constructed with the standard shifted Gr\"unwaldLetnikov formula or Lubich's higer order approximation schemes.

Numerical modelling of qualitative behaviour of solutions to convolution integral equationsWe 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 wellknown PaleyWiener 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 solution for diffusion equations with distributed order in time using a Chebyshev collocation methodIn 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 equationsThis 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 issuesThis journal article discusses the decomposition, by numerical methods, of solutions to mixedtype functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions.

The numerical solution of fractional and distributed order differential equationsFractional 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 noninteger 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 Singleterm and Multiterm FDEs. We then present: 1. a graphical technique for comparing the efficiency of numerical methods. We use this to compare Singleterm and Multiterm methods and give recommendations for which method is best for any given FDE. 2. a new method for the solution of a nonlinear Multiterm 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 EngineeringThis 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 timefractional 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 integerorder 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 nonpolynomial 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 semidiscrete ordinary differential equations in the initial value variable is integrated in time with a nonpolynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a nonpolynomial approximation in the first subinterval 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 nonpolynomial 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 accuracyThis paper discusses the development of efficient algorithms for a certain fractional differential equation.

The numerical solution of fractional differential equations: Speed versus accuracyThis article discusses the development of efficient algorithms for a certain fractional differential equation.

The numerical solution of linear multiterm fractional differential equations: Systems of equationsThis article discusses how the numerical approximation of a linear multiterm 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 Solutions of Fractional Differential Equations by ExtrapolationAn extrapolation algorithm is considered for solving linear fractional differential equations in this paper, which is based on the direct discretization of the fractional differential operator. Numerical results show that the approximate solutions of this numerical method has the expected asymptotic expansions.

Numerical treatment of oscillary functional differential equationsThis preprint is concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear vmethods will also be oscillatory. We conclude with some general theory