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Subjectsnumerical analysis (2)numerical methods (2)bifurcations (1)Caputo fractional derivative (1)computational and mathematical modelling (1)delay equations (1)discrete adjoint equations (1)discrete delay differential equations (1)discrete equations (1)divide and conquer (1)View MoreJournal

Journal of Computational and Applied Mathematics (11)

AuthorsFord, Neville J. (5)Yan, Yubin (4)Baker, Christopher T. H. (3)Asl, Mohammad S. (1)Buckwar, Evelyn (1)Diogo, Teresa (1)Ford, Judith M. (1)Javidi, Mohammad (1)Liang, Zongqi (1)Lima, Pedro M. (1)View MoreTypesArticle (11)

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Superfast solution of linear convolutional Volterra equations using QTT approximation

Roberts, Jason A.; Savostyanov, Dmitry V.; Tyrtyshnikov, Eugene E. (Elsevier, 2014-04)

This article address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O(nlogn) to the speed of superfast Fourier transform, i.e., O(log^2n). The results of the paper are illustrated by numerical examples.

Numerical treatment of oscillary functional differential equations

Ford, Neville J.; Yan, Yubin; Malique, Md A. (Elsevier, 2010-09)

This 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 v-methods will also be oscillatory. We conclude with some general theory

A nonpolynomial collocation method for fractional terminal value problems

Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S. (Elsevier, 2014-06-24)

In this paper we propose a non-polynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α, 0 < α < 1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a non-polynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.

Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions

Liu, Fang; Liang, Zongqi; Yan, Yubin (Elsevier, 2018-12-17)

We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear space-fractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Numerical modelling of qualitative behaviour of solutions to convolution integral equations

Ford, Neville J.; Diogo, Teresa; Ford, Judith M.; Lima, Pedro M. (Elsevier, 2007-08)

Numerical analysis of a two-parameter fractional telegraph equation

Ford, Neville J.; Rodrigues, M. M.; Xiao, Jingyu; Yan, Yubin (Elsevier, 2013-09-26)

In this paper we consider the two-parameter fractional telegraph equation of the form
$$-\, ^CD_{t_0^+}^{\alpha+1} u(t,x) + \, ^CD_{x_0^+}^{\beta+1} u (t,x)- \, ^CD_{t_0^+}^{\alpha}u (t,x)-u(t,x)=0.$$ Here
$\, ^CD_{t_0^+}^{\alpha}$, $\, ^CD_{t_0^+}^{\alpha+1}$, $\, ^CD_{x_0^+}^{\beta+1}$ are
operators of the Caputo-type
fractional derivative, where $0\leq \alpha < 1$ and $0 \leq \beta < 1$. The existence
and uniqueness of the equations are proved by using the Banach
fixed point theorem. A numerical method is introduced to solve this
fractional telegraph equation and stability conditions for the numerical
method are obtained. Numerical examples are given in the final section of the
paper.

A novel high-order algorithm for the numerical estimation of fractional differential equations

Asl, Mohammad S.; Javidi, Mohammad; Yan, Yubin (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.

Noise-induced changes to the behaviour of semi-implicit Euler methods for stochastic delay differential equations undergoing bifurcation

Ford, Neville J.; Norton, Stewart J. (Elsevier, 2009-07-15)

This article discusses estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there may be some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.

Identification of the initial function for discretized delay differential equations

Baker, Christopher T. H.; Parmuzin, Evgeny I. (Elsevier, 2005)

In the present work, we analyze a discrete analogue for the problem of the identification of the initial function for a delay differential equation (DDE) discussed by Baker and Parmuzin in 2004. The basic problem consists of finding an initial function that gives rise to a solution of a discretized DDE, which is a close fit to observed data.

Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

Baker, Christopher T. H.; Buckwar, Evelyn (Elsevier, 2004-02)

This article carries out an analysis which proceeds as follows: showing that an inequality of Halanay type (derivable via comparison theory) can be employed to derive conditions for p-th mean stability of a solution; producing a discrete analogue of the Halanay-type theory, that permits the development of a p-th mean stability analysis of analogous stochastic difference equations. The application of the theoretical results is illustrated by deriving mean-square stability conditions for solutions and numerical solutions of a constant-coefficient linear test equation.

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